New variants of bundle methods

In this paper we describe a number of new variants of bundle methods for nonsmooth unconstrained and constrained convex optimization, convex—concave games and variational inequalities. We outline the ideas underlying these methods and present rate-of-convergence estimates.Corresponding author.     

Computational study of a column generation algorithm for bin packing and cutting stock problems

This paper reports on our attempt to design an efficient exact algorithm based on column generation for the cutting stock problem. The main focus of the research is to study the extend to which standard branch-and-bound enhancement features such as variable fixing, the tightening of the formulation with cutting planes, early branching, and rounding heuristics can be usefully incorporated in a branch-and-price algorithm. We review and compare lower bounds for the cutting stock problem. We propose a pseudo-polynomial heuristic. We discuss the implementation of the important features of the integer programming column generation algorithm and, in particular, the implementation of the branching scheme. Our computational results demonstrate the efficiency of the resulting algorithm for various classes of bin packing and cutting stock problems. Received October 18, 1996 / Revised version received May 14, 1998?Published online July 19, 1999     

Cutting planes and column generation techniques with the projective algorithm

The problem studied is that of solving linear programs defined recursively by column generation techniques or cutting plane techniques using, respectively, the primal projective method or the dual projective method.This research has been supported in part by FCAR of Quebec, Grant Nos. CE-130 and EQ-3078, by NSERC of Canada, Grant No. A4152, and by the Fonds National Suisse de la Recherche Scientifique, Grant No. 1.467.0.86.     

Entropic proximal decomposition methods for convex programs and variational inequalities

We consider convex optimization and variational inequality problems with a given separable structure. We propose a new decomposition method for these problems which combines the recent logarithmic-quadratic proximal theory introduced by the authors with a decomposition method given by Chen-Teboulle for convex problems with particular structure. The resulting method allows to produce for the first time provably convergent decomposition schemes based on C ^∞ Lagrangians for solving convex structured problems. Under the only assumption that the primal-dual problems have nonempty solution sets, global convergence of the primal-dual sequences produced by the algorithm is established. Received: October 6, 1999 / Accepted: February 2001?Published online September 17, 2001     

A geometric study of duality gaps, with applications

Lagrangian relaxation is often an efficient tool to solve (large-scale) optimization problems, even nonconvex. However it introduces a duality gap, which should be small for the method to be really efficient. Here we make a geometric study of the duality gap. Given a nonconvex problem, we formulate in a first part a convex problem having the same dual. This formulation involves a convexification in the product of the three spaces containing respectively the variables, the objective and the constraints. We apply our results to several relaxation schemes, especially one called “Lagrangean decomposition” in the combinatorial-optimization community, or “operator splitting” elsewhere. We also study a specific application, highly nonlinear: the unit-commitment problem. Received: June 1997 / Accepted: December 2000?Published online April 12, 2001     

Existence of optimal solutions and duality results under weak conditions

In this paper we consider an ordinary convex program with no qualification conditions (such as Slater’s condition) and for which the optimal set is neither required to be compact, nor to be equal to the sum of a compact set and a linear space. It is supposed only that the infimum α is finite. A very wide class of convex functions is exhibited for which the optimum is always attained and α is equal to the supremum of the ordinary dual program. Additional results concerning the existence of optimal solutions in the non convex case are also given. Received: February 1, 1999 / Accepted: December 15, 1999?Published online February 23, 2000     

An efficient algorithm for globally minimizing a quadratic function under convex quadratic constraints

In this paper we investigate two approaches to minimizing a quadratic form subject to the intersection of finitely many ellipsoids. The first approach is the d.c. (difference of convex functions) optimization algorithm (abbr. DCA) whose main tools are the proximal point algorithm and/or the projection subgradient method in convex minimization. The second is a branch-and-bound scheme using Lagrangian duality for bounding and ellipsoidal bisection in branching. The DCA was first introduced by Pham Dinh in 1986 for a general d.c. program and later developed by our various work is a local method but, from a good starting point, it provides often a global solution. This motivates us to combine the DCA and our branch and bound algorithm in order to obtain a good initial point for the DCA and to prove the globality of the DCA. In both approaches we attempt to use the ellipsoidal constrained quadratic programs as the main subproblems. The idea is based upon the fact that these programs can be efficiently solved by some available (polynomial and nonpolynomial time) algorithms, among them the DCA with restarting procedure recently proposed by Pham Dinh and Le Thi has been shown to be the most robust and fast for large-scale problems. Several numerical experiments with dimension up to 200 are given which show the effectiveness and the robustness of the DCA and the combined DCA-branch-and-bound algorithm. Received: April 22, 1999 / Accepted: November 30, 1999?Published online February 23, 2000     

