Chemical Equilibrium
ChemistryLab computes thermodynamic equilibrium by minimising the Gibbs free energy of the system subject to element-conservation constraints. The workflow always follows the same four steps:
- Build a
ChemicalSystem(species + stoichiometric matrix). - Create an initial
ChemicalState(temperature, pressure, initial amounts). - Call
equilibrate(or useEquilibriumSolverexplicitly). - Inspect the resulting
ChemicalState.
Minimal workflow
The convenience function equilibrate handles everything with sensible defaults. The example below computes the equilibrium state of calcite (CaCO₃) dissolving in mildly acidic water — a standard geochemical benchmark.
using Optimization, OptimizationIpopt
using ChemistryLab
using DynamicQuantities
substances = build_species("../../../data/slop98-inorganic-thermofun.json")
# Select the carbonate-system species, calcite and its dissolution product Ca²⁺
dict = Dict(symbol(s) => s for s in substances)
species = [dict[sym] for sym in split("H2O@ H+ OH- CO2@ HCO3- CO3-2 Ca+2 Cal")]
cs = ChemicalSystem(species, ["H2O@", "H+", "Ca+2", "CO3-2", "Zz"])8-element ChemicalSystem{Species{Int64}, AbstractReaction, StoichMatrix{Int64, Symbol, Vector{Symbol}, Matrix{Int64}, Species{Int64}}, StoichMatrix{Int64, Species{Int64}, Vector{Species{Int64}}, Matrix{Int64}, Species{Int64}}, Nothing}:
H2O@ {Water HGK} [H2O@ ◆ H₂O@]
H+ {H+} [H+ ◆ H⁺]
OH- {OH- hydroXyl ion} [OH- ◆ OH⁻]
CO2@ {CO2,aq (+ H2O = H2CO3,aq )} [CO2@ ◆ CO₂@]
HCO3- {HCO3- bicarbonate ion} [HCO3- ◆ HCO₃⁻]
CO3-2 {CO3-2 carbonate ion} [CO3-2 ◆ CO₃²⁻]
Ca+2 {Ca+2 ion} [Ca+2 ◆ Ca²⁺]
Cal {CALCITE} [CaCO3 ◆ CaCO₃]state = ChemicalState(cs)
# 1 mmol calcite dissolved in 1 L of acidic water (initial pH ≈ 4)
set_quantity!(state, "Cal", 1e-3u"mol")
set_quantity!(state, "H2O@", 1.0u"kg")
V = volume(state)
set_quantity!(state, "H+", 1e-4u"mol/L" * V.liquid) # pH = 4
set_quantity!(state, "OH-", 1e-10u"mol/L" * V.liquid) # charge seed
state_eq = equilibrate(state)ChemicalState{Species{Int64}, AbstractReaction, DynamicQuantities.Quantity{Float64, DynamicQuantities.SymbolicDimensions{DynamicQuantities.FRInt32}}}
┌────────────────────────────────────────────────────────────────────────────────────────────────┐
│ T : 298.15 K │
│ P : 1.0 bar │
╞════════════════════════════════════════════════════════════════════════════════════════════════╡
│ # liquid #│ n [mol]│ m [g]│ V [cm³]│ c [mol/L]│
├┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┤
│ tot. liquid│ 55.5096│ 1000.02│ 1002.96│ │
├┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┤
│ H2O@│ 55.5093│ 999.999│ 1002.96│ 55.3452│
│ Ca+2│ 0.00015563│ 0.00623736│ -0.00286963│ 0.000155171│
│ HCO3-│ 0.000134262│ 0.00819215│ 0.00325061│ 0.000133866│
│ OH-│ 3.41433e-5│ 0.000580675│ -0.000160741│ 3.40424e-5│
│ CO3-2│ 2.12763e-5│ 0.00127675│ -0.000128886│ 2.12134e-5│
│ CO2@│ 9.19514e-8│ 4.04669e-6│ 3.01662e-6│ 9.16797e-8│
│ H+│ 3.83001e-9│ 3.86065e-9│ 0.0│ 3.81869e-9│
╞════════════════════════════════════════════════════════════════════════════════════════════════╡
│ # solid #│ n [mol]│ m [g]│ V [cm³]│ │
├┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┤
│ tot. solid│ 0.00084437│ 0.0845096│ 0.0311859│ │
├┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┤
│ Cal│ 0.00084437│ 0.0845096│ 0.0311859│ │
╞════════════════════════════════════════════════════════════════════════════════════════════════╡
│ # TOTAL #│ n [mol]│ m [g]│ V [cm³]│ │
├┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┤
│ │ 55.5105│ 1000.1│ 1002.99│ │
╞════════════════════════════════════════════════════════════════════════════════════════════════╡
│ pH : 9.5274 │
│ pOH : 4.468 │
│ porosity : 0.999969 │
│ saturation : 1.0 │
└────────────────────────────────────────────────────────────────────────────────────────────────┘
Calling equilibrate(state) with no extra arguments uses sensible defaults and is usually sufficient for aqueous geochemical problems.
Choosing a solver
ChemistryLab provides two solver extensions. Load whichever fits your workflow:
| Extension packages | Default solver | When to use |
|---|---|---|
Optimization, OptimizationIpopt | IpoptOptimizer | general-purpose, robust |
OptimaSolver | OptimaOptimizer | preferred when available |
When both are loaded, OptimaSolver always takes priority for equilibrate(state).
Explicit solver (always works)
Pass the solver as the second positional argument:
using Optimization, OptimizationIpopt
state_eq = equilibrate(state, IpoptOptimizer())
using OptimaSolver
state_eq = equilibrate(state, OptimaOptimizer())Default shortcut
When only one extension is loaded:
using Optimization, OptimizationIpopt
state_eq = equilibrate(state) # → IpoptOptimizer
using OptimaSolver
state_eq = equilibrate(state) # → OptimaOptimizerWith both loaded, OptimaSolver always wins:
using Optimization, OptimizationIpopt
using OptimaSolver
state_eq = equilibrate(state) # → OptimaOptimizer (priority)Inspecting the equilibrium state
The returned ChemicalState carries all derived thermodynamic quantities:
println("pH = ", pH(state_eq))
println("pOH = ", pOH(state_eq))
println("porosity = ", porosity(state_eq))
println("saturation = ", saturation(state_eq))pH = 9.527415206123138
pOH = 4.4679798329298395
porosity = 0.9999689071703629
saturation = 1.0Phase volumes and mole amounts are accessible via named tuples:
v = volume(state_eq)
println("V liquid = ", v.liquid)
println("V solid = ", v.solid)
println("V total = ", v.total)
m = moles(state_eq)
println("n liquid = ", m.liquid)
println("n solid = ", m.solid)V liquid = 0.0010029635119444432 m³
V solid = 3.1185943268259964e-8 m³
V total = 0.0010029946978877115 m³
n liquid = 55.50960918339471 mol
n solid = 0.0008443695256573842 molIndividual species amounts (in mol):
cs_eq = state_eq.system
for (i, sp) in enumerate(cs_eq.species)
n_i = state_eq.n[i]
println(rpad(symbol(sp), 20), ustrip(n_i), " mol")
endH2O@ 55.50926377533243 mol
H+ 3.830008572261524e-9 mol
OH- 3.414328479796852e-5 mol
CO2@ 9.195137148453631e-8 mol
HCO3- 0.00013426225206497052 mol
CO3-2 2.127626686915876e-5 mol
Ca+2 0.00015563047716080978 mol
Cal 0.0008443695256573842 molScaling and normalisation
It is often useful to express a composition relative to a reference amount — per mole, per kilogram, or per cubic metre of system. Two mechanisms are provided.
Scalar multiplication
A ChemicalState can be multiplied or divided by a real number. All molar amounts are scaled proportionally; temperature, pressure, and the chemical system are unchanged. The operation is non-mutating — a new state is returned:
state2 = state_eq * 2.0 # double all amounts
state_m = state_eq / 1000 # millimolar scalerescale! — rescale to a target total
rescale! scales all molar amounts in-place so that the total of the matching physical quantity equals target:
target dimension | Quantity brought to target |
|---|---|
| mol | moles(state).total |
| kg (mass) | mass(state).total |
| m³ (volume) | volume(state).total |
All derived quantities (pH, porosity, volume, …) are recomputed automatically after scaling.
# Express the equilibrium composition per kilogram of total system
state_pkg = copy(state_eq)
rescale!(state_pkg, 1.0u"kg")
println("Ca²⁺ = ", moles(state_pkg, "Ca+2"), " mol/kg")
println("pH = ", pH(state_pkg)) # intensive quantities are invariantCa²⁺ = 0.00015561488655492893 mol mol/kg
pH = 9.527415206123138pH, porosity, and saturation are intensive — they are invariant under homothety and remain unchanged after rescale! or scalar multiplication.
Controlling the solver
Variable space: :linear vs :log
equilibrate accepts a variable_space keyword that selects the optimisation variable space:
variable_space | Variables | Recommended when |
|---|---|---|
Val(:linear) | mole amounts nᵢ ≥ 0 | most systems, default |
Val(:log) | log nᵢ | systems spanning many orders of magnitude |
state_eq_log = equilibrate(state; variable_space=Val(:log))Solving a system of equations in chemistry can be a difficult undertaking. The orders of magnitude can vary greatly, and convergence is not guaranteed.
Tolerances
Tighter tolerances are passed directly as keyword arguments and forwarded to the underlying Ipopt solver:
state_eq_tight = equilibrate(state; abstol=1e-12, reltol=1e-12)Using EquilibriumSolver explicitly
For batch calculations where many different initial states share the same system and activity model, construct an EquilibriumSolver once and reuse it:
using Optimization, OptimizationIpopt
opt = IpoptOptimizer(
acceptable_tol = 1e-12,
dual_inf_tol = 1e-12,
acceptable_iter = 1000,
constr_viol_tol = 1e-12,
warm_start_init_point = "no",
)
solver = EquilibriumSolver(
cs,
DiluteSolutionModel(),
opt;
variable_space = Val(:linear),
abstol = 1e-10,
reltol = 1e-10,
)Once built, solver is called with any compatible ChemicalState:
state_eq2 = solve(solver, state)ChemicalState{Species{Int64}, AbstractReaction, DynamicQuantities.Quantity{Float64, DynamicQuantities.SymbolicDimensions{DynamicQuantities.FRInt32}}}
┌────────────────────────────────────────────────────────────────────────────────────────────────┐
│ T : 298.15 K │
│ P : 1.0 bar │
╞════════════════════════════════════════════════════════════════════════════════════════════════╡
│ # liquid #│ n [mol]│ m [g]│ V [cm³]│ c [mol/L]│
├┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┤
│ tot. liquid│ 55.5096│ 1000.02│ 1002.96│ │
├┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┤
│ H2O@│ 55.5093│ 999.999│ 1002.96│ 55.3452│
│ Ca+2│ 0.00015562│ 0.00623692│ -0.00286943│ 0.00015516│
│ HCO3-│ 0.000134259│ 0.00819197│ 0.00325053│ 0.000133863│
│ OH-│ 3.41393e-5│ 0.000580607│ -0.000160722│ 3.40384e-5│
│ CO3-2│ 2.12722e-5│ 0.0012765│ -0.000128861│ 2.12094e-5│
│ CO2@│ 8.79874e-8│ 3.87224e-6│ 2.88657e-6│ 8.77274e-8│
│ H+│ 3.34877e-10│ 3.37556e-10│ 0.0│ 3.33888e-10│
╞════════════════════════════════════════════════════════════════════════════════════════════════╡
│ # solid #│ n [mol]│ m [g]│ V [cm³]│ │
├┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┤
│ tot. solid│ 0.00084438│ 0.0845107│ 0.0311863│ │
├┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┤
│ Cal│ 0.00084438│ 0.0845107│ 0.0311863│ │
╞════════════════════════════════════════════════════════════════════════════════════════════════╡
│ # TOTAL #│ n [mol]│ m [g]│ V [cm³]│ │
├┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┤
│ │ 55.5105│ 1000.1│ 1002.99│ │
╞════════════════════════════════════════════════════════════════════════════════════════════════╡
│ pH : 9.5274 │
│ pOH : 4.468 │
│ porosity : 0.999969 │
│ saturation : 1.0 │
└────────────────────────────────────────────────────────────────────────────────────────────────┘
The potential function μ(n, p) is compiled once during EquilibriumSolver construction. Repeated calls to solve(solver, ...) with different states reuse it, avoiding redundant compilation overhead.
