Gurobi 9.5.2

[Pablo Tatts]

TheNEOS Server offers the GurobiOptimizer for the solution of linear programming (LP) problems,global (GO) problems, mixed-integer linear programming (MILP)problems, and second-order conic programming (SOCP) problems. LPs andMILPs can be submitted to Gurobi on the NEOS server in AMPL, GAMS, LP,or MPS format. SOCPs can be submitted in AMPL, GAMS or NL format.

The Gurobi Optimizer is a state-of-the-art solver for mathematicalprogramming. The solvers in the Gurobi Optimizer were designed fromthe ground up to exploit modern architectures and multi-coreprocessors. For more information on Gurobi products and services, seethe Gurobi website.

The problem must be specified in a model file. A data file and commandsfiles may also be provided. If the commands file is specified, it mustcontain the AMPL solve command; however, it must not containthe model or data commands. The model and datafiles are renamed internally by NEOS.

The commands file may include option settings for the solver. To specifysolver options, add option gurobi_options 'OPTIONS';where OPTIONS is a list of one or more ofthe availablesolver options for AMPL.

The Gurobi suite of optimization products include state-of-the-art simplex and parallel barrier solvers for linear programming (LP) and quadratic programming (QP), parallel barrier solver for quadratically constrained programming (QCP), as well as parallel mixed-integer linear programming (MILP), mixed-integer quadratic programming (MIQP), mixed-integer quadratically constrained programming (MIQCP) and (mixed-integer) nonlinear programming (NLP) solvers.

The Gurobi MIP solver includes shared memory parallelism, capable of simultaneously exploiting any number of processors and cores per processor. The implementation is deterministic: two separate runs on the same model will produce identical solution paths.

While numerous solving options are available, Gurobi automatically calculates and sets most options at the best values for specific problems. All Gurobi options available through GAMS/Gurobi are summarized at the end of this chapter.

Gurobi can solve LP and convex QP problems using several alternative algorithms, while the only choice for solving convex QCP is the parallel barrier algorithm. The majority of LP problems solve best using Gurobi's state-of-the-art dual simplex algorithm, while most convex QP problems solve best using the parallel barrier algorithm. Certain types of LP problems benefit from using the parallel barrier or the primal simplex algorithms, while for some types of QP, the dual or primal simplex algorithm can be a better choice. If you are solving LP problems on a multi-core system, you should also consider using the concurrent optimizer. It runs different optimization algorithms on different cores, and returns when the first one finishes.

GAMS/Gurobi also provides access to the Gurobi infeasibility finder. The infeasibility finder takes an infeasible linear program and produces an irreducibly inconsistent set of constraints (IIS). An IIS is a set of constraints and variable bounds which is infeasible but becomes feasible if any one member of the set is dropped. GAMS/Gurobi reports the IIS in terms of GAMS equation and variable names and includes the IIS report as part of the normal solution listing. The infeasibility finder is activated by the option IIS. Another option for analyzing infeasible model the FeasOpt option which instructs GAMS/Gurobi to find a minimal feasible relaxation of an infeasible model. See section Feasible Relaxation for details.

GAMS/Gurobi supports sensitivity analysis (post-optimality analysis) for linear programs which allows one to find out more about an optimal solution for a problem. In particular, objective ranging and constraint ranging give information about how much an objective coefficient or a right-hand-side and variable bounds can change without changing the optimal basis. In other words, they give information about how sensitive the optimal basis is to a change in the objective function or the bounds and right-hand side. GAMS/Gurobi reports the sensitivity information as part of the normal solution listing. Sensitivity analysis is activated by the option Sensitivity.

The Gurobi presolve can sometimes diagnose a problem as being infeasible or unbounded. When this happens, GAMS/Gurobi can, in order to get better diagnostic information, rerun the problem with presolve turned off. The rerun without presolve is controlled by the option ReRun. In default mode only problems that are small (i.e. demo sized) will be rerun.

Gurobi can either presolve a model or start from an advanced basis or primal/dual solution pair. Often the solve from scratch of a presolved model outperforms a solve from an unpresolved model started from an advanced basis/solution. It is impossible to determine a priori if presolve or starting from a given advanced basis/solution without presolve will be faster. By default, GAMS/Gurobi will automatically use an advanced basis or solution from a previous solve statement. The GAMS BRatio option can be used to specify when not to use an advanced basis/solution. The GAMS/Gurobi option UseBasis can be used to ignore or force a basis/solution passed on by GAMS (it overrides BRatio). In case of multiple solves in a row and slow performance of the second and subsequent solves, the user is advised to set the GAMS BRatio option to 1.

