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.     

Integrating SQP and Branch-and-Bound for Mixed Integer Nonlinear Programming

This paper considers the solution of Mixed Integer Nonlinear Programming (MINLP) problems. Classical methods for the solution of MINLP problems decompose the problem by separating the nonlinear part from the integer part. This approach is largely due to the existence of packaged software for solving Nonlinear Programming (NLP) and Mixed Integer Linear Programming problems.In contrast, an integrated approach to solving MINLP problems is considered here. This new algorithm is based on branch-and-bound, but does not require the NLP problem at each node to be solved to optimality. Instead, branching is allowed after each iteration of the NLP solver. In this way, the nonlinear part of the MINLP problem is solved whilst searching the tree. The nonlinear solver that is considered in this paper is a Sequential Quadratic Programming solver.A numerical comparison of the new method with nonlinear branch-and-bound is presented and a factor of up to 3 improvement over branch-and-bound is observed.     

A Finite Branch-and-Bound Algorithm for Linear Multiplicative Programming

On the basis of Soland's rectangular branch-and-bound, we develop an algorithm for minimizing a product of p (2) affine functions over a polytope. To tighten the lower bound on the value of each subproblem, we install a second-stage bounding procedure, which requires O(p) additional time in each iteration but remarkably reduces the number of branching operations. Computational results indicate that the algorithm is practical if p is less than 15, both in finding an exact optimal solution and an approximate solution.     

An Algorithm for Global Minimization of Linearly Constrained Quadratic Functions*

A branch and bound algorithm is proposed for finding an approximate global optimum of quadratic functions over a bounded polyhedral set. The algorithm uses Lagrangian duality to obtain lower bounds. Preliminary computational results are reported.     

求不定二次规划全局解的一个新算法

本文提出了一个求不定二次规划问题全局最优解的新算法.首先,给出了三种计算下界的方法:线性逼近法、凸松弛法和拉格朗日松弛法;并且证明了拉格朗日对偶界与通过凸松弛得到的下界是相等的;然后建立了基于拉格朗日对偶界和矩形两分法的分枝定界算法,并给出了初步的数值试验结果.     

On Copositive Programming and Standard Quadratic Optimization Problems

A standard quadratic problem consists of finding global maximizers of a quadratic form over the standard simplex. In this paper, the usual semidefinite programming relaxation is strengthened by replacing the cone of positive semidefinite matrices by the cone of completely positive matrices (the positive semidefinite matrices which allow a factorization FF ^T where F is some non-negative matrix). The dual of this cone is the cone of copositive matrices (i.e., those matrices which yield a non-negative quadratic form on the positive orthant). This conic formulation allows us to employ primal-dual affine-scaling directions. Furthermore, these approaches are combined with an evolutionary dynamics algorithm which generates primal-feasible paths along which the objective is monotonically improved until a local solution is reached. In particular, the primal-dual affine scaling directions are used to escape from local maxima encountered during the evolutionary dynamics phase.     

Convexity Properties Associated with Nonconvex Quadratic Matrix Functions and Applications to Quadratic Programming

We establish several convexity results which are concerned with nonconvex quadratic matrix (QM) functions: strong duality of quadratic matrix programming problems, convexity of the image of mappings comprised of several QM functions and existence of a corresponding S-lemma. As a consequence of our results, we prove that a class of quadratic problems involving several functions with similar matrix terms has a zero duality gap. We present applications to robust optimization, to solution of linear systems immune to implementation errors and to the problem of computing the Chebyshev center of an intersection of balls. This research was partially supported by the Israel Science Foundation under Grant ISF 489/06.     

Exact Solutions of Some Nonconvex Quadratic Optimization Problems via SDP and SOCP Relaxations

We show that SDP (semidefinite programming) and SOCP (second order cone programming) relaxations provide exact optimal solutions for a class of nonconvex quadratic optimization problems. It is a generalization of the results by S. Zhang for a subclass of quadratic maximization problems that have nonnegative off-diagonal coefficient matrices of quadratic objective functions and diagonal coefficient matrices of quadratic constraint functions. A new SOCP relaxation is proposed for the class of nonconvex quadratic optimization problems by extracting valid quadratic inequalities for positive semidefinite cones. Its effectiveness to obtain optimal values is shown to be the same as the SDP relaxation theoretically. Numerical results are presented to demonstrate that the SOCP relaxation is much more efficient than the SDP relaxation.     

Canonical Duality Theory and Solutions to Constrained Nonconvex Quadratic Programming

This paper presents a perfect duality theory and a complete set of solutions to nonconvex quadratic programming problems subjected to inequality constraints. By use of the canonical dual transformation developed recently, a canonical dual problem is formulated, which is perfectly dual to the primal problem in the sense that they have the same set of KKT points. It is proved that the KKT points depend on the index of the Hessian matrix of the total cost function. The global and local extrema of the nonconvex quadratic function can be identified by the triality theory [11]. Results show that if the global extrema of the nonconvex quadratic function are located on the boundary of the primal feasible space, the dual solutions should be interior points of the dual feasible set, which can be solved by deterministic methods. Certain nonconvex quadratic programming problems in {\open {R}}^{n} can be converted into a dual problem with only one variable. It turns out that a complete set of solutions for quadratic programming over a sphere is obtained as a by-product. Several examples are illustrated.     

凸二次规划问题逆问题的模型与解法

本文分别考虑带非负约束和不带大量负约束凸二次规划问题逆问题。首先得到各个逆问题的数学模型,然后对不同的模型给出不同的求解方法。     

A Decomposition Method for Global and Local Quadratic Minimization

We present a decomposition method for indefinite quadratic programming problems having n variables and m linear constraints. The given problem is decomposed into at most m QP subproblems each having m linear constraints and n-1 variables. All global minima, all isolated local minima and some of the non-isolated local minima for the given problem are obtained from those of the lower dimensional subproblems. One way to continue solving the given problem is to apply the decomposition method again to the subproblems and repeatedly doing so until subproblems of dimension 1 are produced and these can be solved directly. A technique to reduce the potentially large number of subproblems is formulated.     