Temperature dependence (10–30 °C)
Calcite solubility varies with temperature. Using the solver built above, we sweep from 10 to 30 °C and track pH, dissolved calcium and remaining solid calcite:
using Plots
temperatures = 10:30 # °C
pH_vals = Float64[]
nCa_vals = Float64[] # mmol
nCal_vals = Float64[] # mmol
i_Ca = findfirst(sp -> symbol(sp) == "Ca+2", cs.species)
i_Cal = findfirst(sp -> symbol(sp) == "Cal", cs.species)
s = ChemicalState(cs)
for θ in temperatures
set_temperature!(s, (273.15 + θ) * u"K")
s_eq = solve(solver, s)
push!(pH_vals, pH(s_eq))
push!(nCa_vals, ustrip(s_eq.n[i_Ca]) * 1e3)
push!(nCal_vals, ustrip(s_eq.n[i_Cal]) * 1e3)
endThis is Ipopt version 3.14.19, running with linear solver MUMPS 5.8.2.
Number of nonzeros in equality constraint Jacobian...: 32
Number of nonzeros in inequality constraint Jacobian.: 0
Number of nonzeros in Lagrangian Hessian.............: 36
Total number of variables............................: 8
variables with only lower bounds: 0
variables with lower and upper bounds: 8
variables with only upper bounds: 0
Total number of equality constraints.................: 4
Total number of inequality constraints...............: 0
inequality constraints with only lower bounds: 0
inequality constraints with lower and upper bounds: 0
inequality constraints with only upper bounds: 0
iter objective inf_pr inf_du lg(mu) ||d|| lg(rg) alpha_du alpha_pr ls
0 -1.4956935e-13 0.00e+00 3.32e+00 -1.0 0.00e+00 - 0.00e+00 0.00e+00 0
1 -1.2025432e-13 1.48e-31 1.50e+08 -1.0 3.54e-08 - 1.00e+00 1.37e-08f 12
2 -1.2047221e-13 1.48e-31 1.53e-05 -2.5 6.25e-18 - 1.00e+00 1.00e+00 0
3 -1.2047226e-13 9.86e-32 3.85e-07 -11.0 6.87e-22 - 1.00e+00 1.00e+00 0
4 -4.9364991e-13 9.37e-31 2.46e+00 -11.0 8.10e-15 - 1.00e+00 1.00e+00f 1
5 -1.0668167e-12 1.63e-30 1.95e+00 -11.0 1.67e-14 14.0 1.00e+00 1.00e+00f 1
6 -3.2338793e-12 1.10e-29 9.77e-01 -11.0 7.72e-14 - 1.00e+00 1.00e+00f 1
7 -3.5455090e-12 8.83e-30 8.84e-02 -11.0 1.40e-14 - 1.00e+00 1.00e+00h 1
8 -3.5476952e-12 9.66e-30 6.00e-04 -11.0 1.94e-15 - 1.00e+00 1.00e+00 0
9 -3.5477255e-12 4.78e-30 6.40e-08 -11.0 1.49e-17 - 1.00e+00 1.00e+00 0
iter objective inf_pr inf_du lg(mu) ||d|| lg(rg) alpha_du alpha_pr ls
10 -3.5477255e-12 1.01e-29 1.78e-15 -11.0 1.57e-21 - 1.00e+00 1.00e+00T 0
Number of Iterations....: 10
(scaled) (unscaled)
Objective...............: -7.4056197044188541e-13 -3.5477254668958915e-12
Dual infeasibility......: 1.7763568394002505e-15 8.5097893882865893e-15
Constraint violation....: 1.0107280348144214e-29 1.0107280348144214e-29
Variable bound violation: 1.0040723276967796e-13 1.0040723276967796e-13
Complementarity.........: 9.0909090909090936e-12 4.3550777634191928e-11
Overall NLP error.......: 9.0909090909090936e-12 4.3550777634191928e-11
Number of objective function evaluations = 23
Number of objective gradient evaluations = 11
Number of equality constraint evaluations = 23
Number of inequality constraint evaluations = 0
Number of equality constraint Jacobian evaluations = 11
Number of inequality constraint Jacobian evaluations = 0
Number of Lagrangian Hessian evaluations = 10
Total seconds in IPOPT = 0.003
EXIT: Optimal Solution Found.
This is Ipopt version 3.14.19, running with linear solver MUMPS 5.8.2.
Number of nonzeros in equality constraint Jacobian...: 32
Number of nonzeros in inequality constraint Jacobian.: 0
Number of nonzeros in Lagrangian Hessian.............: 36
Total number of variables............................: 8
variables with only lower bounds: 0
variables with lower and upper bounds: 8
variables with only upper bounds: 0
Total number of equality constraints.................: 4
Total number of inequality constraints...............: 0
inequality constraints with only lower bounds: 0
inequality constraints with lower and upper bounds: 0
inequality constraints with only upper bounds: 0
iter objective inf_pr inf_du lg(mu) ||d|| lg(rg) alpha_du alpha_pr ls
0 -1.4904656e-13 0.00e+00 3.33e+00 -1.0 0.00e+00 - 0.00e+00 0.00e+00 0
1 -1.1975977e-13 4.93e-32 1.50e+08 -1.0 3.53e-08 - 1.00e+00 1.37e-08f 12
2 -1.1997762e-13 4.93e-32 3.29e-05 -2.5 6.25e-18 - 1.00e+00 1.00e+00 0
3 -1.1997767e-13 4.93e-32 5.60e-07 -11.0 6.88e-22 - 1.00e+00 1.00e+00 0
4 -4.9231687e-13 7.40e-31 2.46e+00 -11.0 8.10e-15 - 1.00e+00 1.00e+00f 1
5 -1.0648554e-12 3.20e-30 1.95e+00 -11.0 1.67e-14 14.0 1.00e+00 1.00e+00f 1
6 -3.2265479e-12 1.10e-29 9.99e-01 -11.0 7.72e-14 - 1.00e+00 1.00e+00f 1
7 -3.5392518e-12 3.99e-30 6.61e-02 -11.0 1.39e-14 - 1.00e+00 1.00e+00h 1
8 -3.5412989e-12 9.66e-30 4.19e-04 -11.0 1.51e-15 - 1.00e+00 1.00e+00 0
9 -3.5413213e-12 8.83e-30 3.12e-08 -11.0 1.04e-17 - 1.00e+00 1.00e+00 0
iter objective inf_pr inf_du lg(mu) ||d|| lg(rg) alpha_du alpha_pr ls
10 -3.5413213e-12 7.84e-30 1.24e-14 -11.0 7.67e-22 - 1.00e+00 1.00e+00T 0
Number of Iterations....: 10
(scaled) (unscaled)
Objective...............: -7.4177754309316537e-13 -3.5413213022949307e-12
Dual infeasibility......: 1.2434497875801753e-14 5.9363555315110397e-14
Constraint violation....: 7.8393052456338048e-30 7.8393052456338048e-30
Variable bound violation: 1.0049188974296617e-13 1.0049188974296617e-13
Complementarity.........: 9.0909090909090936e-12 4.3400922986447909e-11
Overall NLP error.......: 9.0909090909090936e-12 4.3400922986447909e-11
Number of objective function evaluations = 23
Number of objective gradient evaluations = 11
Number of equality constraint evaluations = 23
Number of inequality constraint evaluations = 0
Number of equality constraint Jacobian evaluations = 11
Number of inequality constraint Jacobian evaluations = 0
Number of Lagrangian Hessian evaluations = 10
Total seconds in IPOPT = 0.003
EXIT: Optimal Solution Found.
This is Ipopt version 3.14.19, running with linear solver MUMPS 5.8.2.
Number of nonzeros in equality constraint Jacobian...: 32
Number of nonzeros in inequality constraint Jacobian.: 0
Number of nonzeros in Lagrangian Hessian.............: 36
Total number of variables............................: 8
variables with only lower bounds: 0
variables with lower and upper bounds: 8
variables with only upper bounds: 0
Total number of equality constraints.................: 4
Total number of inequality constraints...............: 0
inequality constraints with only lower bounds: 0
inequality constraints with lower and upper bounds: 0
inequality constraints with only upper bounds: 0
iter objective inf_pr inf_du lg(mu) ||d|| lg(rg) alpha_du alpha_pr ls
0 -1.4852741e-13 0.00e+00 3.33e+00 -1.0 0.00e+00 - 0.00e+00 0.00e+00 0
1 -1.1926831e-13 9.86e-32 1.50e+08 -1.0 3.52e-08 - 1.00e+00 1.38e-08f 12
2 -1.1948613e-13 4.93e-32 2.53e-05 -2.5 6.25e-18 - 1.00e+00 1.00e+00 0
3 -1.1948617e-13 4.93e-32 2.69e-07 -11.0 6.89e-22 - 1.00e+00 1.00e+00 0
4 -4.9100649e-13 8.38e-31 2.46e+00 -11.0 8.11e-15 - 1.00e+00 1.00e+00f 1
5 -1.0629479e-12 2.96e-30 1.96e+00 -11.0 1.68e-14 14.0 1.00e+00 1.00e+00f 1
6 -3.2194026e-12 1.19e-29 1.02e+00 -11.0 7.73e-14 - 1.00e+00 1.00e+00f 1
7 -3.5331753e-12 4.68e-30 4.41e-02 -11.0 1.38e-14 - 1.00e+00 1.00e+00h 1
8 -3.5351184e-12 5.57e-30 2.74e-04 -11.0 1.08e-15 - 1.00e+00 1.00e+00 0
9 -3.5351341e-12 4.63e-30 1.33e-08 -11.0 6.75e-18 - 1.00e+00 1.00e+00T 0
Number of Iterations....: 9
(scaled) (unscaled)
Objective...............: -7.4302889907565039e-13 -3.5351340795147369e-12
Dual infeasibility......: 1.3347269955943375e-08 6.3502764089576249e-08
Constraint violation....: 4.6345578181734444e-30 4.6345578181734444e-30
Variable bound violation: 1.0057899537809294e-13 1.0057899537809294e-13
Complementarity.........: 9.0909090909087608e-12 4.3252130006009164e-11
Overall NLP error.......: 1.3347269955943375e-08 6.3502764089576249e-08
Number of objective function evaluations = 22
Number of objective gradient evaluations = 10
Number of equality constraint evaluations = 22
Number of inequality constraint evaluations = 0
Number of equality constraint Jacobian evaluations = 10
Number of inequality constraint Jacobian evaluations = 0
Number of Lagrangian Hessian evaluations = 10
Total seconds in IPOPT = 0.003
EXIT: Search Direction is becoming Too Small.
