A convex analysis approach for convex multiplicative programming

Global optimization problems involving the minimization of a product of convex functions on a convex set are addressed in this paper. Elements of convex analysis are used to obtain a suitable representation of the convex multiplicative problem in the outcome space, where its global solution is reduced to the solution of a sequence of quasiconcave minimizations on polytopes. Computational experiments illustrate the performance of the global optimization algorithm proposed.      

Duality for quasi-concave programs with explicit constraints

The idea of duality is now well established in the theory of concave programming. The basis of this duality is the concave conjugate transform. This has been exemplified in the development of generalised geometric programming. Much of the current research in duality theory is focused on relaxing the requirement of concavity. Here we develop a duality theory for mathematical programs with a quasi concave objective function and explicit quasi concave constraints. Generalisations of the concave conjugate transform are introduced which pair quasi concave functions as the concave conjugate transform does for concave functions. Optimality conditions are derived relating the primal quasi concave program to its dual. This duality theory was motivated by and has implications in certain problems of mathematical economics. An application to economics is given.     

An interior-point method for multifractional programs with convex constraints

We present an interior-point method for a family of multi-fractional programs with convex constraints. The programs under consideration consist of minimizing the maximum of a finite number of linear fractions over some convex set. First, we present a simple shortstep algorithm for solving such multifractional programs, and we show that, under suitable assumptions, the convergence of the short-step algorithm is weakly polynomial in a sense specified below. Then, we describe a practical implementation of the proposed method, and we report results of numerical experiments with this algorithm. These results suggest that the proposed method is a viable alternative to the standard Dinkelbach-type algorithms for solving multifractional programs.The authors would like to thank Professor A. S. Nemirovsky for stimulating discussions via electronic mail. We are grateful to two anonymous referees for comments and suggestions that improved our paper.     

A unifying approach to solve some classes of rank-three multiplicative and fractional programs involving linear functions

The aim of this paper is to propose a solution algorithm for solving a class of low-rank programs involving linear functions and having a polyhedral feasible region. In particular, the proposed solution method solves in an unifying approach some classes of rank-three multiplicative and fractional programs. The algorithm is based on the so called optimal level solutions method. Some optimality conditions are used to improve the performance of the proposed algorithm. Results of a computational test are provided.     

Branch-and-bound decomposition approach for solving quasiconvex-concave programs

A class of branch-and-bound methods is proposed for minimizing a quasiconvex-concave function subject to convex and quasiconvex-concave inequality constraints. Several important special cases where the subproblems involved by the bounding-and-branching operations can be solved quite effectively include certain d.c. programming problems, indefinite quadratic programming with one negative eigenvalue, affine multiplicative problems, and fractional multiplicative optimization.This research was accomplished while the second author was a Fellow of the Alexander von Humboldt Foundation at the University of Trier, Trier, Germany.     

A global optimization approach for solving the convex multiplicative programming problem

We consider a convex multiplicative programming problem of the form% MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9qq-f0-yqaqVeLsFr0-vr% 0-vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaGG7bGaam% OzamaaBaaaleaacaaIXaaabeaakiaacIcacaWG4bGaaiykaiabgwSi% xlaadAgadaWgaaWcbaGaaGOmaaqabaGccaGGOaGaamiEaiaacMcaca% GG6aGaamiEaiabgIGiolaadIfacaGG9baaaa!4A08!\[\{ f_1 (x) \cdot f_2 (x):x \in X\} \]where X is a compact convex set of ⁿ and f ₁, f ₂ are convex functions which have nonnegative values over X.Using two additional variables we transform this problem into a problem with a special structure in which the objective function depends only on two of the (n+2) variables. Following a decomposition concept in global optimization we then reduce this problem to a master problem of minimizing a quasi-concave function over a convex set in ² ₂. This master problem can be solved by an outer approximation method which requires performing a sequence of simplex tableau pivoting operations. The proposed algorithm is finite when the functions f _i, (i=1, 2) are affine-linear and X is a polytope and it is convergent for the general convex case.Partly supported by the Deutsche Forschungsgemeinschaft Project CONMIN.     

New strong duality results for convex programs with separable constraints

It is known that convex programming problems with separable inequality constraints do not have duality gaps. However, strong duality may fail for these programs because the dual programs may not attain their maximum. In this paper, we establish conditions characterizing strong duality for convex programs with separable constraints. We also obtain a sub-differential formula characterizing strong duality for convex programs with separable constraints whenever the primal problems attain their minimum. Examples are given to illustrate our results.     

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 reverse convex constraint

A method is presented for solving a class of global optimization problems of the form (P): minimizef(x), subject toxD,g(x)0, whereD is a closed convex subset ofR ⁿ andf,g are convex finite functionsR ⁿ . Under suitable stability hypotheses, it is shown that a feasible point is optimal if and only if 0=max{g(x):xD,f(x)f( )}. On the basis of this optimality criterion, the problem is reduced to a sequence of subproblemsQ _k ,k=1, 2, ..., each of which consists in maximizing the convex functiong(x) over some polyhedronS _k . The method is similar to the outer approximation method for maximizing a convex function over a compact convex set.     