二次半定规划的原始对偶内点算法的H..K..M搜索方向的存在唯一性

主要是将半定规划(Semidefinite Programming,简称SDP)的内点算法推广到二次半定规划(Quadratic Semidefinite Programming,简称QSDP),重点讨论了其中搜索方向的产生方法.首先利用Wolfe对偶理论推导得到了求解二次半定规划的非线性方程组,利用牛顿法求解该方程组,得到了求解QSDP的内点算法的H..K..M搜索方向,接着证明了该搜索方向的存在唯一性,最后给出了搜索方向的具体计算方法.     

Ellipsoidal Approach to Box-Constrained Quadratic Problems

We present a new heuristic for the global solution of box constrained quadratic problems, based on the classical results which hold for the minimization of quadratic problems with ellipsoidal constraints. The approach is tested on several problems randomly generated and on graph instances from the DIMACS challenge, medium size instances of the Maximum Clique Problem. The numerical results seem to suggest some effectiveness of the proposed approach.     

Global and Local Quadratic Minimization

We present a method which when applied to certain non-convex QP will locatethe globalminimum, all isolated local minima and some of the non-isolated localminima. The method proceeds by formulating a (multi) parametric convex QP interms ofthe data of the given non-convex QP. Based on the solution of the parametricQP,an unconstrained minimization problem is formulated. This problem ispiece-wisequadratic. A key result is that the isolated local minimizers (including theglobalminimizer) of the original non-convex problem are in one-to-one correspondencewiththose of the derived unconstrained problem.The theory is illustrated with several numerical examples. A numericalprocedure isdeveloped for a special class of non-convex QP's. It is applied to a problemfrom theliterature and verifies a known global optimum and in addition, locates apreviously unknown local minimum.     

Minimizing an indefinite quadratic function subject to a single indefinite quadratic constraint

In this paper, we consider the problem of minimizing an indefinite quadratic function subject to a single indefinite quadratic constraint. A key difficulty with this problem is its nonconvexity. Using Lagrange duality, we show that under a mild assumption, this problem can be solved by solving a linearly constrained convex univariate minimization problem. Finally, the superior efficiency of the new approach compared to the known semidefinite relaxation and a known approach from the literature is demonstrated by solving several randomly generated test problems.     

Optimal Multisections in Interval Branch-and-Bound Methods of Global Optimization

In this paper we define multisections of intervals that yield sharp lower bounds in branch-and-bound type methods for interval global optimization. A so called 'generalized kite', defined for differentiable univariate functions, is built simultaneously with linear boundary forms and suitably chosen centered forms. Proofs for existence and uniqueness of optimal cuts are given. The method described may be used either as an accelerating device or in a global optimization algorithm with an efficient pruning effect. A more general principle for decomposition of boxes is suggested.     

Branch-and-Bound Variant of an Outcome-Based Algorithm for Optimizing over the Efficient Set of a Bicriteria Linear Programming Problem

The paper presents a finite branch-and-bound variant of an outcome-based algorithm proposed by Benson and Lee for minimizing a lower-semicontinuous function over the efficient set of a bicriteria linear programming problem. Similarly to the Benson-Lee algorithm, we work primarily in the outcome space. Dissimilarly, instead of constructing a sequence of consecutive efficient edges in the outcome space, we use the idea of generating a refining sequence of partitions covering the at most two-dimensional efficient set in the outcome space. Computational experience is also presented.     

Global Optimization of a Quadratic Functional with Quadratic Equality Constraints, Part 2

In this paper, we investigate a constrained optimization problem with a quadratic cost functional and two quadratic equality constraints. It is assumed that the cost functional is positive definite and that the constraints are both feasible and regular (but otherwise they are unrestricted quadratic functions). Thus, the existence of a global constrained minimum is assured. We develop a necessary and sufficient condition that completely characterizes the global minimum cost. Such a condition is of essential importance in iterative numerical methods for solving the constrained minimization problem, because it readily distinguishes between local minima and global minima and thus provides a stopping criterion for the computation. The result is similar to one obtained previously by the authors. In the previous result, we gave a characterization of the global minimum of a constrained quadratic minimization problem in which the cost functional was an arbitrary quadratic functional (as opposed to positive-definite here) and the constraints were at least positive-semidefinite quadratic functions (as opposed to essentially unrestricted here).     

解带有二次约束非凸二次规划问题的一个分枝缩减方法

在这篇论文里,有机地把外逼近方法与分枝定界技术结合起来,提出了解带有二次约束非凸二次规划问题的一个分枝缩减方法;给出了原问题的一个新的线性规划松弛,以便确定它在超矩形上全局最优值的一个下界;利用超矩形的一个深度二级剖分方法,以及超矩形的缩减和删除技术,提高算法的收敛速度;证明了在知道原问题可行点的条件下,该算法在有限步里就可以获得原问题的一个全局最优化解,并且用一个例子说明了该算法是有效的.     

Global Optimization Techniques for Solving the General Quadratic Integer Programming Problem

We consider the problem of minimizing a general quadratic function over a polytope in the n-dimensional space with integrality restrictions on all of the variables. (This class of problems contains, e.g., the quadratic 0-1 program as a special case.) A finite branch and bound algorithm is established, in which the branching procedure is the so-called integral rectangular partition, and the bound estimation is performed by solving a concave programming problem with a special structure. Three methods for solving this special concave program are proposed.