This is Ipopt version 3.14.19, running with linear solver MUMPS 5.8.2.
Number of nonzeros in equality constraint Jacobian...: 32
Number of nonzeros in inequality constraint Jacobian.: 0
Number of nonzeros in Lagrangian Hessian.............: 36
Total number of variables............................: 8
variables with only lower bounds: 0
variables with lower and upper bounds: 8
variables with only upper bounds: 0
Total number of equality constraints.................: 4
Total number of inequality constraints...............: 0
inequality constraints with only lower bounds: 0
inequality constraints with lower and upper bounds: 0
inequality constraints with only upper bounds: 0
iter objective inf_pr inf_du lg(mu) ||d|| lg(rg) alpha_du alpha_pr ls
0 -1.4801185e-13 0.00e+00 3.33e+00 -1.0 0.00e+00 - 0.00e+00 0.00e+00 0
1 -1.1877992e-13 4.93e-32 1.50e+08 -1.0 3.51e-08 - 1.00e+00 1.38e-08f 12
2 -1.1899770e-13 9.86e-32 1.24e-05 -2.5 6.25e-18 - 1.00e+00 1.00e+00 0
3 -1.1899775e-13 9.86e-32 3.12e-07 -11.0 6.90e-22 - 1.00e+00 1.00e+00 0
4 -4.8971827e-13 1.38e-30 2.46e+00 -11.0 8.11e-15 - 1.00e+00 1.00e+00f 1
5 -1.0610933e-12 1.77e-30 1.96e+00 -11.0 1.68e-14 14.0 1.00e+00 1.00e+00f 1
6 -3.2124399e-12 5.08e-30 1.04e+00 -11.0 7.73e-14 - 1.00e+00 1.00e+00f 1
7 -3.5272760e-12 7.15e-30 2.25e-02 -11.0 1.37e-14 - 1.00e+00 1.00e+00f 1
8 -3.5291493e-12 6.95e-30 1.63e-04 -11.0 7.22e-16 - 1.00e+00 1.00e+00 0
9 -3.5291594e-12 2.96e-30 4.68e-09 -11.0 3.98e-18 - 1.00e+00 1.00e+00T 0
Number of Iterations....: 9
(scaled) (unscaled)
Objective...............: -7.4431558784044539e-13 -3.5291594320398967e-12
Dual infeasibility......: 4.6814605525469233e-09 2.2197063899574444e-08
Constraint violation....: 2.9582283945787943e-30 2.9582283945787943e-30
Variable bound violation: 1.0066849641161203e-13 1.0066849641161203e-13
Complementarity.........: 9.0909090909083569e-12 4.3104387558351086e-11
Overall NLP error.......: 4.6814605525469233e-09 2.2197063899574444e-08
Number of objective function evaluations = 22
Number of objective gradient evaluations = 10
Number of equality constraint evaluations = 22
Number of inequality constraint evaluations = 0
Number of equality constraint Jacobian evaluations = 10
Number of inequality constraint Jacobian evaluations = 0
Number of Lagrangian Hessian evaluations = 10
Total seconds in IPOPT = 0.003
EXIT: Search Direction is becoming Too Small.
This is Ipopt version 3.14.19, running with linear solver MUMPS 5.8.2.
Number of nonzeros in equality constraint Jacobian...: 32
Number of nonzeros in inequality constraint Jacobian.: 0
Number of nonzeros in Lagrangian Hessian.............: 36
Total number of variables............................: 8
variables with only lower bounds: 0
variables with lower and upper bounds: 8
variables with only upper bounds: 0
Total number of equality constraints.................: 4
Total number of inequality constraints...............: 0
inequality constraints with only lower bounds: 0
inequality constraints with lower and upper bounds: 0
inequality constraints with only upper bounds: 0
iter objective inf_pr inf_du lg(mu) ||d|| lg(rg) alpha_du alpha_pr ls
0 -1.4749985e-13 0.00e+00 3.34e+00 -1.0 0.00e+00 - 0.00e+00 0.00e+00 0
1 -1.1829458e-13 4.93e-32 1.50e+08 -1.0 3.49e-08 - 1.00e+00 1.39e-08f 12
2 -1.1851233e-13 4.93e-32 2.15e-05 -2.5 6.25e-18 - 1.00e+00 1.00e+00 0
3 -1.1851237e-13 4.93e-32 3.90e-07 -11.0 6.91e-22 - 1.00e+00 1.00e+00 0
4 -4.8845173e-13 1.43e-30 2.47e+00 -11.0 8.12e-15 - 1.00e+00 1.00e+00f 1
5 -1.0592903e-12 1.63e-30 1.97e+00 -11.0 1.68e-14 14.0 1.00e+00 1.00e+00f 1
6 -3.2056568e-12 1.64e-29 1.06e+00 -11.0 7.74e-14 - 1.00e+00 1.00e+00f 1
7 -3.5215504e-12 4.78e-30 2.20e-02 -11.0 1.36e-14 - 1.00e+00 1.00e+00h 1
8 -3.5233876e-12 1.03e-29 8.21e-05 -11.0 5.15e-16 - 1.00e+00 1.00e+00 0
9 -3.5233932e-12 8.83e-30 1.18e-09 -11.0 1.99e-18 - 1.00e+00 1.00e+00T 0
Number of Iterations....: 9
(scaled) (unscaled)
Objective...............: -7.4563719284938099e-13 -3.5233932144404266e-12
Dual infeasibility......: 1.1758363172020836e-09 5.5562326303635941e-09
Constraint violation....: 8.8253813771600696e-30 8.8253813771600696e-30
Variable bound violation: 1.0076034198984540e-13 1.0076034198984540e-13
Complementarity.........: 9.0909090909086558e-12 4.2957684663769971e-11
Overall NLP error.......: 1.1758363172020836e-09 5.5562326303635941e-09
Number of objective function evaluations = 22
Number of objective gradient evaluations = 10
Number of equality constraint evaluations = 22
Number of inequality constraint evaluations = 0
Number of equality constraint Jacobian evaluations = 10
Number of inequality constraint Jacobian evaluations = 0
Number of Lagrangian Hessian evaluations = 10
Total seconds in IPOPT = 0.003
EXIT: Search Direction is becoming Too Small.
This is Ipopt version 3.14.19, running with linear solver MUMPS 5.8.2.
Number of nonzeros in equality constraint Jacobian...: 32
Number of nonzeros in inequality constraint Jacobian.: 0
Number of nonzeros in Lagrangian Hessian.............: 36
Total number of variables............................: 8
variables with only lower bounds: 0
variables with lower and upper bounds: 8
variables with only upper bounds: 0
Total number of equality constraints.................: 4
Total number of inequality constraints...............: 0
inequality constraints with only lower bounds: 0
inequality constraints with lower and upper bounds: 0
inequality constraints with only upper bounds: 0
iter objective inf_pr inf_du lg(mu) ||d|| lg(rg) alpha_du alpha_pr ls
0 -1.4699138e-13 0.00e+00 3.34e+00 -1.0 0.00e+00 - 0.00e+00 0.00e+00 0
1 -1.1781226e-13 4.93e-32 1.50e+08 -1.0 3.48e-08 - 1.00e+00 1.39e-08f 12
2 -1.1802998e-13 4.93e-32 1.48e-05 -2.5 6.25e-18 - 1.00e+00 1.00e+00 0
3 -1.1803003e-13 1.48e-31 6.34e-07 -11.0 6.92e-22 - 1.00e+00 1.00e+00 0
4 -4.8720641e-13 1.38e-30 2.47e+00 -11.0 8.12e-15 - 1.00e+00 1.00e+00f 1
5 -1.0575380e-12 3.99e-30 1.97e+00 -11.0 1.68e-14 14.0 1.00e+00 1.00e+00f 1
6 -3.1990503e-12 4.63e-30 1.07e+00 -11.0 7.74e-14 - 1.00e+00 1.00e+00f 1
7 -3.5159957e-12 5.67e-30 2.34e-02 -11.0 1.35e-14 - 1.00e+00 1.00e+00f 1
8 -3.5178297e-12 4.63e-30 3.00e-05 -11.0 4.05e-16 - 1.00e+00 1.00e+00 0
9 -3.5178315e-12 4.63e-30 1.48e-10 -11.0 7.02e-19 - 1.00e+00 1.00e+00T 0
Number of Iterations....: 9
(scaled) (unscaled)
Objective...............: -7.4699333051500644e-13 -3.5178314936023297e-12
Dual infeasibility......: 1.4754597543742420e-10 6.9484138337647392e-10
Constraint violation....: 4.6345578181734444e-30 4.6345578181734444e-30
Variable bound violation: 1.0085448390580427e-13 1.0085448390580427e-13
Complementarity.........: 9.0909090909091324e-12 4.2812010494695322e-11
Overall NLP error.......: 1.4754597543742420e-10 6.9484138337647392e-10
Number of objective function evaluations = 22
Number of objective gradient evaluations = 10
Number of equality constraint evaluations = 22
Number of inequality constraint evaluations = 0
Number of equality constraint Jacobian evaluations = 10
Number of inequality constraint Jacobian evaluations = 0
Number of Lagrangian Hessian evaluations = 10
Total seconds in IPOPT = 0.003
EXIT: Search Direction is becoming Too Small.
This is Ipopt version 3.14.19, running with linear solver MUMPS 5.8.2.