Approximations in proximal bundle methods and decomposition of convex programs

A proximal bundle method is presented for minimizing a nonsmooth convex functionf. At each iteration, it requires only one approximate evaluation off and its -subgradient, and it finds a search direction via quadratic programming. When applied to the Lagrangian decomposition of convex programs, it allows for inexact solutions of decomposed subproblems; yet, increasing their required accuracy automatically, it asymptotically finds both the primal and dual solutions. It is an implementable approximate version of the proximal point algorithm. Some encouraging numerical experience is reported.The author thanks two anonymous referees for their valuable comments.Research supported by the State Committee for Scientific Research under Grant 8550502206.     

Models and algorithms for the 2-dimensional cell suppression problem in statistical disclosure control

Lagrangean dualization and subgradient optimization techniques are frequently used within the field of computational optimization for finding approximate solutions to large, structured optimization problems. The dual subgradient scheme does not automatically produce primal feasible solutions; there is an abundance of techniques for computing such solutions (via penalty functions, tangential approximation schemes, or the solution of auxiliary primal programs), all of which require a fair amount of computational effort. We consider a subgradient optimization scheme applied to a Lagrangean dual formulation of a convex program, and construct, at minor cost, an ergodic sequence of subproblem solutions which converges to the primal solution set. Numerical experiments performed on a traffic equilibrium assignment problem under road pricing show that the computation of the ergodic sequence results in a considerable improvement in the quality of the primal solutions obtained, compared to those generated in the basic subgradient scheme. Received February 11, 1997 / Revised version received June 19, 1998?Published online June 28, 1999     

Convergence and efficiency of subgradient methods for quasiconvex minimization

We study a general subgradient projection method for minimizing a quasiconvex objective subject to a convex set constraint in a Hilbert space. Our setting is very general: the objective is only upper semicontinuous on its domain, which need not be open, and various subdifferentials may be used. We extend previous results by proving convergence in objective values and to the generalized solution set for classical stepsizes t _k →0, ∑t _k =∞, and weak or strong convergence of the iterates to a solution for {t _k }∈ℓ₂∖ℓ₁ under mild regularity conditions. For bounded constraint sets and suitable stepsizes, the method finds ε-solutions with an efficiency estimate of O(ε^-2), thus being optimal in the sense of Nemirovskii. Received: October 4, 1998 / Accepted: July 24, 2000?Published online January 17, 2001     

A study of exponential neighborhoods for the Travelling Salesman Problem and for the Quadratic Assignment Problem

This paper deals with exponential neighborhoods for combinatorial optimization problems. Exponential neighborhoods are large sets of feasible solutions whose size grows exponentially with the input length. We are especially interested in exponential neighborhoods over which the TSP (respectively, the QAP) can be solved in polynomial time, and we investigate combinatorial and algorithmical questions related to such neighborhoods.?First, we perform a careful study of exponential neighborhoods for the TSP. We investigate neighborhoods that can be defined in a simple way via assignments, matchings in bipartite graphs, partial orders, trees and other combinatorial structures. We identify several properties of these combinatorial structures that lead to polynomial time optimization algorithms, and we also provide variants that slightly violate these properties and lead to NP-complete optimization problems. Whereas it is relatively easy to find exponential neighborhoods over which the TSP can be solved in polynomial time, the corresponding situation for the QAP looks pretty hopeless: Every exponential neighborhood that is considered in this paper provably leads to an NP-complete optimization problem for the QAP. Received: September 5, 1997 / Accepted: November 15, 1999?Published online February 23, 2000     

Concavity and efficient points of discrete distributions in probabilistic programming

We consider stochastic programming problems with probabilistic constraints involving integer-valued random variables. The concept of a p-efficient point of a probability distribution is used to derive various equivalent problem formulations. Next we introduce the concept of r-concave discrete probability distributions and analyse its relevance for problems under consideration. These notions are used to derive lower and upper bounds for the optimal value of probabilistically constrained stochastic programming problems with discrete random variables. The results are illustrated with numerical examples. Received: October 1998 / Accepted: June 2000?Published online October 18, 2000