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GAMSCplex is a GAMS solver that allows users to combine the high level modeling capabilities of GAMS with the power of Cplex optimizers. Cplex optimizers are designed to solve large, difficult problems quickly and with minimal user intervention. Access is provided (subject to proper licensing) to Cplex solution algorithms for linear, quadratically constrained and mixed integer programming problems. While numerous solving options are available, GAMS/Cplex automatically calculates and sets most options at the best values for specific problems.

Cplex solves LP problems using several alternative algorithms. The majority of LP problems solve best using Cplex's state of the art dual simplex algorithm. Certain types of problems benefit from using the primal simplex algorithm, the network optimizer, the barrier algorithm, or the sifting algorithm. The concurrent option will allow solving with different algorithms in parallel. The solution is returned by the first to finish.

Solving linear programming problems is memory intensive. Even though Cplex manages memory very efficiently, insufficient physical memory is one of the most common problems when running large LPs. When memory is limited, Cplex will automatically make adjustments which may negatively impact performance. If you are working with large models, study the section entitled Physical Memory Limitations carefully.

Cplex is designed to solve the majority of LP problems using default option settings. These settings usually provide the best overall problem optimization speed and reliability. However, there are occasionally reasons for changing option settings to improve performance, avoid numerical difficulties, control optimization run duration, or control output options.

Some problems solve faster with the primal simplex algorithm rather than the default dual simplex algorithm. Very few problems exhibit poor numerical performance in both the primal and the dual. Therefore, consider trying primal simplex if numerical problems occur while using dual simplex.

The barrier algorithm is an alternative to the simplex method for solving linear programs. It employs a primal-dual logarithmic barrier algorithm which generates a sequence of strictly positive primal and dual solutions. Specifying the barrier algorithm may be advantageous for large, sparse problems.

Cplex provides a sifting algorithm which can be effective on problems with many more variables than equations. Sifting solves a sequence of LP subproblems where the results from one subproblem are used to select columns from the original model for inclusion in the next subproblem.

GAMS/Cplex also provides access to the Cplex Conflict Refiner previously known as IIS (Irreducibly Inconsistent Set). The conflict refinder takes an infeasible program and produces an set of conflicting constraints. Such a set consists of constraints and variable bounds which is infeasible but becomes feasible if any one member of the set is dropped. GAMS/Cplex reports the conflict in terms of GAMS equation and variable names and includes the conflict report as part of the normal solution listing.

QP models are a special case that can be reformulated to have a quadratic objective function and only linear constraints. Those are automatically reformulated from GAMS QCP models. When such problems are convex, Cplex normally solves them efficiently in polynomial time. Nonconvex QPs, however, are known to be quite hard. Cplex applies various approaches to those problems, such approaches as barrier algorithms or branch and bound algorithms. Notably, in the branch and bound approach, there is no theoretical guarantee about the complexity of such a problem. Consequently, solution of such a problem (that is, a nonconvex QP) can take many orders of magnitude longer than the solution of a convex QP of comparable dimensions. Therefore, the default is to reject such model. The parameter OptimalityTarget allows to change this behavior.

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

For problems with integer variables, Cplex uses a branch and cut algorithm which solves a series of LP, subproblems. 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.

Cplex can also solve problems of GAMS model type MIQCP. As in the continuous case, if the base model is a QP the Simplex methods can be used and duals will be available at the solution. If the base model is a QCP, only the Barrier method can be used for the nodes and only primal values will be available at the solution.

The Conflict Refiner identifies the causes of infeasibility by means of inconsistent set of constraints. 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/Cplex is known as FeasOpt (for Feasible Optimization) and turned on by parameter FeasOpt in a CPLEX option file. More details can be found in the section entitled Using the Feasibility Relaxation.

This section introduces the solution pool for storing multiple solutions to a mixed integer programming problem (MIP and MIQCP). The chapter also explains techniques for generating and managing those solutions.

The solution pool stores multiple solutions to a mixed integer programming (MIP and MIQCP) model. With this feature, you can direct the algorithm to generate multiple solutions in addition to the optimal solution. For example, some constraints may be difficult to formulate efficiently as linear expressions, or the objective may be difficult to quantify exactly. In such cases, obtaining multiple solutions will help you choose one which best fits all your criteria,including the criteria that could not be expressed easily in a conventional MIP or MIQCP model. For example,

There are two ways to fill the solution pool associated with a model: You can accumulate successive incumbents or generate alternative solutions by populating the solution pool. The method is selected with the parameter SolnPoolPop:

The option SolnPoolReplace designates the strategy for replacing a solution in the solution pool when the solution pool has reached its capacity. The value 0 replaces solutions according to a first-in, first-out policy. The value 1 keeps the solutions with the best objective values. The value 2 replaces solutions in order to build a set of diverse solutions.

The replacement strategy applies only to the subset of solutions created in the current call of populate. Solutions already in the pool are not affected by the replacement strategy. They will not be replaced, even if they satisfy the criterion of the replacement strategy. So with every repeated call of the populate procedure the solution pool will be extended by the newly found solution. After the GAMS program specified in SolnPoolPopRepeat determined to continue the search for alternative solutions, the file specified by option SolnPoolPopDel option is read in. The solution numbers present in this file will be delete from the solution pool before the populate routine is called again. The file is automatically deleted by the GAMS/Cplex link after processing.

There may be an infinite number of possible values for a continuous variable, and it is not practical to enumerate all of them on a finite-precision computer. Therefore, populate gives only one solution for each set of binary and integer variables, even though there may exist several solutions that have the same values for all binary and integer variables but different values for continuous variables.

Cplex uses numerical methods of finite-precision arithmetic. Consequently, the feasibility of a solution depends on the value given to tolerances. Two parameters define the tolerances that assess the feasibility of a solution:

A solution may be considered feasible for one pair of values for these two parameters, and infeasible for a different pair. This phenomenon is especially noticeable in models with numeric difficulties, for example, in models with BigM coefficients.

Since the definition of a feasible solution is subject to tolerances, the total number of solutions to a model may vary, depending on the approach used to enumerate solutions, and on precisely which tolerances are used. In most models, this tolerance issue is not problematic. But, in the presence of numeric difficulties, Cplex may create solutions that are slightly infeasible or integer infeasible, and therefore create more solutions than expected.

If you want to filter solutions based on their difference as compared to a reference solution, use a diversity filter. This filter is practical for most purposes. However, if you require finer control of which solutions to keep and which to eliminate, use the incumbent filter.

A diversity filter allows you to generate solutions that are similar to (or different from) a set of reference values that you specify for a set of binary variables using dot option DivFlt and lower and upper bounds DivFltLo and DivFltUp. In particular, you can use a diversity filter to generate more solutions that are similar to an existing solution or to an existing partial solution. If you need more than one diversity filter, for example, to generate solutions that share the characteristics of several different solutions, additional filters can be specified through a Cplex Filter File using parameter ReadFLT. Details can be found in the example model solnpool in the GAMS model library.

If you need to enforce more complex constraints on solutions (e.g. if you need to enforce nonlinear constraints), you can use the incumbent filtering. The incumbent checking routine is part of the GAMS BCH Facility. It will accept or reject incumbents independent of a solution pool. During the populate or regular optimize procedure, the incumbent checking routine specified by the parameter UserIncbCall is called each time a new solution is found, even if the new solution does not improve the objective value of the incumbent. The incumbent filter allows your application to accept or reject the new solution based on your own criteria. If the GAMS program specified by UserIncbCall terminates normally, the solution is rejected. If this program returns with a compilation or execution error, the incumbent is accepted.

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