A new version of the Improved Primal Simplex for degenerate linear programs

Abstract

The Improved Primal Simplex (IPS) algorithm [Elhallaoui I, Metrane A, Desaulniers G, Soumis F. An Improved Primal Simplex algorithm for degenerate linear programs. SIAM Journal of Optimization, submitted for publication] is a dynamic constraint reduction method particularly effective on degenerate linear programs. It is able to achieve a reduction in CPU time of over a factor of three on some problems compared to the commercial implementation of the simplex method CPLEX. We present a number of further improvements and effective parameter choices for IPS. On certain types of degenerate problems, our improvements yield CPU times lower than those of CPLEX by a factor of 12.

Introduction

We consider the solution of linear programs in standard form

$$\underset{x\in {\mathbb{R}}^n}{\mathrm{minimize}}{c}^{T}x\text{subject to}\mathit{Ax}=b,x⩾0\text{,}$$

where $c\in {\mathbb{R}}^n$ is the cost, $A$ is the $m\times n$ constraint matrix and $b\in {\mathbb{R}}^m$ is the right-hand side. We are particularly interested in the so-called degenerate problems, on which the simplex algorithm [3] is likely to encounter degenerate pivots and possibly cycle.

In the presence of degeneracy, a common strategy to ensure convergence consists in imposing specific rules on the choice of a new basis, e.g., [1], [10], [15]. A different approach, proposed in [2] and developed in [16], [14], perturbs the right-hand side associated to degenerate variables. Although both strategies alleviate cycling, they do not improve the performance of the simplex algorithm on degenerate problems.

On the other hand, Pan uses a reduced basis; one with a smaller number of variables and constraints [13]. His method starts with a feasible initial guess and identifies its $p$ nonzero components. The $m-p$ constraints corresponding to the zero variables are removed, to leave a $p\times p$ nondegenerate basis. To preserve feasibility, variables that cannot become basic without violating one of the $m-p$ eliminated constraints must be temporarily removed from the problem. Such variables are said to be incompatible—a definition which will be generalized in Section 2.1. The resulting reduced problem is solved and the reduced costs are computed by means of its dual variables. By convention, dual variables of (LP) corresponding to the $m-p$ eliminated constraints are set to zero. At the next iteration, if an incompatible variable is to become basic, one of the eliminated constraints must be added to the reduced problem. When compared to his own primal simplex, Pan reports gains of a factor of 4–5.

Omer [12] improves on the method of [13] by allowing further reductions while the reduced problem is being solved. This dynamic reduction method appears promising in terms of number of iterations. However, its implementation does not lend itself to comparisons with commercial implementations of the simplex algorithm. It is interesting to note that most problems tested in [12], [13] have vanishing optimal dual variables and this characteristic gives advantage to their algorithms.

In the same vein, [5] developed a dynamic constraint aggregation method, DCA, for partitioning problems. The method groups constraints that become identical with respect to nonzero basic variables, a stage referred to as aggregation. In the reduction phase, a single row per constraint group remains in the reduced problem. Once the reduced problem is solved, dual variables are recovered by disaggregating which corresponds to solving a shortest-path problem. The dual variables then allow to compute the reduced costs of incompatible variables. As in [12], DCA makes provision for reaggregation, which allows to further reduce the reduced problem if the latter becomes too degenerate.

The multi-phase dynamic constraint aggregation, MDCA [6], further improves on the DCA. The MDCA examines incompatible variables about to be added to the reduced problem in order of their number of incompatibilities. First, variables with a single incompatibility are considered. If none of them is selected to become basic, attention turns to variables with two incompatibilities. If again none of those is selected, all incompatible variables are added to the reduced problem. On a set of large bus driver scheduling problems in a column-generation framework, MDCA reduces the solution time by a factor of more than five over the classical column-generation method.

The Improved Primal Simplex IPS [8] combines ideas from [13] and DCA. Instead of disaggregating the dual variables to compute the reduced costs, the incompatible variables that are to become basic are obtained from the solution of a complementary problem. The authors of [8] show that when the reduced problem is not degenerate, the objective function decrease at each iteration. In practice, reducing the problem until all degeneracy disappears is computationally costly. Partial reduction, however, yields good results.

In this paper, we provide a number of strategies and parameter values that speed up IPS. We pay attention to the number of variables in the reduced problem and hence, on the number of variables selected by the complementary problem. We also discuss how the former should be solved. To some extent, we optimize the number of further reductions of the reduced problem and propose a procedure that only adds independent rows to the reduced problem. Compared to the standard simplex method, we obtain a reduction of CPU time of a factor of 12 on some instances of fleet assignment (FA) problems.

The rest of this paper is organized as follows. Section 2 summarizes the results of [13], [8]. In Section 3 we compare different methods to find feasible solution to the complementary problem and to solve it. In Section 4, we present refinements of IPS. Numerical experimentations on realistic instances is reported in Section 5. Finally, we discuss our results and conclude in Section 6.

All tests in this paper are performed on a 2.8 GHz PC running Linux and all comparisons are made with CPLEX 9.1.3.

If $x\in {\mathbb{R}}^n$ and $I\subseteq \{1,\dots ,n\}$ is an index set, we denote $x_I$ the subvector of $x$ indexed by $I$. Similarly, if $A$ is an $m\times n$ matrix, we denote $A_I$ the $m\times \left|I\right|$ matrix whose columns are index by $I$. If $J=\{1,\dots ,n\}⧹I$, we allow ourselves to write $x=(x_I,x_J)$ even though the indices in $I$ and $J$ may not appear in order. In the same way, we denote with superior indices the subset of rows associated with the variables in the index set.

The vector of all ones with dimension dictated by the context is denoted $e$ and we note $A^{-T}$ the inverse of the transpose of $A$.