Global Optimization Algorithms for Linearly Constrained Indefinite Quadratic Problems

1. I N T R O D U C T I O N Global optimization of constrained quadratic problems has been the subject of active research during the last two decades. Quadratic programming is a very old and important problem of optimization. It has numerous applications in many diverse fields of science and technology and plays a key role in many nonlinear programming methods. Nonconvex quadratic programming refers to problems where the objective function is concave or indefinite. A substantial literature exists for the problem of finding the global minimum of a concave function subject to linear and nonlinear constraints. A survey of deterministic methods for global concave minimization problems can be found in [32,59] and in the recent monograph by Pardalos and Rosen [61]. In this paper we consider the problem of finding a globally optimal solution to nonconvex quadratic problems of the form • 1 T global ~ f (z) = cT x + ~ z Qx, (1.1) where Qn×n is an indefinite symmetric matrix, e, x G R n, and P is a bounded polyhedron in R n. The matrix Q is indefinite iff it has at least one negative and one positive eigenvalue. Apart from its importance as a mathematical programming problem, indefinite quadratic programming arises in several practical applications including production planning, microeconomic theory, and transportation problems. Recently it found applications in VLSI chip design [36,41,85]. Certain aspects of physical chip design can be formulated as an indefinite quadratic problem of special structure. We are concerned here with several aspects of indefinite quadratic programming. Methods that have been proposed include Benders decomposition, concave programming approaches, bilinear programming, enumeration, and gradient projection methods. Algorithms for structured problems such as linear complementarity, product of linear forms, and problems with network constraints have also been developed.