Number of nonzeros in equality constraint Jacobian...: 32
Number of nonzeros in inequality constraint Jacobian.: 0
Number of nonzeros in Lagrangian Hessian.............: 36
Total number of variables............................: 8
variables with only lower bounds: 0
variables with lower and upper bounds: 8
variables with only upper bounds: 0
Total number of equality constraints.................: 4
Total number of inequality constraints...............: 0
inequality constraints with only lower bounds: 0
inequality constraints with lower and upper bounds: 0
inequality constraints with only upper bounds: 0
iter objective inf_pr inf_du lg(mu) ||d|| lg(rg) alpha_du alpha_pr ls
0 -1.4648640e-13 0.00e+00 3.34e+00 -1.0 0.00e+00 - 0.00e+00 0.00e+00 0
1 -1.1733293e-13 9.86e-32 1.50e+08 -1.0 3.47e-08 - 1.00e+00 1.39e-08f 12
2 -1.1755064e-13 9.86e-32 1.86e-05 -2.5 6.25e-18 - 1.00e+00 1.00e+00 0
3 -1.1755068e-13 1.48e-31 7.54e-07 -11.0 6.92e-22 - 1.00e+00 1.00e+00 0
4 -4.8598186e-13 1.63e-30 2.47e+00 -11.0 8.13e-15 - 1.00e+00 1.00e+00f 1
5 -1.0558352e-12 2.12e-30 1.98e+00 -11.0 1.69e-14 14.0 1.00e+00 1.00e+00f 1
6 -3.1926173e-12 3.35e-30 1.09e+00 -11.0 7.75e-14 - 1.00e+00 1.00e+00f 1
7 -3.5106090e-12 8.83e-30 3.99e-02 -11.0 1.36e-14 - 1.00e+00 1.00e+00f 1
8 -3.5124719e-12 8.83e-30 3.95e-06 -11.0 6.81e-16 - 1.00e+00 1.00e+00 0
9 -3.5124705e-12 9.66e-30 1.08e-12 -11.0 1.16e-19 - 1.00e+00 1.00e+00 0
iter objective inf_pr inf_du lg(mu) ||d|| lg(rg) alpha_du alpha_pr ls
10 -3.5124705e-12 4.63e-30 8.88e-15 -11.0 4.34e-26 - 1.00e+00 1.00e+00T 0
Number of Iterations....: 10
(scaled) (unscaled)
Objective...............: -7.4838364751532561e-13 -3.5124705325103086e-12
Dual infeasibility......: 8.8817841970012523e-15 4.1685845717845905e-14
Constraint violation....: 4.6345578181734444e-30 4.6345578181734444e-30
Variable bound violation: 1.0095087629021677e-13 1.0095087629021677e-13
Complementarity.........: 9.0909090909090936e-12 4.2667354373071559e-11
Overall NLP error.......: 9.0909090909090936e-12 4.2667354373071559e-11
Number of objective function evaluations = 23
Number of objective gradient evaluations = 11
Number of equality constraint evaluations = 23
Number of inequality constraint evaluations = 0
Number of equality constraint Jacobian evaluations = 11
Number of inequality constraint Jacobian evaluations = 0
Number of Lagrangian Hessian evaluations = 10
Total seconds in IPOPT = 0.003
EXIT: Optimal Solution Found.
This is Ipopt version 3.14.19, running with linear solver MUMPS 5.8.2.
Number of nonzeros in equality constraint Jacobian...: 32
Number of nonzeros in inequality constraint Jacobian.: 0
Number of nonzeros in Lagrangian Hessian.............: 36
Total number of variables............................: 8
variables with only lower bounds: 0
variables with lower and upper bounds: 8
variables with only upper bounds: 0
Total number of equality constraints.................: 4
Total number of inequality constraints...............: 0
inequality constraints with only lower bounds: 0
inequality constraints with lower and upper bounds: 0
inequality constraints with only upper bounds: 0
iter objective inf_pr inf_du lg(mu) ||d|| lg(rg) alpha_du alpha_pr ls
0 -1.4598488e-13 0.00e+00 3.35e+00 -1.0 0.00e+00 - 0.00e+00 0.00e+00 0
1 -1.1685659e-13 9.86e-32 1.50e+08 -1.0 3.46e-08 - 1.00e+00 1.40e-08f 12
2 -1.1707427e-13 4.93e-32 2.05e-05 -2.5 6.25e-18 - 1.00e+00 1.00e+00 0
3 -1.1707432e-13 4.93e-32 5.47e-07 -11.0 6.93e-22 - 1.00e+00 1.00e+00 0
4 -4.8477766e-13 8.38e-31 2.47e+00 -11.0 8.13e-15 - 1.00e+00 1.00e+00f 1
5 -1.0541812e-12 1.63e-30 1.98e+00 -11.0 1.69e-14 14.0 1.00e+00 1.00e+00f 1
6 -3.1863551e-12 5.67e-30 1.11e+00 -11.0 7.75e-14 - 1.00e+00 1.00e+00f 1
7 -3.5053876e-12 9.27e-30 6.00e-02 -11.0 1.37e-14 - 1.00e+00 1.00e+00f 1
8 -3.5073107e-12 3.35e-30 2.24e-06 -11.0 9.51e-16 - 1.00e+00 1.00e+00 0
9 -3.5073068e-12 3.90e-30 1.38e-12 -11.0 2.61e-19 - 1.00e+00 1.00e+00 0
iter objective inf_pr inf_du lg(mu) ||d|| lg(rg) alpha_du alpha_pr ls
10 -3.5073068e-12 5.47e-30 8.88e-15 -11.0 1.64e-25 - 1.00e+00 1.00e+00T 0
Number of Iterations....: 10
(scaled) (unscaled)
Objective...............: -7.4980781877477722e-13 -3.5073067774767852e-12
Dual infeasibility......: 8.8817841970012523e-15 4.1545501567496582e-14
Constraint violation....: 5.4727225299707694e-30 5.4727225299707694e-30
Variable bound violation: 1.0104947545874047e-13 1.0104947545874047e-13
Complementarity.........: 9.0909090909090936e-12 4.2523705767795002e-11
Overall NLP error.......: 9.0909090909090936e-12 4.2523705767795002e-11
Number of objective function evaluations = 23
Number of objective gradient evaluations = 11
Number of equality constraint evaluations = 23
Number of inequality constraint evaluations = 0
Number of equality constraint Jacobian evaluations = 11
Number of inequality constraint Jacobian evaluations = 0
Number of Lagrangian Hessian evaluations = 10
Total seconds in IPOPT = 0.003
EXIT: Optimal Solution Found.
This is Ipopt version 3.14.19, running with linear solver MUMPS 5.8.2.
Number of nonzeros in equality constraint Jacobian...: 32
Number of nonzeros in inequality constraint Jacobian.: 0
Number of nonzeros in Lagrangian Hessian.............: 36
Total number of variables............................: 8
variables with only lower bounds: 0
variables with lower and upper bounds: 8
variables with only upper bounds: 0
Total number of equality constraints.................: 4
Total number of inequality constraints...............: 0
inequality constraints with only lower bounds: 0
inequality constraints with lower and upper bounds: 0
inequality constraints with only upper bounds: 0
iter objective inf_pr inf_du lg(mu) ||d|| lg(rg) alpha_du alpha_pr ls
0 -1.4548678e-13 0.00e+00 3.35e+00 -1.0 0.00e+00 - 0.00e+00 0.00e+00 0
1 -1.1638320e-13 9.86e-32 1.50e+08 -1.0 3.45e-08 - 1.00e+00 1.40e-08f 12
2 -1.1660087e-13 4.93e-32 2.19e-05 -2.5 6.25e-18 - 1.00e+00 1.00e+00 0
3 -1.1660091e-13 4.93e-32 2.80e-07 -11.0 6.94e-22 - 1.00e+00 1.00e+00 0
4 -4.8359340e-13 1.38e-30 2.47e+00 -11.0 8.14e-15 - 1.00e+00 1.00e+00f 1
5 -1.0525750e-12 3.35e-30 1.99e+00 -11.0 1.69e-14 14.0 1.00e+00 1.00e+00f 1
6 -3.1802611e-12 1.01e-29 1.12e+00 -11.0 7.76e-14 - 1.00e+00 1.00e+00f 1
7 -3.5003290e-12 8.83e-30 7.97e-02 -11.0 1.37e-14 - 1.00e+00 1.00e+00f 1
8 -3.5023429e-12 1.01e-29 2.15e-05 -11.0 1.32e-15 - 1.00e+00 1.00e+00 0
9 -3.5023368e-12 3.35e-30 7.46e-11 -11.0 5.12e-19 - 1.00e+00 1.00e+00 0
iter objective inf_pr inf_du lg(mu) ||d|| lg(rg) alpha_du alpha_pr ls
10 -3.5023368e-12 3.99e-30 1.07e-14 -11.0 1.90e-24 - 1.00e+00 1.00e+00T 0
Number of Iterations....: 10
(scaled) (unscaled)
Objective...............: -7.5126554560901991e-13 -3.5023368463937066e-12
Dual infeasibility......: 1.0658141036401503e-14 4.9687357930928674e-14
Constraint violation....: 3.9936083326813723e-30 3.9936083326813723e-30
Variable bound violation: 1.0115023977162965e-13 1.0115023977162965e-13
Complementarity.........: 9.0909090909090920e-12 4.2381054292188414e-11
Overall NLP error.......: 9.0909090909090920e-12 4.2381054292188414e-11
Number of objective function evaluations = 23
Number of objective gradient evaluations = 11
Number of equality constraint evaluations = 23
Number of inequality constraint evaluations = 0
Number of equality constraint Jacobian evaluations = 11
Number of inequality constraint Jacobian evaluations = 0
Number of Lagrangian Hessian evaluations = 10
Total seconds in IPOPT = 0.003
EXIT: Optimal Solution Found.
This is Ipopt version 3.14.19, running with linear solver MUMPS 5.8.2.
Number of nonzeros in equality constraint Jacobian...: 32
Number of nonzeros in inequality constraint Jacobian.: 0
Number of nonzeros in Lagrangian Hessian.............: 36
Total number of variables............................: 8
variables with only lower bounds: 0
variables with lower and upper bounds: 8
variables with only upper bounds: 0
Total number of equality constraints.................: 4
Total number of inequality constraints...............: 0
inequality constraints with only lower bounds: 0
inequality constraints with lower and upper bounds: 0
inequality constraints with only upper bounds: 0
iter objective inf_pr inf_du lg(mu) ||d|| lg(rg) alpha_du alpha_pr ls
0 -1.4499207e-13 0.00e+00 3.35e+00 -1.0 0.00e+00 - 0.00e+00 0.00e+00 0
1 -1.1591275e-13 4.93e-32 1.50e+08 -1.0 3.44e-08 - 1.00e+00 1.41e-08f 12
2 -1.1613039e-13 9.86e-32 1.96e-05 -2.5 6.25e-18 - 1.00e+00 1.00e+00 0
3 -1.1613044e-13 9.86e-32 4.03e-07 -11.0 6.95e-22 - 1.00e+00 1.00e+00 0
4 -4.8242869e-13 1.63e-30 2.48e+00 -11.0 8.14e-15 - 1.00e+00 1.00e+00f 1
5 -1.0510157e-12 1.77e-30 2.00e+00 -11.0 1.70e-14 14.0 1.00e+00 1.00e+00f 1
6 -3.1743326e-12 5.67e-30 1.14e+00 -11.0 7.76e-14 - 1.00e+00 1.00e+00f 1
7 -3.4954307e-12 9.66e-30 9.90e-02 -11.0 1.38e-14 - 1.00e+00 1.00e+00f 1
8 -3.4975653e-12 4.78e-30 6.13e-05 -11.0 1.69e-15 - 1.00e+00 1.00e+00 0
9 -3.4975575e-12 7.84e-30 6.70e-10 -11.0 1.50e-18 - 1.00e+00 1.00e+00 0
iter objective inf_pr inf_du lg(mu) ||d|| lg(rg) alpha_du alpha_pr ls
10 -3.4975575e-12 9.27e-30 1.07e-14 -11.0 1.65e-23 - 1.00e+00 1.00e+00T 0
Number of Iterations....: 10
(scaled) (unscaled)
Objective...............: -7.5275655403263730e-13 -3.4975575179789092e-12
Dual infeasibility......: 1.0658141036401503e-14 4.9521271000357649e-14
Constraint violation....: 9.2691156363468887e-30 9.2691156363468887e-30
Variable bound violation: 1.0125312950575730e-13 1.0125312950575730e-13
Complementarity.........: 9.0909090909090936e-12 4.2239389701538660e-11
Overall NLP error.......: 9.0909090909090936e-12 4.2239389701538660e-11
Number of objective function evaluations = 23
Number of objective gradient evaluations = 11
Number of equality constraint evaluations = 23
Number of inequality constraint evaluations = 0
Number of equality constraint Jacobian evaluations = 11
Number of inequality constraint Jacobian evaluations = 0
Number of Lagrangian Hessian evaluations = 10
Total seconds in IPOPT = 0.003
EXIT: Optimal Solution Found.
