Reverse Convex Programming Research Articles

On measuring the inefficiency with the inner-product norm in data envelopment analysis

A technique for assessing the sensitivity of efficiency classifications in Data Envelopment Analysis (DEA) is presented. It extends the technique proposed by Charnes et al. (A. Charnes, J.J. Rousseau, J.H. Semple, Journal of Productivity Analysis 7 (1996) 5–18). An organization's input–output vector serves as the center for a cell within which the organization's classification remains unchanged under perturbations of the data. The maximal radius among such cells can be interpreted as a stability measure of the classification. Our approach adopts the inner-product norm for the radius, while the previous work does the polyhedral norms. For an efficient organization, the maximal-radius problem is a convex program. On the other hand, for an inefficient organization, it is reduced to a nonconvex program whose feasible region is the complement of a convex polyhedral set. We show that the latter nonconvex problem can be transformed into a linear reverse convex program. Our formulations and algorithms are valid not only in the CCR model but in its variants.

A Cutting Plane Algorithm for Linear Reverse Convex Programs

In this paper, global optimization of linear programs with an additional reverse convex constraint is considered. This type of problem arises in many applications such as engineering design, communications networks, and many management decision support systems with budget constraints and economies-of-scale. The main difficulty with this type of problem is the presence of the complicated reverse convex constraint, which destroys the convexity and possibly the connectivity of the feasible region, putting the problem in a class of difficult and mathematically intractable problems. We present a cutting plane method within the scope of a branch-and-bound scheme that efficiently partitions the polytope associated with the linear constraints and systematically fathoms these portions through the use of the bounds. An upper bound and a lower bound for the optimal value is found and improved at each iteration. The algorithm terminates when all the generated subdivisions have been fathomed.

Inner Approximation Method for a Reverse Convex Programming Problem

In this paper, we consider a reverse convex programming problem constrained by a convex set and a reverse convex set, which is defined by the complement of the interior of a compact convex set X. We propose an inner approximation method to solve the problem in the case where X is not necessarily a polytope. The algorithm utilizes an inner approximation of X by a sequence of polytopes to generate relaxed problems. It is shown that every accumulation point of the sequence of optimal solutions of the relaxed problems is an optimal solution of the original problem.

A New Approach to Optimization Under Monotonic Constraint

A new efficient branch and bound method is proposed for solving convex programs with an additional monotonic nonconvex constraint. Computational experiments demonstrated that this method is quite practical for solving rank k reverse convex programs with much higher values of k than previously considered in the literature and can be applied to a wider class of nonconvex problems.

Remarks on an algorithm for reverse convex programs

Remarks on an algorithm for reverse convex programs

How to Extend the Concept of Convexity Cuts to Derive Deeper Cutting Planes

A new type of cutting plane, termed a decomposition cut, is introduced that can be constructed under the same assumptions as the well-known convexity cut. Therefore it can be applied in algorithms (e.g. cutting plane, branch-and-cut) for various problems of global optimization, such as concave minimization, bilinear programming, reverse-convex programming, and integer programming. In computational tests with cutting plane algorithms for concave minimization, decomposition cuts were shown to be superior to convexity cuts.

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.

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 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.

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/.

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.

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

Comments on a reverse convex programming algorithm

Comments on a reverse convex programming algorithm

Construction of test problems for a class of reverse convex programs

A method of constructing test problems with known global solution for a class of reverse convex programs or linear programs with an additional reverse convex constraint is presented. The initial polyhedron is assumed to be a hypercube. The method then systematically generates cuts that slice the cube in such a way that a prespecified global solution on its edge remains intact. The proposed method does not require the solution of linear programs or systems of linear equations as is often required by existing techniques.

A GLOBAL OPTIMIZATION APPROACH TO RATIONALLY CONSTRAINED RATIONAL PROGRAMMING

Abstract The rationally constrained rational programming (RCR) problem is shown, for the first time, to be equivalent to the quadratically constrained quadratic programming problem with convex objective function and constraints that are all convex except for one that is concave and separable. This equivalence is then used in developing a novel implementation of the Generalized Benders Decomposition (GBDA) which, unlike all earlier implementations, is guaranteed to identify the global optimum of the RCRP problem. It is also shown, that the critical step in the proposed GBDA implementation is the solution of the master problem which is a quadratically constrained, separable, reverse convex programming problem that must be solved globally. Algorithmic approaches to the solution of such problems are discussed and illustrative examples are presented.

On nonconvex optimization problems with separated nonconvex variables

A mathematical programming problem is said to have separated nonconvex variables when the variables can be divided into two groups: x=(x1,...,xn) and y=( y1,...,yn), such that the objective function and any constraint function is a sum of a convex function of (x, y) jointly and a nonconvex function of x alone. A method is proposed for solving a class of such problems which includes Lipschitz optimization, reverse convex programming problems and also more general nonconvex optimization problems.

On the use of cuts in reverse convex programs

A cutting plane method, using the idea of Tuy cuts, has been suggested in earlier papers as a possible means of solving reverse convex programs. However, the method is fraught with theoretical and numerical difficulties. Stringent sufficient conditions for convergence inn dimensions are given. However, examples of nonconvergence are given and reasons for this nonconvergence are developed. A result of the discussion is a convergent algorithm which combines the idea of the cutting plane method with vertex enumeration procedures in order to numerically improve upon the edge search procedure of Hillestad.

A level set algorithm for a class of reverse convex programs

A new algorithm is presented for minimizing a linear function subject to a set of linear inequalities and one additional reverse convex constraint. The algorithm utilizes a conical partition of the convex polytope in conjuction with its facets in order to remain on the level surface of the reverse convex constraint. The algorithm does not need to solve linear programs on a set of cones which converges to a line segment.

On solving general reverse convex programming problems by a sequence of linear programs and line searches

Many multiextremal global optimization problems can be formulated as problems of minimizing a linear function over the intersection of a convex set with the complement of a convex set (so-called canonical d.c. programs or general reverse convex programming problems). In this paper it is shown that these general reverse convex programming problems can be solved by a sequence of linear programs and univariate convex minimization problems (line searches).

Deterministic methods in constrained global optimization: Some recent advances and new fields of application

Recent developments in deterministic global optimization methods have considerably enlarged the fields of optimization where those methods can be successfully applied. It is the purpose of the present article to give a brief survey of both some of the most promising methods and new fields of application. The methods considered comprise branch and bound and outer approximation as well as combinations of branch and bound with outer approximation. The fields of applications to be discussed include concave minimization, reverse convex programming, d.c. programming, Lipschitzian optimization, systems of equations, and (or) inequalities and global integer programming.