Linear programming with inequality constraints via entropic perturbation

H.-S Jacob Tsao,Shu‐Cherng Fang

doi:10.1155/s0161171296000257

Abstract

A dual convex programming approach to solving linear programs with inequality constraints through entropic perturbation is derived. The amount of perturbation required depends on the desired accuracy of the optimum. The dual program contains only non-positivity constraints. Anϵ-optimal solution to the linear program can be obtained effortlessly from the optimal solution of the dual program. Since cross-entropy minimization subject to linear inequality constraints is a special case of the perturbed linear program, the duality result becomes readily applicable. Many standard constrained optimization techniques can be specialized to solve the dual program. Such specializations, made possible by the simplicity of the constraints, significantly reduce the computational effort usually incurred by these methods. Immediate applications of the theory developed include an entropic path-following approach to solving linear semi-infinite programs with an infinite number of inequality constraints and the widely used entropy optimization models with linear inequality and/or equality constraints.

Highlights

Since Karmarkar's projective scaling algorithm was introduced in 1984 [1], various interior point methods [2,3] have been proposed to compete with the classical simplex method [4] for linear programs
The work was extended for linear programming problems in standard form [7] with a qua dratically convergent global algorithm, based on the curved search methods [8]
This paper further extends the approach to solve linear programming problems with inequality constraints directly without a conversion to the standard form

Summary

INTRODUCTION

Since Karmarkar's projective scaling algorithm was introduced in 1984 [1], various interior point methods [2,3] have been proposed to compete with the classical simplex method [4] for linear programs. Note that this dual program differs from the one obtained for standard-form linear programs in [7] only in the extra non-positivity requirements While it is usually the case and easy to see that, in the Lagrangian max-min denvatwn, a change of stgn m a primal constraint re,ults m a change of range of the corresponding dual variable, thts causal relationship i' not apparent in the geometric programming denvation. The theory can be extended for linear programs with both inequality and equality constraints in the following form: Program P': where c and x are n-dimensional column vectors, A1 is an m 1 x n (m1 ::; n) matrix, A2 is an m2 x n (m2 ::; n) matrix, b 1 is an m1-dimensional column vector, b2 is an mTdimensional column vector, and 0 is the n-dimensional zero column vector. Where 't =max{ 1/e, IMlnMI }, the optimal solution of Program P~ is an £-optimal solution of Program P'

CROSS-ENTROPY MINIMIZATION SUBJECT TO INEQUALITY CONSTRAINTS

CONCLUSION