Global minimization of large-scale constrained concave quadratic problems by separable programming

Abstract

The global minimization of a large-scale linearly constrained concave quadratic problem is considered. The concave quadratic part of the objective function is given in terms of the nonlinear variables x ∈Rn, while the linear part is in terms of y ∈Rk. For large-scale problems we may have k much larger than n. The original problem is reduced to an equivalent separable problem by solving a multiple-cost-row linear program with 2 n cost rows. The solution of one additional linear program gives an incumbent vertex which is a candidate for the global minimum, and also gives a bound on the relative error in the function value of this incumbent. An a priori bound on this relative error is obtained, which is shown to be ≤ 0.25, in important cases. If the incumbent is not a satisfactory approximation to the global minimum, a guaranteed ε-approximate solution is obtained by solving a single liner zero-one mixed integer programming problem. This integer problem is formulated by a simple piecewise-linear underestimation of the separable problem. © 1986 The Mathematical Programming Society, Inc.