Reverse Convex Research Articles

Lagrangian duality theorems for reverse convex infimization

For the reverse convex infimization problem (P) inf f(X\\G)where G convex subset of a locally convex space X and is a function, we.carry ouexplicitly one of the main ideas of our previous paper [11], namely, we obtain some new Lagrangian duality theorems for (P) from the surrogate duality theorems and of our recen paper [16].

Technical efficiency and distance to a reverse convex set

The problem of finding the distance to a reverse (or complement of a) convex subset in a normed vector space is considered. This nonconvex and, in general, nonsmooth optimization problem arises in quantitative economics in the theory of measuring the technical efficiency of production units. In this context, applying a suitable duality theorem similar to the Nirenberg's one known for the distance to a convex subset, the problem reduces to a finite number of independent linear programming problems.

Duality in quasi-convex supremization and reverse convex infimization via abstract convex analysis,and applications to approximation **

We give duality theorems and dual characterizations of optimal solutions for abstract quasi-convex supremization problems and infimization problems with abstract reverse convex constraint sets. Our main tools are dualities between families of subsets, conjugations of type Lau associated to them, and subdifferentials with respect to conjugations of type Lau. These tools permit us to give explicitly the relation between.the constraint sets, and the relation between the objective functions, of the primal problem and the dual problem. As applications, we obtain duality theorems for quasi-convex supremization and reverse convex infimization in locally convex spaces and, in particular, for worst and best approximation in normed linear spaces.

A Deterministic Approach to Linear Programs with Several Additional Multiplicative Constraints

We consider a global optimization problem of minimizing a linear function subject to p linear multiplicative constraints as well as ordinary linear constraints. We show that this problem can reduce to a 2p-dimensional reverse convex program, and present an algorithm for solving the resulting problem. Our algorithm converges to a globally optimal solution and yields an e-approximate solution in finite time for any e > 0. We also report some results of computational experiment.

Duality in Reverse Convex Optimization

A duality theorem for the general problem of minimizing an extended real-valued convex function on a locally convex linear space under a reverse convex constraint is considered. In the particular case of the distance to a reverse convex subset in a normed linear space, we recover as a corollary a duality theorem due to C. Franchetti and I. Singer [Boll. Un. Mat. Ital. B (5), 17 (1980), pp. 33--43] similar to the one known for the distance to a convex subset. The general theorem also contains the duality principle of Toland--Singer for D. C. optimization.

Construction of Test Problems for Concave Minimization Under Linear and Nonlinear Constraints

This paper presents a method for constructing test problems with known global solutions for concave minimization under linear constraints with an additional convex constraint and for reverse convex programs with an additional convex constraint. The importance of such a construction can be realized from the fact that the well known d.c. programming problem can be formulated in this form. Then, the method is further extended to generate test problems with more than one convex constraint, tight or untight at the global solution.

A Formula on the Conjugate of the Max of a Convex Function and a Concave Function

Given two extended-real-valued convex functionsg,h:X→R̄, withh∈Γ(X), we express the Legendre-Fenchel conjugate (max(g,−h))* in terms ofg* andh* and derive new duality results in reverse convex optimization.

Testing the Re - strategy for a Reverse Convex Problem

This paper is devoted to solving a reverse-convex problem. The approach presented here is based on Global Optimality Conditions. We propose a general conception of a Global Search Algorithm and develop each part of it. The results of numerical experiments with the dimension up to 400 are also given.

A Finite Algorithm for Globally Optimizing a Class of Rank-Two Reverse Convex Programs

In this paper, we propose an algorithm for solving a linear program with an additional rank-two reverse convex constraint. Unlike the existing methods which generate approximately optimal solutions, the algorithm provides a rigorous optimal solution to this nonconvex problem by a finite number of dual pivot operations. Computational results indicate that the algorithm is practical and can solve fairly large scale problems.

Lagrange Duality and Partitioning Techniques in Nonconvex Global Optimization

It is shown that, for very general classes of nonconvex global optimization problems, the duality gap obtained by solving a corresponding Lagrangian dual in reduced to zero in the limit when combined with suitably refined partitioning of the feasible set. A similar result holds for partly convex problems where exhaustive partitioning is applied only in the space of nonconvex variables. Applications include branch-and-bound approaches for linearly constrained problems where convex envelopes can be computed, certain generalized bilinear problems, linearly constrained optimization of the sum of ratios of affine functions, and concave minimization under reverse convex constraints.

