J.-Ph. Vial

A polynomial method of approximate centers for linear programming

We present a path-following algorithm for the linear programming problem with a surprisingly simple and elegant proof of its polynomial behaviour. This is done both for the problem in standard form and for its dual problem. We also discuss some implementation strategies.

Solving the p-Median Problem with a Semi-Lagrangian Relaxation

Lagrangian relaxation is commonly used in combinatorial optimization to generate lower bounds for a minimization problem. We study a modified Lagrangian relaxation which generates an optimal integer solution. We call it semi-Lagrangian relaxation and illustrate its practical value by solving large-scale instances of the p-median problem.

Confidence level solutions for stochastic programming

We propose an alternative approach to stochastic programming based on Monte-Carlo sampling and stochastic gradient optimization. The procedure is by essence probabilistic and the computed solution is a random variable. We propose a solution concept in which the probability that the random algorithm produces a solution with an expected objective value departing from the optimal one by more than @e is small enough. We derive complexity bounds on the number of iterations of this process. We show that by repeating the basic process on independent samples, one can significantly reduce the number of iterations.

Primal-dual target-following algorithms for linear programming

In this paper, we propose a method for linear programming with the property that, starting from an initial non-central point, it generates iterates that simultaneously get closer to optimality and closer to centrality. The iterates follow paths that in the limit are tangential to the central path. Together with the convergence analysis, we provide a general framework which enables us to analyze various primal-dual algorithms in the literature in a short and uniform way.

Homogeneous Analytic Center Cutting Plane Methods for Convex Problems and Variational Inequalities

In this paper we consider a new analytic center cutting plane method in an extended space. We prove the efficiency estimates for the general scheme and show that these results can be used in the analysis of a feasibility problem, the variational inequality problem, and the problem of constrained minimization. Our analysis is valid even for problems whose solution belongs to the boundary of the domain.

A practical implementation of stochastic programming: an application to the evaluation of option contracts in supply chains

Stochastic programming is a powerful analytical method in order to solve sequential decision-making problems under uncertainty. We describe an approach to build such stochastic linear programming models. We show that algebraic modeling languages make it possible for non-specialist users to formulate complex problems and have solved them by powerful commercial solvers. We illustrate our point in the case of option contracts in supply chain management and propose a numerical analysis of performance. We propose easy-to-implement discretization procedures of the stochastic process in order to limit the size of the event tree in a multi-period environment.

Using central prices in the decomposition of linear programs

Abstract This paper deals with a decomposition technique for linear programs which proposes a new treatment of the master program in the classical Dantzig-Wolfe algorithm. This approach exploits (a) the relationship between the master program and the minimization of a convex piecewise linear function and (b) a recently proposed cutting plane algorithm for convex nondifferentiable optimization. The new algorithm seeks to achieve ‘deep’ cuts by generating them at, or near to, the analytic centre of a set of localization containing the solution. Therefore each iteration entails a sizeable decrease of the set of localization and the overall algorithm shows fast convergence. This algorithm has been used to solve reputedly difficult convex nondifferentiable optimization problems. Its implementation, as a decomposition algorithm for large-scale structured linear program, seems to be a promising alternative to the standard Dantzig-Wolfe approach which suffers from very slow convergence in many practical problems. The new approach differs from the Dantzig-Wolfe algorithm through the fact that it does not use prices corresponding to an optimal combination of the past proposals of the subproblem, but rather ‘central’ prices (i.e. analytic centres) which balance them all. This property seems to explain the good behaviour of the new algorithm.

An extension of Karmarkar`s algorithm for solving a system of linear homogeneousequations on the simplex

We present an extension of Karmarkars algorithm for solving a system of linear homogeneous equations on the simplex. It is shown that in at most O(nL) steps, the algorithm produces a feasible point or proves that the problem has no solution. The complexity is O(n2m2L) arithmetic operations. The algorithm is endowed with two new powerful stopping criteria.

Interior-Point Methodology for Linear Programming: Duality, Sensitivity Analysis and Computational Aspects

In this paper we use the interior point methodology to cover the main issues in linear programming-duality theory, parametric and sensitivity analysis, and algorithmic and computational aspects. The aim is to provide a global view on the subject matter.

A Complexity Reduction for the Long-Step Path-Following Algorithm for Linear Programming

A modification of previously published long-step path-following algorithms for the solution of the linear programming problem is presented. This modification uses the simple Goldstein–Armijo rule. A