The s-monotone index selection rules for pivot algorithms of linear programming

In this paper we introduce the concept of s-monotone index selection rule for linear programming problems. We show that several known anti-cycling pivot rules like the minimal index, Last-In–First-Out and the most-often-selected-variable pivot rules are s-monotone index selection rules. Furthermore, we show a possible way to define new s-monotone pivot rules. We prove that several known algorithms like the primal (dual) simplex, MBU-simplex algorithms and criss-cross algorithm with s-monotone pivot rules are finite methods. We implemented primal simplex and primal MBU-simplex algorithms, in MATLAB, using three s-monotone index selection rules, the minimal-index, the Last-In–First-Out (LIFO) and the Most-Often-Selected-Variable (MOSV) index selection rule. Numerical results demonstrate the viability of the above listed s-monotone index selection rules in the framework of pivot algorithms

Strongly polynomial primal monotonic build-up simplex algorithm for maximal flow problems

The maximum flow problem (MFP) is a fundamental model in operations research. The network simplex algorithm is one of the most efficient solution methods for MFP in practice. The theoretical properties of established pivot algorithms for MFP is less understood. Variants of the primal simplex and dual simplex methods for MFP have been proven strongly polynomial, but no similar result exists for other pivot algorithms like the monotonic build-up or the criss-cross simplex algorithm. The monotonic build-up simplex algorithm (MBUSA) starts with a feasible solution, and fixes the dual feasibility one variable a time, temporarily losing primal feasibility. In the case of maximum flow problems, pivots in one such iteration are all dual degenerate, bar the last one. Using a labelling technique to break these ties we show a variant that solves the maximum flow problem in 2|V||A|2 pivots

Copositivity and constrained fractional quadratic programs

Abstract We provide Completely Positive and Copositive Optimization formulations for the Constrained Fractional Quadratic Problem (CFQP) and Standard Fractional Quadratic Problem (StFQP). Based on these formulations, Semidefinite Programming (SDP) relaxations are derived for finding good lower bounds to these fractional programs, which can be used in a global optimization branch-and-bound approach. Applications of the CFQP and StFQP, related with the correction of infeasible linear systems and eigenvalue complementarity problems are also discussed

Un algorithme du simplexe primal amélioré pour des programmes linéaires dégénérés

Algorithme du simplexe -- Réduction du problème -- Dégénérescence -- A new version of the improved primal simplex for degenerate linear programs -- Background -- Solving the complementary problem -- New strategies and parameter values to speed up IPS -- Improved primal simplex method version 3 : cold start, generalization for bounded variables problems, new implementation -- Contributions -- Computational experiments -- A pricing criterion to identify non degenerate pivots -- Reduced problem -- Positive edge rule -- Applications