A Parallel Algorithm for Linear Programs with an Additional Reverse Convex Constraint

A parallel method for globally minimizing a linear program with an additional reverse convex constraint is proposed which combines the outer approximation technique and the cutting plane method. Basicallyp(≤n) processors are used for a problem withnvariables and a globally optimal solution is found effectively in a finite number of steps. Computational results are presented for test problems with a number of variables up to 80 and 63 linear constraints (plus nonnegativity constraints). These results were obtained on a distributed-memory MIMD parallel computer, DELTA, by running both serial and parallel algorithms with double precision. Also, based on 40 randomly generated problems of the same size, with 16 variables and 32 linear constraints (plusx≥ 0), the numerical results from different number processors are reported, including the serial algorithm's.

A parametric approach for maximum flow problems with an additional reverse convex constraint

In this paper, we consider maximum integral flow problems with an additional reverse convex constraint involving one or two nonlinear variables. Based on a parametric approach, we propose a polynomial-time algorithm for computing an integral flow globally optimal to the problem with a single nonlinear variable. We extend this idea and solve the problem with two nonlinear variables. The algorithm solves a sequence of ordinary minimum cost flow problems by using a conventional method and yields a globally optimal solution in pseudo-polynomial time.

Lagrangian duality of concave minimization subject to linear constraints and an additional facial reverse convex constraint

This paper is concerned with the global optimization problem of minimizing a concave function subject to linear constraints and an additional facial reverse convex constraint. Here, the feasible set is the union of some faces of the polyhedron determined by the linear constraints. Several well-known mathematical problems can be written or transformed into the form considered. The paper addresses the Lagrangian duality of the problem. It is shown that, under slight assumptions, the duality gap can be closed with a finite dual multiplier. Finite methods based on solving concave minimization problems are also proposed. We deal with the advantages accrued when outer approximation, cutting plane, or branch-and-bound methods are used for solving these subproblems.

Computational experience using an edge search algorithm for linear reverse convex programs

This paper presents computational experience with a rather straight forward implementation of an edge search algorithm for obtaining the globally optimal solution for linear programs with an additional reverse convex constraint. The paper's purpose is to provide a collection of problems, with known optimal solutions, and performance information for an edge search implementation so that researchers may have some benchmarks with which to compare new methods for reverse convex programs or concave minimization problems. There appears to be nothing in the literature that provides computational experience with a basic edge search procedure. The edge search implementation uses a depth first strategy. As such, this paper's implementation of the edge search algorithm is a modification of Hillestad's algorithm [11]. A variety of test problems is generated by using a modification of the method of Sung and Rosen [20], as well as a new method that is presented in this paper. Test problems presented may be obtained at ftp://newton.ee.ucla.edu/nonconvex/pub/.

Canonical DC programming problem: Outer approximation methods revisited

For solving the canonical DC programming problem (i.e. a convex program with an additional reverse convex constraint) several cutting plane procedures have been developed in the past few years which suffer from serious defects regarding efficiency or convergence. We discuss a general conceptual framework of outer approximation unifying various conjunctive cutting plane approaches and on this basis propose modifications to overcome these defects.

Deletion-by-infeasibility rule for DC-constrained global optimization

In Ref. 1, a general class of branch-and-bound methods was proposed by Horst for solving global optimization problems. One of the main contributions of Ref. 1 was the opportunity of handling partition elements whose feasibility is not known. Deletion-by-infeasibility rules were presented for problems where the feasible set is convex, is defined by finitely many convex and reverse convex constraints, or is defined by Lipschitzian inequalities. In this note, we propose a new deletion-by-infeasibility rule for problems whose feasible set is defined by functions representable as differences of convex functions.

Convex programs with an additional constraint on the product of several convex functions

In this paper, we consider a special type of global optimization problem in which a convex function is minimized on X ν Y, where X is a convex set and Y is a nonconvex set represented in terms of a product of p convex functions. It is shown that this problem can be reduced to a p-dimensional reverse convex program. Two outer approximation algorithms are proposed for solving the resulting problem. Computational experiments indicate that these algorithms can generate ϵ-optimal solutions of fairly large scale linear programs with a constraint on the product of up to 4 linear functions.

Comments on a reverse convex programming algorithm

Comments on a reverse convex programming algorithm

A generalization of the construction of test problems for nonconvex optimization

In this paper we adopt and generalize the basic idea of the method presented in [3] and [4] to construct test problems that involve arbitrary, not necessarily quadratic, concave functions, for both Concave Minimization and Reverse Convex Programs

Linear programs with an additional rank two reverse convex constraint

An efficient algorithm is developed for solving linear programs with an additional reverse convex constraint having a special structure. Computational results are presented and discussed.