The methods used to solve pure integer and mixed integer programming problems require dramatically more mathematical computation than those for similarly sized pure linear or quadratic programs. Many relatively small integer programming models take enormous amounts of time to solve.

For problems with discrete variables, Gurobi uses a branch and cut algorithm which solves a series of subproblems, LP subproblems for MILP, QP subproblems for MIQP, and QCP subproblems or LP outer approximation subproblems for MIQCP. Because a single mixed integer problem generates many subproblems, even small mixed integer problems can be very compute intensive and require significant amounts of physical memory. With option nonConvex Gurobi can also solve nonconvex (MI)QP and (MI)QCP problems using a spatial branch-and-bound method.

If you specify some or all values for the discrete variables together with GAMS/Gurobi option MipStart, Gurobi will check the validity of the values as an integer-feasible solution. If this process succeeds, the solution will be treated as an integer solution of the current problem.

Gurobi can solve (mixed-integer) nonlinear programs to global optimality either directly or by approximating the model by piecewise-linear functions and/or reformulating nonlinear constraints into supported linear and/or quadratic constraints (see also funcnonlinear.)

For Gurobi to accept a nonlinear constraint, it has to be in one of the forms listed below. Furthermore, GAMS/Gurobi can automatically reformulate a nonlinear constraint into the supported form by enabling nlreform.

The Infeasibility Finder identifies the causes of infeasibility by means of inconsistent set of constraints (IIS). However, you may want to go beyond diagnosis to perform automatic correction of your model and then proceed with delivering a solution. One approach for doing so is to build your model with explicit slack variables and other modeling constructs, so that an infeasible outcome is never a possibility. An automated approach offered in GAMS/Gurobi is known as FeasOpt (for Feasible Optimization) and turned on by parameter FeasOpt in a GAMS/Gurobi option file.

With the FeasOpt option GAMS/Gurobi accepts an infeasible model and selectively relaxes the bounds and constraints in a way that minimizes a weighted penalty function. In essence, the feasible relaxation tries to suggest the least change that would achieve feasibility. It returns an infeasible solution to GAMS and marks the relaxations of bounds and constraints with the INFES marker in the solution section of the listing file.

By default all equations are candidates for relaxation and weighted equally but none of the variables can be relaxed. This default behavior can be modified by assigning relaxation preferences to variable bounds and constraints. These preferences can be conveniently specified with the .feaspref option. The input value denotes the users willingness to relax a constraint or bound. The larger the preference, the more likely it will be that a given bound or constraint will be relaxed. More precisely, the reciprocal of the specified value is used to weight the relaxation of that constraint or bound. The user may specify a preference value less than or equal to 0 (zero), which denotes that the corresponding constraint or bound must not be relaxed. It is not necessary to specify a unique preference for each bound or range. In fact, it is conventional to use only the values 0 (zero) and 1 (one) except when your knowledge of the problem suggests assigning explicit preferences.

First we turn the feasible relaxtion on. Futhermore, we specify that all variables v(i,j) have preference of 1, except variables over set element i1, which have a preference of 2. The variable over set element i1 and j2 has preference 0. Note that preferences are assigned in a procedural fashion so that preferences assigned later overwrite previous preferences. The same syntax applies for assigning preferences to equations as demonstrated above. If you want to assign a preference to all variables or equations in a model, use the keywords variables or equations instead of the individual variable and equations names (e.g. variables.feaspref 1).

The parameter FeasOptMode allows different strategies in finding feasible relaxation in one or two phases. In its first phase, it attempts to minimize its relaxation of the infeasible model. That is, it attempts to find a feasible solution that requires minimal change. In its second phase, it finds an optimal solution (using the original objective) among those that require only as much relaxation as it found necessary in the first phase. Values of the parameter FeasOptMode indicate two aspects: (1) whether to stop in phase one or continue to phase two and (2) how to measure the relaxation (as a sum of required relaxations; as the number of constraints and bounds required to be relaxed; as a sum of the squares of required relaxations). Please check description of parameter FeasOptMode for details. Also check example models feasopt* in the GAMS Model library.

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