This is Ipopt version 3.14.19, running with linear solver MUMPS 5.8.2.
Number of nonzeros in equality constraint Jacobian...: 32
Number of nonzeros in inequality constraint Jacobian.: 0
Number of nonzeros in Lagrangian Hessian.............: 36
Total number of variables............................: 8
variables with only lower bounds: 0
variables with lower and upper bounds: 8
variables with only upper bounds: 0
Total number of equality constraints.................: 4
Total number of inequality constraints...............: 0
inequality constraints with only lower bounds: 0
inequality constraints with lower and upper bounds: 0
inequality constraints with only upper bounds: 0
iter objective inf_pr inf_du lg(mu) ||d|| lg(rg) alpha_du alpha_pr ls
0 -1.4450072e-13 0.00e+00 3.35e+00 -1.0 0.00e+00 - 0.00e+00 0.00e+00 0
1 -1.1544520e-13 1.48e-31 1.50e+08 -1.0 3.42e-08 - 1.00e+00 1.41e-08f 12
2 -1.1566284e-13 4.93e-32 2.15e-05 -2.5 6.25e-18 - 1.00e+00 1.00e+00 0
3 -1.1566288e-13 4.93e-32 4.20e-07 -11.0 6.96e-22 - 1.00e+00 1.00e+00 0
4 -4.8128315e-13 1.77e-30 2.48e+00 -11.0 8.15e-15 - 1.00e+00 1.00e+00f 1
5 -1.0495024e-12 1.63e-30 2.00e+00 -11.0 1.70e-14 14.0 1.00e+00 1.00e+00f 1
6 -3.1685672e-12 3.11e-30 1.15e+00 -11.0 7.77e-14 - 1.00e+00 1.00e+00f 1
7 -3.4906907e-12 9.66e-30 1.18e-01 -11.0 1.39e-14 - 1.00e+00 1.00e+00f 1
8 -3.4929749e-12 7.94e-30 1.19e-04 -11.0 2.06e-15 - 1.00e+00 1.00e+00 0
9 -3.4929657e-12 1.02e-29 2.62e-09 -11.0 2.94e-18 - 1.00e+00 1.00e+00 0
iter objective inf_pr inf_du lg(mu) ||d|| lg(rg) alpha_du alpha_pr ls
10 -3.4929657e-12 1.01e-29 1.07e-14 -11.0 6.38e-23 - 1.00e+00 1.00e+00T 0
Number of Iterations....: 10
(scaled) (unscaled)
Objective...............: -7.5428059319056088e-13 -3.4929657218121374e-12
Dual infeasibility......: 1.0658141036401503e-14 4.9356329242033616e-14
Constraint violation....: 1.0107280348144214e-29 1.0107280348144214e-29
Variable bound violation: 1.0135810673510399e-13 1.0135810673510399e-13
Complementarity.........: 9.0909090909090936e-12 4.2098701890681467e-11
Overall NLP error.......: 9.0909090909090936e-12 4.2098701890681467e-11
Number of objective function evaluations = 23
Number of objective gradient evaluations = 11
Number of equality constraint evaluations = 23
Number of inequality constraint evaluations = 0
Number of equality constraint Jacobian evaluations = 11
Number of inequality constraint Jacobian evaluations = 0
Number of Lagrangian Hessian evaluations = 10
Total seconds in IPOPT = 0.003
EXIT: Optimal Solution Found.
This is Ipopt version 3.14.19, running with linear solver MUMPS 5.8.2.
Number of nonzeros in equality constraint Jacobian...: 32
Number of nonzeros in inequality constraint Jacobian.: 0
Number of nonzeros in Lagrangian Hessian.............: 36
Total number of variables............................: 8
variables with only lower bounds: 0
variables with lower and upper bounds: 8
variables with only upper bounds: 0
Total number of equality constraints.................: 4
Total number of inequality constraints...............: 0
inequality constraints with only lower bounds: 0
inequality constraints with lower and upper bounds: 0
inequality constraints with only upper bounds: 0
iter objective inf_pr inf_du lg(mu) ||d|| lg(rg) alpha_du alpha_pr ls
0 -1.4401269e-13 0.00e+00 3.36e+00 -1.0 0.00e+00 - 0.00e+00 0.00e+00 0
1 -1.1498054e-13 1.48e-31 1.50e+08 -1.0 3.41e-08 - 1.00e+00 1.42e-08f 12
2 -1.1519816e-13 9.86e-32 2.10e-05 -2.5 6.25e-18 - 1.00e+00 1.00e+00 0
3 -1.1519821e-13 4.93e-32 3.90e-07 -11.0 6.97e-22 - 1.00e+00 1.00e+00 0
4 -4.8015643e-13 1.38e-30 2.48e+00 -11.0 8.16e-15 - 1.00e+00 1.00e+00f 1
5 -1.0480345e-12 1.68e-30 2.01e+00 -11.0 1.70e-14 14.0 1.00e+00 1.00e+00f 1
6 -3.1629626e-12 5.67e-30 1.17e+00 -11.0 7.77e-14 - 1.00e+00 1.00e+00f 1
7 -3.4861066e-12 2.96e-30 1.37e-01 -11.0 1.40e-14 - 1.00e+00 1.00e+00f 1
8 -3.4885690e-12 5.57e-30 1.94e-04 -11.0 2.42e-15 - 1.00e+00 1.00e+00h 1
9 -3.4885585e-12 3.90e-30 7.02e-09 -11.0 4.79e-18 - 1.00e+00 1.00e+00 0
iter objective inf_pr inf_du lg(mu) ||d|| lg(rg) alpha_du alpha_pr ls
10 -3.4885585e-12 3.35e-30 8.88e-15 -11.0 1.71e-22 - 1.00e+00 1.00e+00T 0
Number of Iterations....: 10
(scaled) (unscaled)
Objective...............: -7.5583743392296687e-13 -3.4885585291872333e-12
Dual infeasibility......: 8.8817841970012523e-15 4.0993767474616745e-14
Constraint violation....: 3.3526588471893002e-30 3.3526588471893002e-30
Variable bound violation: 1.0146513522023081e-13 1.0146513522023081e-13
Complementarity.........: 9.0909090909090936e-12 4.1958980891635633e-11
Overall NLP error.......: 9.0909090909090936e-12 4.1958980891635633e-11
Number of objective function evaluations = 23
Number of objective gradient evaluations = 11
Number of equality constraint evaluations = 23
Number of inequality constraint evaluations = 0
Number of equality constraint Jacobian evaluations = 11
Number of inequality constraint Jacobian evaluations = 0
Number of Lagrangian Hessian evaluations = 10
Total seconds in IPOPT = 0.003
EXIT: Optimal Solution Found.
This is Ipopt version 3.14.19, running with linear solver MUMPS 5.8.2.
Number of nonzeros in equality constraint Jacobian...: 32
Number of nonzeros in inequality constraint Jacobian.: 0
Number of nonzeros in Lagrangian Hessian.............: 36
Total number of variables............................: 8
variables with only lower bounds: 0
variables with lower and upper bounds: 8
variables with only upper bounds: 0
Total number of equality constraints.................: 4
Total number of inequality constraints...............: 0
inequality constraints with only lower bounds: 0
inequality constraints with lower and upper bounds: 0
inequality constraints with only upper bounds: 0
iter objective inf_pr inf_du lg(mu) ||d|| lg(rg) alpha_du alpha_pr ls
0 -1.4352795e-13 0.00e+00 3.36e+00 -1.0 0.00e+00 - 0.00e+00 0.00e+00 0
1 -1.1451874e-13 4.93e-32 1.50e+08 -1.0 3.40e-08 - 1.00e+00 1.42e-08f 12
2 -1.1473636e-13 9.86e-32 1.96e-05 -2.5 6.25e-18 - 1.00e+00 1.00e+00 0
3 -1.1473640e-13 4.93e-32 5.08e-07 -11.0 6.98e-22 - 1.00e+00 1.00e+00 0
4 -4.7904817e-13 1.73e-30 2.48e+00 -11.0 8.16e-15 - 1.00e+00 1.00e+00f 1
5 -1.0466111e-12 1.48e-30 2.01e+00 -11.0 1.71e-14 14.0 1.00e+00 1.00e+00f 1
6 -3.1575165e-12 1.62e-29 1.18e+00 -11.0 7.78e-14 - 1.00e+00 1.00e+00f 1
7 -3.4816766e-12 1.01e-29 1.55e-01 -11.0 1.41e-14 - 1.00e+00 1.00e+00f 1
8 -3.4843446e-12 3.90e-30 2.85e-04 -11.0 2.77e-15 - 1.00e+00 1.00e+00h 1
9 -3.4843331e-12 5.67e-30 1.52e-08 -11.0 7.01e-18 - 1.00e+00 1.00e+00 0
iter objective inf_pr inf_du lg(mu) ||d|| lg(rg) alpha_du alpha_pr ls
10 -3.4843331e-12 1.01e-29 5.33e-15 -11.0 3.69e-22 - 1.00e+00 1.00e+00T 0
Number of Iterations....: 10
(scaled) (unscaled)
Objective...............: -7.5742686743267756e-13 -3.4843331446189855e-12
Dual infeasibility......: 5.3290705182007514e-15 2.4514917327287913e-14
Constraint violation....: 1.0107280348144214e-29 1.0107280348144214e-29
Variable bound violation: 1.0157418030581493e-13 1.0157418030581493e-13
Complementarity.........: 9.0909090909090936e-12 4.1820216871284997e-11
Overall NLP error.......: 9.0909090909090936e-12 4.1820216871284997e-11
Number of objective function evaluations = 23
Number of objective gradient evaluations = 11
Number of equality constraint evaluations = 23
Number of inequality constraint evaluations = 0
Number of equality constraint Jacobian evaluations = 11
Number of inequality constraint Jacobian evaluations = 0
Number of Lagrangian Hessian evaluations = 10
Total seconds in IPOPT = 0.003
EXIT: Optimal Solution Found.
This is Ipopt version 3.14.19, running with linear solver MUMPS 5.8.2.
