Undominated d.c. Decompositions of Quadratic Functions and Applications to Branch-and-Bound Approaches

In this paper we analyze difference-of-convex (d.c.) decompositions for indefinite quadratic functions. Given a quadratic function, there are many possible ways to decompose it as a difference of two convex quadratic functions. Some decompositions are dominated, in the sense that other decompositions exist with a lower curvature. Obviously, undominated decompositions are of particular interest. We provide three different characterizations of such decompositions, and show that there is an infinity of undominated decompositions for indefinite quadratic functions. Moreover, two different procedures will be suggested to find an undominated decomposition starting from a generic one. Finally, we address applications where undominated d.c.d.s may be helpful: in particular, we show how to improve bounds in branch-and-bound procedures for quadratic optimization problems.     

A continuous approch for globally solving linearly constrained quadratic

In a continuous approach we propose an efficient method for globally solving linearly constrained quadratic zero-one programming considered as a d.c. (difference of onvex functions) program. A combination of the d.c. optimization algorithm (DCA) which has a finite convergence, and the branch-and-bound scheme was studied. We use rectangular bisection in the branching procedure while the bounding one proceeded by applying d.c.algorithms from a current best feasible point (for the upper bound) and by minimizing a well tightened convex underestimation of the objective function on the current subdivided domain (for the lower bound). DCA generates a sequence of points in the vertex set of a new polytope containing the feasible domain of the problem being considered. Moreover if an iterate is integral then all following iterates are integral too.Our combined algorithm converges so quite often to an integer approximate solution.Finally, we present computational results of several test problems with up to 1800

variables which prove the efficiency of our method, in particular, for linear zero-one programming     

求解中大规模复杂凸二次整数规划问题的新型分枝定界算法

针对现有分枝定界算法在求解高维复杂二次整数规划问题时所存在的诸多不足，本文通过充分挖掘二次整数规划问题的结构特性来设计选择分枝变量与分枝方向的新方法，并将HNF算法与原问题松弛问题的求解相结合来寻求较好的初始整数可行解，由此导出可用于有效求解中大规模复杂二次整数规划问题的改进型分枝定界算法．数值试验结果表明所给算法大大改进了已有相关的分枝定界算法，并具有较好的稳定性与广泛的适用性．     

A Branch and Bound Method via d.c. Optimization Algorithms and Ellipsoidal Technique for Box Constrained Nonconvex Quadratic Problems

In this paper we propose a new branch and bound algorithm using a rectangular partition and ellipsoidal technique for minimizing a nonconvex quadratic function with box constraints. The bounding procedures are investigated by d.c. (difference of convex functions) optimization algorithms, called DCA. This is based upon the fact that the application of the DCA to the problems of minimizing a quadratic form over an ellipsoid and/or over a box is efficient. Some details of computational aspects of the algorithm are reported. Finally, numerical experiments on a lot of test problems showing the efficiency of our algorithm are presented.     

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     

Maximization of the Ratio of Two Convex Quadratic Functions over a Polytope

In this paper, we will develop an algorithm for solving a quadratic fractional programming problem which was recently introduced by Lo and MacKinlay to construct a maximal predictability portfolio, a new approach in portfolio analysis. The objective function of this problem is defined by the ratio of two convex quadratic functions, which is a typical global optimization problem with multiple local optima. We will show that a well-designed branch-and-bound algorithm using (i) Dinkelbach's parametric strategy, (ii) linear overestimating function and (iii) -subdivision strategy can solve problems of practical size in an efficient way. This algorithm is particularly efficient for Lo-MacKinlay's problem where the associated nonconvex quadratic programming problem has low rank nonconcave property.     

An eigenvalue decomposition based branch-and-bound algorithm for nonconvex quadratic programming problems with convex quadratic constraints

In this paper, we propose a branch-and-bound algorithm for finding a global optimal solution for a nonconvex quadratic program with convex quadratic constraints (NQPCQC). We first reformulate NQPCQC by adding some nonconvex quadratic constraints induced by eigenvectors of negative eigenvalues associated with the nonconvex quadratic objective function to Shor’s semidefinite relaxation. Under the assumption of having a bounded feasible domain, these nonconvex quadratic constraints can be further relaxed into linear ones to form a special semidefinite programming relaxation. Then an efficient branch-and-bound algorithm branching along the eigendirections of negative eigenvalues is designed. The theoretic convergence property and the worst-case complexity of the proposed algorithm are proved. Numerical experiments are conducted on several types of quadratic programs to show the efficiency of the proposed method.     

