## APM.SOLVER - APMonitor Option

Type: Integer, Input Default Value: 3 Description: Solver options 0 = Benchmark All Solvers 1-5 = Available Solvers Depends on License

SOLVER selects the solver to use in an attempt to find a solution. There are free solvers: 1: APOPT, 2: BPOPT, 3: IPOPT distributed with the public version of the software. There are additional solvers that are not included with the public version and require a commercial license. IPOPT is generally the best for problems with large numbers of degrees of freedom or when starting without a good initial guess. BPOPT has been found to be the best for systems biology applications. APOPT is generally the best when warm-starting from a prior solution or when the number of degrees of freedom (Number of Variables - Number of Equations) is less than 2000. APOPT is also the only solver that handles Mixed Integer problems. Use option 0 to compare all available solvers.

Options for solvers can be set using configuration parameters such as MAX_ITER or else by creating an options file such as **ipopt.opt** or **apopt.opt**. Below is an example of setting options for the APOPT solver for a mixed integer nonlinear programming solution. The *File...End File* section creates a new **apopt.opt** file when APMonitor compiles the model file.

File apopt.opt minlp_maximum_iterations 10000 minlp_max_iter_with_int_sol 500 minlp_as_nlp 1 minlp_branch_method 3 minlp_gap_tol 1.0e-2 minlp_integer_tol 1.0e-2 minlp_integer_max 2.0e9 minlp_integer_leaves 1 minlp_print_level 1 nlp_maximum_iterations 500 objective_convergence_tolerance 1.0e-6 constraint_convergence_tolerance 1.0e-6 End File

In the GEKKO Optimization Suite the solver options are changed directly within Python.

m.solver_options = ['linear_solver ma57']

m.options.solver = 3

# multiple options as one list

m.solver_options = ['minlp_gap_tol 1.0e-2',\

'minlp_maximum_iterations 10000',\

'minlp_max_iter_with_int_sol 500']

m.options.solver = 1

#### APOPT Options

The APOPT solver has a number of options that are available for tuning the solver performance. Some of the available options and default values are listed below:

**minlp_maximum_iterations 10000**- maximum number of nlp solutions from the branch and bound method. A successful solution is returned if there is an integer solution upon reaching the maximum number of iterations. Otherwise, the solution is not considered to be successful and an error message is returned with the failed solution.**minlp_max_iter_with_int_sol 500**- maximum number of nlp solutions when a candidate integer solution is found**minlp_as_nlp 1**- solve minlp problem as a continuous nlp problem, ignoring integer constraints**minlp_branch_method 3**- 1=depth first (find integer solution faster), 2=breadth first, 3=lowest objective leaf, 4=highest objective leaf**minlp_gap_tol 1.0e-2**- gap is the spread between the lowest candidate leaf (obj_r=non-integer solution) and the best integer solution (obj_i). When the gap is below the minlp_gap_tol, the best integer solution is returned. The gap is defined as

$$gap=\frac{obj_i-obj_r}{\max \left( \frac{\lvert obj_i\rvert+\lvert obj_r\rvert}{2},1 \right)}$$

**minlp_integer_tol 1.0e-2**- amount that a candidate solution variable can deviate from an integer solution and still be considered an integer.**minlp_integer_max 2.0e9**- maximum value to be considered as an integer. Values over 2147483647 or below -2147483648 not stored correctly with an internal integer variable type because of the number of bits used to store an integer.**minlp_integer_leaves 1**- add additional integer leaves, 0=off, 1=integer leaves with inequality on branching, 2=integer leaves with equality constraint on branching.**minlp_print_level 1**- print level (0-10). Development version has additional advanced diagnostics.**nlp_maximum_iterations 500**- maximum number of iterations for each nlp sub-problem. Reducing the nlp maximum iterations can improve the solution speed because less computational time is spent on candidate solutions that may not converge**objective_convergence_tolerance 1.0e-6**- convergence tolerance for the objective function. Values lower than 1.0e-10 sometimes run into covergence problems because of numerical scaling and cannot achieve the requested accuracy.**constraint_convergence_tolerance 1.0e-6**- convergence tolerance for the constraints. A lower convergence tolerance typically adds only a couple additional iterations to the solution but the solution also does not change significantly.

Refer to Options for apopt.opt for additional details for the **APOPT** solver.

#### IPOPT Options

Refer to Options for ipopt.opt for additional details for the **IPOPT** solver. The IPOPT solver links to free and open-source (e.g. MUMPS) linear solvers or others that require a license (e.g. HSL MA57).

m = GEKKO(remote=True)

m.options.SOLVER=3 # Select IPOPT

m.solver_options = ['linear_solver ma57',\

'mu_strategy adaptive']

The selection of the linear solver in IPOPT affects the speed and sometimes the ability of IPOPT to solve the problem, especially for ill-conditioned problems. IPOPT is available through the default public server. However, IPOPT and certain linear solvers are not currently available (track issue on GitHub) for local solution without an internet connection.

m = GEKKO(remote=False)

m.options.SOLVER=3 # Select IPOPT

m.solver_options = ['linear_solver mumps',\

'mu_strategy adaptive']

When switching to a local solution, the solver selection will switch to one that is available (e.g. APOPT) or else use IPOPT but with a linear solver that can be distributed freely. This change of solvers sometimes leads to a different solution when using *remote=False*.

#### IPOPT Output

See the COIN-OR documentation for a detailed description of IPOPT print columns.

- iter: The current iteration count. The letter "r" is appended for iterations in restoration phase.
- objective: The unscaled objective value. During the restoration phase, this value remains the unscaled objective value for the original problem.
- inf_pr: The unscaled constraint violation. This quantity is the infinity-norm (max) of the (unscaled) constraints (gLâ‰¤g(x)â‰¤gU). During the restoration phase, this value remains the constraint violation of the original problem at the current point.
- inf_du: The scaled dual infeasibility at the current point. This quantity measures the infinity-norm (max) of the internal dual infeasibility, including inequality constraints reformulated using slack variables and problem scaling. During the restoration phase, this is the value of the dual infeasibility for the restoration phase problem.
- lg(mu): log10 of the value of the barrier parameter Î¼.
- ||d||: The infinity norm (max) of the primal step (for the original variables x and the internal slack variables s). During the restoration phase, this value includes the values of additional variables, p and n.
- lg(rg): log10 of the value of the regularization term for the Hessian of the Lagrangian in the augmented system. A dash ("-") indicates that no regularization was done.
- alpha_du: The stepsize for the dual variables.
- alpha_pr: The stepsize for the primal variables.