Number of nonzeros in equality constraint Jacobian...: 32
Number of nonzeros in inequality constraint Jacobian.: 0
Number of nonzeros in Lagrangian Hessian.............: 36
Total number of variables............................: 8
variables with only lower bounds: 0
variables with lower and upper bounds: 8
variables with only upper bounds: 0
Total number of equality constraints.................: 4
Total number of inequality constraints...............: 0
inequality constraints with only lower bounds: 0
inequality constraints with lower and upper bounds: 0
inequality constraints with only upper bounds: 0
iter objective inf_pr inf_du lg(mu) ||d|| lg(rg) alpha_du alpha_pr ls
0 -1.4304647e-13 0.00e+00 3.36e+00 -1.0 0.00e+00 - 0.00e+00 0.00e+00 0
1 -1.1405977e-13 4.93e-32 1.50e+08 -1.0 3.39e-08 - 1.00e+00 1.43e-08f 12
2 -1.1427739e-13 4.93e-32 7.63e-06 -2.5 6.25e-18 - 1.00e+00 1.00e+00 0
3 -1.1427743e-13 4.93e-32 4.76e-07 -11.0 6.98e-22 - 1.00e+00 1.00e+00 0
4 -4.7795804e-13 1.63e-30 2.49e+00 -11.0 8.17e-15 - 1.00e+00 1.00e+00f 1
5 -1.0452315e-12 1.68e-30 2.02e+00 -11.0 1.71e-14 14.0 1.00e+00 1.00e+00f 1
6 -3.1522267e-12 1.35e-29 1.19e+00 -11.0 7.78e-14 - 1.00e+00 1.00e+00f 1
7 -3.4773986e-12 5.47e-30 1.73e-01 -11.0 1.42e-14 - 1.00e+00 1.00e+00f 1
8 -3.4802993e-12 3.35e-30 3.89e-04 -11.0 3.11e-15 - 1.00e+00 1.00e+00h 1
9 -3.4802869e-12 8.83e-30 2.86e-08 -11.0 9.57e-18 - 1.00e+00 1.00e+00 0
iter objective inf_pr inf_du lg(mu) ||d|| lg(rg) alpha_du alpha_pr ls
10 -3.4802869e-12 6.95e-30 7.11e-15 -11.0 6.93e-22 - 1.00e+00 1.00e+00T 0
Number of Iterations....: 10
(scaled) (unscaled)
Objective...............: -7.5904870406757780e-13 -3.4802868980467698e-12
Dual infeasibility......: 7.1054273576010019e-15 3.2578839282861406e-14
Constraint violation....: 6.9518367272601665e-30 6.9518367272601665e-30
Variable bound violation: 1.0168520882625093e-13 1.0168520882625093e-13
Complementarity.........: 9.0909090909090936e-12 4.1682400129107372e-11
Overall NLP error.......: 9.0909090909090936e-12 4.1682400129107372e-11
Number of objective function evaluations = 23
Number of objective gradient evaluations = 11
Number of equality constraint evaluations = 23
Number of inequality constraint evaluations = 0
Number of equality constraint Jacobian evaluations = 11
Number of inequality constraint Jacobian evaluations = 0
Number of Lagrangian Hessian evaluations = 10
Total seconds in IPOPT = 0.003
EXIT: Optimal Solution Found.
This is Ipopt version 3.14.19, running with linear solver MUMPS 5.8.2.
Number of nonzeros in equality constraint Jacobian...: 32
Number of nonzeros in inequality constraint Jacobian.: 0
Number of nonzeros in Lagrangian Hessian.............: 36
Total number of variables............................: 8
variables with only lower bounds: 0
variables with lower and upper bounds: 8
variables with only upper bounds: 0
Total number of equality constraints.................: 4
Total number of inequality constraints...............: 0
inequality constraints with only lower bounds: 0
inequality constraints with lower and upper bounds: 0
inequality constraints with only upper bounds: 0
iter objective inf_pr inf_du lg(mu) ||d|| lg(rg) alpha_du alpha_pr ls
0 -1.4256822e-13 0.00e+00 3.37e+00 -1.0 0.00e+00 - 0.00e+00 0.00e+00 0
1 -1.1360363e-13 1.48e-31 1.50e+08 -1.0 3.38e-08 - 1.00e+00 1.43e-08f 12
2 -1.1382124e-13 4.93e-32 2.24e-05 -2.5 6.25e-18 - 1.00e+00 1.00e+00 0
3 -1.1382129e-13 4.93e-32 2.15e-07 -11.0 6.99e-22 - 1.00e+00 1.00e+00 0
4 -4.7688572e-13 1.77e-30 2.49e+00 -11.0 8.17e-15 - 1.00e+00 1.00e+00f 1
5 -1.0438950e-12 2.61e-30 2.02e+00 -11.0 1.71e-14 14.0 1.00e+00 1.00e+00f 1
6 -3.1470912e-12 1.36e-29 1.21e+00 -11.0 7.79e-14 - 1.00e+00 1.00e+00f 1
7 -3.4732708e-12 3.80e-30 1.91e-01 -11.0 1.43e-14 - 1.00e+00 1.00e+00f 1
8 -3.4764306e-12 3.80e-30 5.07e-04 -11.0 3.45e-15 - 1.00e+00 1.00e+00h 1
9 -3.4764172e-12 3.80e-30 4.88e-08 -11.0 1.24e-17 - 1.00e+00 1.00e+00 0
iter objective inf_pr inf_du lg(mu) ||d|| lg(rg) alpha_du alpha_pr ls
10 -3.4764172e-12 6.95e-30 7.11e-15 -11.0 1.18e-21 - 1.00e+00 1.00e+00T 0
Number of Iterations....: 10
(scaled) (unscaled)
Objective...............: -7.6070277218596222e-13 -3.4764172375721984e-12
Dual infeasibility......: 7.1054273576010019e-15 3.2471855039122509e-14
Constraint violation....: 6.9518367272601665e-30 6.9518367272601665e-30
Variable bound violation: 1.0179818901789385e-13 1.0179818901789385e-13
Complementarity.........: 9.0909090909090936e-12 4.1545521094949129e-11
Overall NLP error.......: 9.0909090909090936e-12 4.1545521094949129e-11
Number of objective function evaluations = 23
Number of objective gradient evaluations = 11
Number of equality constraint evaluations = 23
Number of inequality constraint evaluations = 0
Number of equality constraint Jacobian evaluations = 11
Number of inequality constraint Jacobian evaluations = 0
Number of Lagrangian Hessian evaluations = 10
Total seconds in IPOPT = 0.003
EXIT: Optimal Solution Found.
This is Ipopt version 3.14.19, running with linear solver MUMPS 5.8.2.
Number of nonzeros in equality constraint Jacobian...: 32
Number of nonzeros in inequality constraint Jacobian.: 0
Number of nonzeros in Lagrangian Hessian.............: 36
Total number of variables............................: 8
variables with only lower bounds: 0
variables with lower and upper bounds: 8
variables with only upper bounds: 0
Total number of equality constraints.................: 4
Total number of inequality constraints...............: 0
inequality constraints with only lower bounds: 0
inequality constraints with lower and upper bounds: 0
inequality constraints with only upper bounds: 0
iter objective inf_pr inf_du lg(mu) ||d|| lg(rg) alpha_du alpha_pr ls
0 -1.4209316e-13 0.00e+00 3.37e+00 -1.0 0.00e+00 - 0.00e+00 0.00e+00 0
1 -1.1315027e-13 4.93e-32 1.50e+08 -1.0 3.37e-08 - 1.00e+00 1.44e-08f 12
2 -1.1336789e-13 9.86e-32 1.67e-05 -2.5 6.25e-18 - 1.00e+00 1.00e+00 0
3 -1.1336793e-13 9.86e-32 3.89e-07 -11.0 7.00e-22 - 1.00e+00 1.00e+00 0
4 -4.7583090e-13 1.38e-30 2.49e+00 -11.0 8.18e-15 - 1.00e+00 1.00e+00f 1
5 -1.0426010e-12 2.51e-30 2.03e+00 -11.0 1.72e-14 14.0 1.00e+00 1.00e+00f 1
6 -3.1421081e-12 9.27e-30 1.22e+00 -11.0 7.80e-14 - 1.00e+00 1.00e+00f 1
7 -3.4692915e-12 3.35e-30 2.08e-01 -11.0 1.44e-14 - 1.00e+00 1.00e+00f 1
8 -3.4727359e-12 3.90e-30 6.36e-04 -11.0 3.79e-15 - 1.00e+00 1.00e+00h 1
9 -3.4727217e-12 1.01e-29 7.72e-08 -11.0 1.56e-17 - 1.00e+00 1.00e+00 0
iter objective inf_pr inf_du lg(mu) ||d|| lg(rg) alpha_du alpha_pr ls
10 -3.4727217e-12 5.18e-30 8.88e-15 -11.0 1.86e-21 - 1.00e+00 1.00e+00T 0
Number of Iterations....: 10
(scaled) (unscaled)
Objective...............: -7.6238891711407427e-13 -3.4727217227606241e-12
Dual infeasibility......: 8.8817841970012523e-15 4.0456995406693684e-14
Constraint violation....: 5.1768996905128900e-30 5.1768996905128900e-30
Variable bound violation: 1.0191309043790804e-13 1.0191309043790804e-13
Complementarity.........: 9.0909090909090936e-12 4.1409570326844450e-11
Overall NLP error.......: 9.0909090909090936e-12 4.1409570326844450e-11
Number of objective function evaluations = 23
Number of objective gradient evaluations = 11
Number of equality constraint evaluations = 23
Number of inequality constraint evaluations = 0
Number of equality constraint Jacobian evaluations = 11
Number of inequality constraint Jacobian evaluations = 0
Number of Lagrangian Hessian evaluations = 10
Total seconds in IPOPT = 0.003
EXIT: Optimal Solution Found.
This is Ipopt version 3.14.19, running with linear solver MUMPS 5.8.2.
Number of nonzeros in equality constraint Jacobian...: 32
Number of nonzeros in inequality constraint Jacobian.: 0
Number of nonzeros in Lagrangian Hessian.............: 36
Total number of variables............................: 8
variables with only lower bounds: 0
variables with lower and upper bounds: 8
variables with only upper bounds: 0
Total number of equality constraints.................: 4
Total number of inequality constraints...............: 0
inequality constraints with only lower bounds: 0
inequality constraints with lower and upper bounds: 0
inequality constraints with only upper bounds: 0
iter objective inf_pr inf_du lg(mu) ||d|| lg(rg) alpha_du alpha_pr ls
0 -1.4162126e-13 0.00e+00 3.37e+00 -1.0 0.00e+00 - 0.00e+00 0.00e+00 0
1 -1.1269968e-13 9.86e-32 1.50e+08 -1.0 3.36e-08 - 1.00e+00 1.44e-08f 12
2 -1.1291730e-13 9.86e-32 1.72e-05 -2.5 6.25e-18 - 1.00e+00 1.00e+00 0
3 -1.1291735e-13 1.48e-31 5.06e-07 -11.0 7.01e-22 - 1.00e+00 1.00e+00 0
4 -4.7479327e-13 1.38e-30 2.49e+00 -11.0 8.19e-15 - 1.00e+00 1.00e+00f 1
5 -1.0413487e-12 2.12e-30 2.04e+00 -11.0 1.72e-14 14.0 1.00e+00 1.00e+00f 1
6 -3.1372754e-12 3.80e-30 1.23e+00 -11.0 7.80e-14 - 1.00e+00 1.00e+00f 1
7 -3.4654589e-12 2.96e-30 2.25e-01 -11.0 1.45e-14 - 1.00e+00 1.00e+00f 1
8 -3.4692131e-12 9.27e-30 7.76e-04 -11.0 4.11e-15 - 1.00e+00 1.00e+00h 1
9 -3.4691980e-12 9.66e-30 1.16e-07 -11.0 1.90e-17 - 1.00e+00 1.00e+00 0
iter objective inf_pr inf_du lg(mu) ||d|| lg(rg) alpha_du alpha_pr ls
10 -3.4691980e-12 6.95e-30 1.24e-14 -11.0 2.79e-21 - 1.00e+00 1.00e+00T 0
Number of Iterations....: 10
(scaled) (unscaled)
Objective...............: -7.6410700017651567e-13 -3.4691980184057811e-12
Dual infeasibility......: 1.2434497875801753e-14 5.6455097755467659e-14
Constraint violation....: 6.9518367272601665e-30 6.9518367272601665e-30
Variable bound violation: 1.0202988388879412e-13 1.0202988388879412e-13
Complementarity.........: 9.0909090909090936e-12 4.1274538508878113e-11
Overall NLP error.......: 9.0909090909090936e-12 4.1274538508878113e-11
Number of objective function evaluations = 23
Number of objective gradient evaluations = 11
Number of equality constraint evaluations = 23
Number of inequality constraint evaluations = 0
Number of equality constraint Jacobian evaluations = 11
Number of inequality constraint Jacobian evaluations = 0
Number of Lagrangian Hessian evaluations = 10
Total seconds in IPOPT = 0.003
EXIT: Optimal Solution Found.