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.     

Global optimization of truss topology with discrete bar areas—Part II: Implementation and numerical results

A classical problem within the field of structural optimization is to find the stiffest truss design subject to a given external static load and a bound on the total volume. The design variables describe the cross sectional areas of the bars. This class of problems is well-studied for continuous bar areas. We consider here the difficult situation that the truss must be built from pre-produced bars with given areas. This paper together with Part I proposes an algorithmic framework for the calculation of a global optimizer of the underlying non-convex mixed integer design problem. In this paper we use the theory developed in Part I to design a convergent nonlinear branch-and-bound method tailored to solve large-scale instances of the original discrete problem. The problem formulation and the needed theoretical results from Part I are repeated such that this paper is self-contained. We focus on the implementation details but also establish finite convergence of the branch-and-bound method. The algorithm is based on solving a sequence of continuous non-convex relaxations which can be formulated as quadratic programs according to the theory in Part I. The quadratic programs to be treated within the branch-and-bound search all have the same feasible set and differ from each other only in the objective function. This is one reason for making the resulting branch-and-bound method very efficient. The paper closes with several large-scale numerical examples. These examples are, to the knowledge of the authors, by far the largest discrete topology design problems solved by means of global optimization.     

A nonconvex quadratic optimization approach to the maximum edge weight clique problem

The maximum edge weight clique (MEWC) problem, defined on a simple edge-weighted graph, is to find a subset of vertices inducing a complete subgraph with the maximum total sum of edge weights. We propose a quadratic optimization formulation for the MEWC problem and study characteristics of this formulation in terms of local and global optimality. We establish the correspondence between local maxima of the proposed formulation and maximal cliques of the underlying graph, implying that the characteristic vector of a MEWC in the graph is a global optimizer of the continuous problem. In addition, we present an exact algorithm to solve the MEWC problem. The algorithm is a combinatorial branch-and-bound procedure that takes advantage of a new upper bound as well as an efficient construction heuristic based on the proposed quadratic formulation. Results of computational experiments on some benchmark instances are also presented.     

Optimizing a polyhedral-semidefinite relaxation of completely positive programs

It has recently been shown (Burer, Math Program 120:479–495, 2009) that a large class of NP-hard nonconvex quadratic programs (NQPs) can be modeled as so-called completely positive programs, i.e., the minimization of a linear function over the convex cone of completely positive matrices subject to linear constraints. Such convex programs are NP-hard in general. A basic tractable relaxation is gotten by approximating the completely positive matrices with doubly nonnegative matrices, i.e., matrices which are both nonnegative and positive semidefinite, resulting in a doubly nonnegative program (DNP). Optimizing a DNP, while polynomial, is expensive in practice for interior-point methods. In this paper, we propose a practically efficient decomposition technique, which approximately solves the DNPs while simultaneously producing lower bounds on the original NQP. We illustrate the effectiveness of our approach for solving the basic relaxation of box-constrained NQPs (BoxQPs) and the quadratic assignment problem. For one quadratic assignment instance, a best-known lower bound is obtained. We also incorporate the lower bounds within a branch-and-bound scheme for solving BoxQPs and the quadratic multiple knapsack problem. In particular, to the best of our knowledge, the resulting algorithm for globally solving BoxQPs is the most efficient to date.     

Copositivity detection by difference-of-convex decomposition and ω-subdivision

We present three new copositivity tests based upon difference-of-convex (d.c.) decompositions, and combine them to a branch-and-bound algorithm of ω-subdivision type. The tests employ LP or convex QP techniques, but also can be used heuristically using appropriate test points. We also discuss the selection of efficient d.c. decompositions and propose some preprocessing ideas based on the spectral d.c. decomposition. We report on first numerical experience with this procedure which are very promising.     

Duality Bound Method for the General Quadratic Programming Problem with Quadratic Constraints

The purpose of this article is to develop a branch-and-bound algorithm using duality bounds for the general quadratically-constrained quadratic programming problem and having the following properties: (i) duality bounds are computed by solving ordinary linear programs; (ii) they are at least as good as the lower bounds obtained by solving relaxed problems, in which each nonconvex function is replaced by its convex envelope; (iii) standard convergence properties of branch-and-bound algorithms for nonconvex global optimization problems are guaranteed. Numerical results of preliminary computational experiments for the case of one quadratic constraint are reported.     