Global minimization of a generalized linear multiplicative programming

This article presents a simplicial branch and bound algorithm for globally solving generalized linear multiplicative programming problem (GLMP). Since this problem does not seem to have been studied previously, the algorithm is apparently the first algorithm to be proposed for solving such problem. In this algorithm, a well known simplicial subdivision is used in the branching procedure and the bound estimation is performed by solving certain linear programs. Convergence of this algorithm is established, and some experiments are reported to show the feasibility of the proposed algorithm.     

Solving a class of multiplicative programming problems via c-programming

In this note we show that many classes of global optimization problems can be treated most satisfactorily by classical optimization theory and conventional algorithms. We focus on the class of problems involving the minimization of the product of several convex functions on a convex set which was studied recently by Kunoet al. [3]. It is shown that these problems are typical composite concave programming problems and thus can be handled elegantly by c-programming [4]–[8] and its techniques.     

A new global optimization approach for convex multiplicative programming

In this paper, by solving the relaxed quasiconcave programming problem in outcome space, a new global optimization algorithm for convex multiplicative programming is presented. Two kinds of techniques are employed to establish the algorithm. The first one is outer approximation technique which is applied to shrink relaxation area of quasiconcave programming problem and to compute appropriate feasible points and to raise the capacity of bounding. And the other one is branch and bound technique which is used to guarantee global optimization. Some numerical results are presented to demonstrate the effectiveness and feasibility of the proposed 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.The author would like to thank Prof. S. E. Jacobsen for his valuable remarks on initial drafts of this paper and the referees for their constructive suggestions.     

A primal-dual Newton-type algorithm for geometric programs with equality constraints

An augmented Lagrangian algorithm is used to find local solutions of geometric programming problems with equality constraints (GPE). The algorithm is Newton's method for unconstrained minimization. The complexity of the algorithm is defined by the number of multiplications and divisions. By analyzing the algorithm we obtain results about the influence of each parameter in the GPE problem on the complexity of an iteration. An attempt is made to estimate the number of iterations needed for convergence. In practice, certain hypotheses are tested, such as the influence of the penalty coefficient update formula, the distance of the starting point from the optimum, and the stopping criterion. For these tests, a random problem generator was constructed, many problems were run, and the results were analyzed by statistical methods.The authors are grateful to Dr. J. Moré, Argonne National Laboratory for his valuable comments.This research was partially funded by the Fund for the Advancement of Research at the Technion and by the Applied Mathematical Sciences Research Program (KC-04-02), Office of Energy Research, US Department of Energy, Contract No. W-31-109-Eng-38.     

关于一些凸规划问题的复杂性研究结果

目前,已发表了大量研究各类不同凸规划的低复杂度的障碍函数方法的文章. 利用自和谐理论,对不同的几类凸规划问题构造相应的对数障碍函数,通过两个引理证明这些凸规划问题相应的对数障碍函数都满足自和谐,根据Nesterov 和Nemirovsky的工作证明了所给问题的内点算法具有多项式复杂性.     

A nonisolated optimal solution for special reverse convex programming problems

In this paper, an efficient algorithm is proposed for globally solving special reverse convex programming problems with more than one reverse convex constraints. The proposed algorithm provides a nonisolated global optimal solution which is also stable under small perturbations of the constraints, and it turns out that such an optimal solution is adequately guaranteed to be feasible and to be close to the actual optimal solution. Convergence of the algorithm is shown and the numerical experiment is given to illustrate the feasibility of the presented algorithm.     

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.     

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.     

A global optimization algorithm for linear fractional and bilinear programs

In this paper a deterministic method is proposed for the global optimization of mathematical programs that involve the sum of linear fractional and/or bilinear terms. Linear and nonlinear convex estimator functions are developed for the linear fractional and bilinear terms. Conditions under which these functions are nonredundant are established. It is shown that additional estimators can be obtained through projections of the feasible region that can also be incorporated in a convex nonlinear underestimator problem for predicting lower bounds for the global optimum. The proposed algorithm consists of a spatial branch and bound search for which several branching rules are discussed. Illustrative examples and computational results are presented to demonstrate the efficiency of the proposed algorithm.     

Mathematical programs with a two-dimensional reverse convex constraint

We consider the problem min {f(x): x G, T(x) int D}, where f is a lower semicontinuous function, G a compact, nonempty set in ⁿ, D a closed convex set in ² with nonempty interior and T a continuous mapping from ⁿ to ². The constraint T(x) int D is a reverse convex constraint, so the feasible domain may be disconnected even when f, T are affine and G is a polytope. We show that this problem can be reduced to a quasiconcave minimization problem over a compact convex set in ² and hence can be solved effectively provided f, T are convex and G is convex or discrete. In particular we discuss a reverse convex constraint of the form c, x · d, x1. We also compare the approach in this paper with the parametric approach.