This is Ipopt version 3.14.19, running with linear solver MUMPS 5.8.2.
Number of nonzeros in equality constraint Jacobian...: 32
Number of nonzeros in inequality constraint Jacobian.: 0
Number of nonzeros in Lagrangian Hessian.............: 36
Total number of variables............................: 8
variables with only lower bounds: 0
variables with lower and upper bounds: 8
variables with only upper bounds: 0
Total number of equality constraints.................: 4
Total number of inequality constraints...............: 0
inequality constraints with only lower bounds: 0
inequality constraints with lower and upper bounds: 0
inequality constraints with only upper bounds: 0
iter objective inf_pr inf_du lg(mu) ||d|| lg(rg) alpha_du alpha_pr ls
0 -1.4115250e-13 0.00e+00 3.38e+00 -1.0 0.00e+00 - 0.00e+00 0.00e+00 0
1 -1.1225184e-13 4.93e-32 1.50e+08 -1.0 3.35e-08 - 1.00e+00 1.45e-08f 12
2 -1.1246947e-13 1.48e-31 2.10e-05 -2.5 6.25e-18 - 1.00e+00 1.00e+00 0
3 -1.1246951e-13 9.86e-32 4.03e-07 -11.0 7.02e-22 - 1.00e+00 1.00e+00 0
4 -4.7377255e-13 1.43e-30 2.50e+00 -11.0 8.19e-15 - 1.00e+00 1.00e+00f 1
5 -1.0401376e-12 3.35e-30 2.04e+00 -11.0 1.73e-14 14.0 1.00e+00 1.00e+00f 1
6 -3.1325913e-12 3.11e-30 1.25e+00 -11.0 7.81e-14 - 1.00e+00 1.00e+00f 1
7 -3.4617715e-12 3.80e-30 2.42e-01 -11.0 1.45e-14 - 1.00e+00 1.00e+00f 1
8 -3.4658599e-12 9.27e-30 9.26e-04 -11.0 4.44e-15 - 1.00e+00 1.00e+00h 1
9 -3.4658439e-12 3.35e-30 1.66e-07 -11.0 2.27e-17 - 1.00e+00 1.00e+00 0
iter objective inf_pr inf_du lg(mu) ||d|| lg(rg) alpha_du alpha_pr ls
10 -3.4658439e-12 9.27e-30 1.07e-14 -11.0 3.98e-21 - 1.00e+00 1.00e+00T 0
Number of Iterations....: 10
(scaled) (unscaled)
Objective...............: -7.6585689780042876e-13 -3.4658438887510452e-12
Dual infeasibility......: 1.0658141036401503e-14 4.8232839689177500e-14
Constraint violation....: 9.2691156363468887e-30 9.2691156363468887e-30
Variable bound violation: 1.0214854134833648e-13 1.0214854134833648e-13
Complementarity.........: 9.0909090909090936e-12 4.1140416449091051e-11
Overall NLP error.......: 9.0909090909090936e-12 4.1140416449091051e-11
Number of objective function evaluations = 23
Number of objective gradient evaluations = 11
Number of equality constraint evaluations = 23
Number of inequality constraint evaluations = 0
Number of equality constraint Jacobian evaluations = 11
Number of inequality constraint Jacobian evaluations = 0
Number of Lagrangian Hessian evaluations = 10
Total seconds in IPOPT = 0.003
EXIT: Optimal Solution Found.
This is Ipopt version 3.14.19, running with linear solver MUMPS 5.8.2.
Number of nonzeros in equality constraint Jacobian...: 32
Number of nonzeros in inequality constraint Jacobian.: 0
Number of nonzeros in Lagrangian Hessian.............: 36
Total number of variables............................: 8
variables with only lower bounds: 0
variables with lower and upper bounds: 8
variables with only upper bounds: 0
Total number of equality constraints.................: 4
Total number of inequality constraints...............: 0
inequality constraints with only lower bounds: 0
inequality constraints with lower and upper bounds: 0
inequality constraints with only upper bounds: 0
iter objective inf_pr inf_du lg(mu) ||d|| lg(rg) alpha_du alpha_pr ls
0 -1.4068684e-13 0.00e+00 3.38e+00 -1.0 0.00e+00 - 0.00e+00 0.00e+00 0
1 -1.1180672e-13 4.93e-32 1.50e+08 -1.0 3.34e-08 - 1.00e+00 1.45e-08f 12
2 -1.1202436e-13 4.93e-32 2.10e-05 -2.5 6.25e-18 - 1.00e+00 1.00e+00 0
3 -1.1202440e-13 4.93e-32 5.12e-07 -11.0 7.03e-22 - 1.00e+00 1.00e+00 0
4 -4.7276845e-13 1.38e-30 2.50e+00 -11.0 8.20e-15 - 1.00e+00 1.00e+00f 1
5 -1.0389671e-12 1.48e-30 2.05e+00 -11.0 1.73e-14 14.0 1.00e+00 1.00e+00f 1
6 -3.1280541e-12 6.95e-30 1.26e+00 -11.0 7.81e-14 - 1.00e+00 1.00e+00f 1
7 -3.4582278e-12 3.90e-30 2.58e-01 -11.0 1.46e-14 - 1.00e+00 1.00e+00f 1
8 -3.4626743e-12 3.90e-30 1.09e-03 -11.0 4.75e-15 - 1.00e+00 1.00e+00h 1
9 -3.4626572e-12 9.66e-30 2.28e-07 -11.0 2.65e-17 - 1.00e+00 1.00e+00 0
iter objective inf_pr inf_du lg(mu) ||d|| lg(rg) alpha_du alpha_pr ls
10 -3.4626572e-12 3.35e-30 1.60e-14 -11.0 5.49e-21 - 1.00e+00 1.00e+00T 0
Number of Iterations....: 10
(scaled) (unscaled)
Objective...............: -7.6763850068398657e-13 -3.4626571921141279e-12
Dual infeasibility......: 1.5987211554602254e-14 7.2114977326004243e-14
Constraint violation....: 3.3526588471893002e-30 3.3526588471893002e-30
Variable bound violation: 1.0226903590442867e-13 1.0226903590442867e-13
Complementarity.........: 9.0909090909090936e-12 4.1007195077427342e-11
Overall NLP error.......: 9.0909090909090936e-12 4.1007195077427342e-11
Number of objective function evaluations = 23
Number of objective gradient evaluations = 11
Number of equality constraint evaluations = 23
Number of inequality constraint evaluations = 0
Number of equality constraint Jacobian evaluations = 11
Number of inequality constraint Jacobian evaluations = 0
Number of Lagrangian Hessian evaluations = 10
Total seconds in IPOPT = 0.003
EXIT: Optimal Solution Found.
This is Ipopt version 3.14.19, running with linear solver MUMPS 5.8.2.
Number of nonzeros in equality constraint Jacobian...: 32
Number of nonzeros in inequality constraint Jacobian.: 0
Number of nonzeros in Lagrangian Hessian.............: 36
Total number of variables............................: 8
variables with only lower bounds: 0
variables with lower and upper bounds: 8
variables with only upper bounds: 0
Total number of equality constraints.................: 4
Total number of inequality constraints...............: 0
inequality constraints with only lower bounds: 0
inequality constraints with lower and upper bounds: 0
inequality constraints with only upper bounds: 0
iter objective inf_pr inf_du lg(mu) ||d|| lg(rg) alpha_du alpha_pr ls
0 -1.4022426e-13 0.00e+00 3.38e+00 -1.0 0.00e+00 - 0.00e+00 0.00e+00 0
1 -1.1136430e-13 4.93e-32 1.50e+08 -1.0 3.32e-08 - 1.00e+00 1.46e-08f 12
2 -1.1158195e-13 4.93e-32 2.34e-05 -2.5 6.25e-18 - 1.00e+00 1.00e+00 0
3 -1.1158199e-13 4.93e-32 7.24e-07 -11.0 7.04e-22 - 1.00e+00 1.00e+00 0
4 -4.7178070e-13 1.77e-30 2.50e+00 -11.0 8.21e-15 - 1.00e+00 1.00e+00f 1
5 -1.0378366e-12 2.96e-30 2.05e+00 -11.0 1.73e-14 14.0 1.00e+00 1.00e+00f 1
6 -3.1236621e-12 3.35e-30 1.27e+00 -11.0 7.82e-14 - 1.00e+00 1.00e+00f 1
7 -3.4548262e-12 2.96e-30 2.75e-01 -11.0 1.47e-14 - 1.00e+00 1.00e+00f 1
8 -3.4596541e-12 1.27e-29 1.25e-03 -11.0 5.06e-15 - 1.00e+00 1.00e+00h 1
9 -3.4596359e-12 6.95e-30 3.06e-07 -11.0 3.06e-17 - 1.00e+00 1.00e+00 0
iter objective inf_pr inf_du lg(mu) ||d|| lg(rg) alpha_du alpha_pr ls
10 -3.4596359e-12 1.06e-29 2.31e-14 -11.0 7.34e-21 - 1.00e+00 1.00e+00T 0
Number of Iterations....: 10
(scaled) (unscaled)
Objective...............: -7.6945171302302977e-13 -3.4596358758783439e-12
Dual infeasibility......: 2.3092638912203256e-14 1.0382993591044413e-13
Constraint violation....: 1.0600318413907346e-29 1.0600318413907346e-29
Variable bound violation: 1.0239134169434724e-13 1.0239134169434724e-13
Complementarity.........: 9.0909090909090936e-12 4.0874865443721925e-11
Overall NLP error.......: 9.0909090909090936e-12 4.0874865443721925e-11
Number of objective function evaluations = 23
Number of objective gradient evaluations = 11
Number of equality constraint evaluations = 23
Number of inequality constraint evaluations = 0
Number of equality constraint Jacobian evaluations = 11
Number of inequality constraint Jacobian evaluations = 0
Number of Lagrangian Hessian evaluations = 10
Total seconds in IPOPT = 0.003
EXIT: Optimal Solution Found.