Solving a Class of Linearly Constrained Indefinite Quadratic Problems by D.C. Algorithms

Linearly constrained indefinite quadratic problems play an important role in global optimization. In this paper we study d.c. theory and its local approachto such problems. The new algorithm, CDA, efficiently produces local optima and sometimes produces global optima. We also propose a decomposition branch andbound method for globally solving these problems. Finally many numericalsimulations are reported.     

Decomposition approach for the global minimization of biconcave functions over polytopes

A decomposition approach is proposed for minimizing biconcave functions over polytopes. Important special cases include concave minimization, bilinear and indefinite quadratic programming for which new algorithms result. The approach introduces a new polyhedral partition and combines branch-and-bound techniques, outer approximation, and projection of polytopes in a suitable way.The authors are indebted to two anonymous reviewers for suggestions which have considerably improved this article.     

A parametric successive underestimation method for convex multiplicative programming problems

This paper addresses itself to the algorithm for minimizing the product of two nonnegative convex functions over a convex set. It is shown that the global minimum of this nonconvex problem can be obtained by solving a sequence of convex programming problems. The basic idea of this algorithm is to embed the original problem into a problem in a higher dimensional space and to apply a branch-and-bound algorithm using an underestimating function. Computational results indicate that our algorithm is efficient when the objective function is the product of a linear and a quadratic functions and the constraints are linear. An extension of our algorithm for minimizing the sum of a convex function and a product of two convex functions is also discussed.     

A new global optimization method for univariate constrained twice-differentiable NLP problems

In this paper, a new global optimization method is proposed for an optimization problem with twice-differentiable objective and constraint functions of a single variable. The method employs a difference of convex underestimator and a convex cut function, where the former is a continuous piecewise concave quadratic function, and the latter is a convex quadratic function. The main objectives of this research are to determine a quadratic concave underestimator that does not need an iterative local optimizer to determine the lower bounding value of the objective function and to determine a convex cut function that effectively detects infeasible regions for nonconvex constraints. The proposed method is proven to have a finite ε-convergence to locate the global optimum point. The numerical experiments indicate that the proposed method competes with another covering method, the index branch-and-bound algorithm, which uses the Lipschitz constant.     

Linear,quadratic, and bilinear programming approaches to the linear complementarity problem

Traditional pivoting procedures for solving the linear complementarity problem can only guarantee convergence for problems having well defined structures. Recently, optimization procedures based on linear, quadratic, and bilinear programming have been developed to extend the class of problems that can be solved efficiently. These latter approaches are the focus of this paper.The strengths and weaknesses of each of the approaches are discussed. The linear programming approach, advanced by Mangasarian, is the most efficient once an appropriate objective function is found. This requires the solution of a system of linear and bilinear equations that is easily solvable only in some cases. Extensions to this approach, due to Shiau, show some promise but are still limited to special cases of the general problem. The quadratic programming approaches discussed here are restricted to specialized procedures for the complementarity problem. One, proposed by Cheng, is based on the levitin-Poljak gradient projection method, and the other, due to Cirina, is based on Karush-Kuhn-Tucker theory. Both are only successful on some problems. The two bilinear programming algorithms discussed are the most general. For any problem, they are guaranteed to find at least one solution or conclude that none exist. One is a specialization of the recent biconvex programming algorithm of Al-Khayyal and Falk and the other is an entirely new implicit enumeration procedure.     

Convergence and restart in branch-and-bound algorithms for global optimization. Application to concave minimization and D.C. Optimization problems

A general branch-and-bound conceptual scheme for global optimization is presented that includes along with previous branch-and-bound approaches also grid-search techniques. The corresponding convergence theory, as well as the question of restart capability for branch-and-bound algorithms used in decomposition or outer approximation schemes are discussed. As an illustration of this conceptual scheme, a finite branch-and-bound algorithm for concave minimization is described and a convergent branch-and-bound algorithm, based on the previous one, is developed for the minimization of a difference of two convex functions.     

Solving an Inverse Problem for an Elliptic Equation by d.c. Programming

An inverse problem of determination of a coefficient in an elliptic equation is considered. This problem is ill-posed in the sense of Hadamard and Tikhonov's regularization method is used for solving it in a stable way. This method requires globally solving nonconvex optimization problems, the solution methods for which have been very little studied in the inverse problems community. It is proved that the objective function of the corresponding optimization problem for our inverse problem can be represented as the difference of two convex functions (d.c. functions), and the difference of convex functions algorithm (DCA) in combination with a branch-and-bound technique can be used to globally solve it. Numerical examples are presented which show the efficiency of the method.