This is Ipopt version 3.14.19, running with linear solver MUMPS 5.8.2.
Number of nonzeros in equality constraint Jacobian...: 32
Number of nonzeros in inequality constraint Jacobian.: 0
Number of nonzeros in Lagrangian Hessian.............: 36
Total number of variables............................: 8
variables with only lower bounds: 0
variables with lower and upper bounds: 8
variables with only upper bounds: 0
Total number of equality constraints.................: 4
Total number of inequality constraints...............: 0
inequality constraints with only lower bounds: 0
inequality constraints with lower and upper bounds: 0
inequality constraints with only upper bounds: 0
iter objective inf_pr inf_du lg(mu) ||d|| lg(rg) alpha_du alpha_pr ls
0 -1.3976471e-13 0.00e+00 3.38e+00 -1.0 0.00e+00 - 0.00e+00 0.00e+00 0
1 -1.1092455e-13 1.48e-31 1.50e+08 -1.0 3.31e-08 - 1.00e+00 1.46e-08f 12
2 -1.1114222e-13 4.93e-32 3.29e-05 -2.5 6.25e-18 - 1.00e+00 1.00e+00 0
3 -1.1114226e-13 9.86e-32 3.23e-07 -11.0 7.04e-22 - 1.00e+00 1.00e+00 0
4 -4.7080904e-13 1.48e-30 2.50e+00 -11.0 8.21e-15 - 1.00e+00 1.00e+00f 1
5 -1.0367456e-12 4.39e-30 2.06e+00 -11.0 1.74e-14 14.0 1.00e+00 1.00e+00f 1
6 -3.1194137e-12 1.33e-29 1.28e+00 -11.0 7.83e-14 - 1.00e+00 1.00e+00f 1
7 -3.4515653e-12 9.66e-30 2.91e-01 -11.0 1.48e-14 - 1.00e+00 1.00e+00f 1
8 -3.4567975e-12 4.68e-30 1.43e-03 -11.0 5.36e-15 - 1.00e+00 1.00e+00h 1
9 -3.4567779e-12 3.35e-30 3.99e-07 -11.0 3.48e-17 - 1.00e+00 1.00e+00 0
iter objective inf_pr inf_du lg(mu) ||d|| lg(rg) alpha_du alpha_pr ls
10 -3.4567780e-12 3.35e-30 2.49e-14 -11.0 9.57e-21 - 1.00e+00 1.00e+00T 0
Number of Iterations....: 10
(scaled) (unscaled)
Objective...............: -7.7129645179475436e-13 -3.4567779718371899e-12
Dual infeasibility......: 2.4868995751603507e-14 1.1145726976419668e-13
Constraint violation....: 3.3526588471893002e-30 3.3526588471893002e-30
Variable bound violation: 1.0251543384822464e-13 1.0251543384822464e-13
Complementarity.........: 9.0909090909090936e-12 4.0743418715728018e-11
Overall NLP error.......: 9.0909090909090936e-12 4.0743418715728018e-11
Number of objective function evaluations = 23
Number of objective gradient evaluations = 11
Number of equality constraint evaluations = 23
Number of inequality constraint evaluations = 0
Number of equality constraint Jacobian evaluations = 11
Number of inequality constraint Jacobian evaluations = 0
Number of Lagrangian Hessian evaluations = 10
Total seconds in IPOPT = 0.003
EXIT: Optimal Solution Found.The figures can then be drawn.
p1 = plot(collect(temperatures), pH_vals,
xlabel = "T (°C)", ylabel = "pH", label = "pH",
marker = :circle, linewidth = 2, title = "pH")
p2 = plot(collect(temperatures), nCa_vals,
xlabel = "T (°C)", ylabel = "n (mmol)", label = "Ca²⁺",
marker = :circle, linewidth = 2, title = "Dissolved species")
plot!(p2, collect(temperatures), nCal_vals,
label = "Cal", marker = :square, linewidth = 2)
plot(p1, p2, layout = (1, 2), size = (700, 350))
Calcite is a retrograde soluble mineral: its solubility decreases with increasing temperature, so less Ca²⁺ is released and pH rises slightly as temperature increases.
Activity models
All activity models inherit from AbstractActivityModel. Three built-in models are provided, covering ideal behaviour through to the extended Debye-Hückel level used by standard geochemical codes.
Choosing a model
| Model | Formula | Valid range | Parameters needed |
|---|---|---|---|
DiluteSolutionModel | Raoult / Henry | I ≪ 1 mol/kg | none |
HKFActivityModel | B-dot extended Debye-Hückel | I ≲ 1 mol/kg | A, B, Ḃ (defaults at 25 °C) |
DaviesActivityModel | Davies equation | I ≲ 0.5 mol/kg | A, b (defaults at 25 °C) |
DiluteSolutionModel (ideal dilute solution)
| Phase | Law | Expression |
|---|---|---|
| Solvent (H₂O) | Raoult | ln a = ln xₛ |
| Aqueous solutes | Henry | ln a = ln(cᵢ / c°), c° = 1 mol/L |
| Crystals | Pure solid | ln a = 0 |
| Gas | Ideal mixture | ln a = ln xᵢ |
state_eq = equilibrate(state) # DiluteSolutionModel is the defaultHKFActivityModel (extended Debye-Hückel B-dot)
Implements the extended Debye-Hückel model of Helgeson (1969) and Helgeson, Kirkham & Flowers (1981), identical to the model used by PHREEQC and EQ3/6.
Ion activity coefficient:
log₁₀ γᵢ = −A zᵢ² √I / (1 + B åᵢ √I) + Ḃ INeutral aqueous species (salting-out):
log₁₀ γᵢ = Kₙ IWater activity is computed from the osmotic coefficient via Gibbs-Duhem (not Raoult), which is accurate up to I ≈ 1 mol/kg.
Ionic radius lookup (priority order):
sp[:å]— explicit value set in species properties.REJ_HKF— Helgeson et al. (1981) Table 3 (27 common ions).REJ_CHARGE_DEFAULT— fallback by formal charge.model.å_default(default: 3.72 Å).
Usage:
# Fixed A, B at 25 °C / 1 bar (fast — suitable for isothermal calculations)
state_eq = equilibrate(state; model=HKFActivityModel())
# Temperature-dependent A and B (recomputed from T, P at each equilibrium solve)
state_eq = equilibrate(state; model=HKFActivityModel(temperature_dependent=true))
# Custom parameters
model = HKFActivityModel(A=0.52, B=0.33, Ḃ=0.04)The A and B parameters depend on the water dielectric constant and density and can be computed explicitly via hkf_debye_huckel_params:
ab = hkf_debye_huckel_params(298.15, 1e5) # → (A=0.5114, B=0.3288)The B-dot model is reliable for I ≲ 1 mol/kg. For higher ionic strengths (brines, evaporites), use the Pitzer model (planned future extension).
DaviesActivityModel (Davies equation)
Simpler alternative with no species-specific ionic radii. Suitable when ionic radii data are unavailable or for rapid screening calculations.
Ion activity coefficient:
log₁₀ γᵢ = −A zᵢ² (√I / (1 + √I) − b I)Water activity uses the Raoult (mole fraction) approximation.
state_eq = equilibrate(state; model=DaviesActivityModel())
# Temperature-dependent A
state_eq = equilibrate(state; model=DaviesActivityModel(temperature_dependent=true))Custom activity models
To implement a custom activity model, define a new subtype and extend activity_model:
struct MyModel <: AbstractActivityModel
# model parameters
end
function ChemistryLab.activity_model(cs::ChemicalSystem, ::MyModel)
# Precompute species indices and constants here (called once)
idx_solvent = only(cs.idx_solvent)
# ...
# Return a closure lna(n, p) -> Vector compatible with ForwardDiff
function lna(n::AbstractVector, p)
# p contains at minimum: p.ΔₐG⁰overT, p.T, p.P, p.ϵ
# n is dimensionless mole vector, same indexing as cs.species
out = zeros(eltype(n), length(n))
# ... fill log-activities ...
return out
end
return lna
endPass your model to equilibrate or EquilibriumSolver:
state_eq = equilibrate(state; model=MyModel(...))Solid solutions
Pure crystalline species have activity ln a = 0. Solid solutions are mineral phases with variable composition (e.g. C-S-H, AFm, hydrogarnet), where the activity of each end-member depends on its mole fraction within the phase.
Defining end-members and phases
End-member species must carry aggregate_state = AS_CRYSTAL. SolidSolutionPhase automatically requalifies any end-member whose class is not already SC_SSENDMEMBER, so database species with SC_COMPONENT can be passed directly.
Workflow A — pass database species directly:
using ChemistryLab
substances = build_species("data/cemdata18-thermofun.json")
dict = Dict(symbol(s) => s for s in substances)
# SolidSolutionPhase requalifies SC_COMPONENT → SC_SSENDMEMBER automatically
ss_afm = SolidSolutionPhase("AFm", [dict["Ms"], dict["Mc"]])Workflow B — automated via build_solid_solutions and a TOML file:
# Load all phases defined in the TOML
ss_phases = build_solid_solutions("data/solid_solutions.toml", dict)See the Databases tutorial for the TOML format and the pre-built data/solid_solutions.toml file shipped with ChemistryLab.
Then pass solid_solutions as a keyword to ChemicalSystem:
cs = ChemicalSystem(
[H2O_sp, dict["Ms"], dict["Mc"], ...],
["H2O@", "Al+3", ...]; # primaries
solid_solutions = [ss_afm], # or solid_solutions = ss_phases
)Activity models for solid solutions
| Model | Formula | Notes |
|---|---|---|
IdealSolidSolutionModel | ln aᵢ = ln xᵢ | Default, any number of end-members |
RedlichKisterModel | ln aᵢ = ln xᵢ + ln γᵢ (Margules) | Binary only (2 end-members), parameters in J/mol |
The solid-solution activity is computed inside the aqueous activity closure — no separate activity model is needed. The existing equilibrate(state) call handles solid solutions automatically.
Ideal solid solution
ss = SolidSolutionPhase("AFm", [em_ms, em_mc]) # IdealSolidSolutionModel() by default
cs = ChemicalSystem([...]; solid_solutions=[ss])
state_eq = equilibrate(state)Non-ideal binary: Redlich-Kister
# Interaction parameters for monosulfoaluminate-monocarboaluminate (example values)
rk = RedlichKisterModel(a0 = 3000.0, a1 = 500.0) # a2 defaults to 0.0
# or 3-parameter: RedlichKisterModel(a0 = 3000.0, a1 = 500.0, a2 = 50.0)
ss = SolidSolutionPhase("AFm", [em_ms, em_mc]; model=rk)Activity coefficients (Guggenheim / ThermoCalc convention):
\[\begin{aligned} \ln \gamma_1 &= \frac{x_2^2}{RT}\bigl[a_0 + a_1(3x_1 - x_2) + a_2(x_1 - x_2)(5x_1 - x_2)\bigr] \\[4pt] \ln \gamma_2 &= \frac{x_1^2}{RT}\bigl[a_0 - a_1(3x_2 - x_1) + a_2(x_2 - x_1)(5x_2 - x_1)\bigr] \end{aligned}\]