Mathematical Programming

, Volume 34, Issue 2, pp 163–174

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

• J. B. Rosen
• P. M. Pardalos
Article

## 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 variablesxR n , while the linear part is in terms ofyRk. For large-scale problems we may havek much larger thann. The original problem is reduced to an equivalent separable problem by solving a multiple-cost-row linear program with 2n 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. Ana 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 guaranteedε-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.

## Key words

Global Minimization Separable Programming Quadratic Programming Large-Scale

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© The Mathematical Programming Society, Inc. 1986

## Authors and Affiliations

• J. B. Rosen
• 1
• P. M. Pardalos
• 1
1. 1.Computer Science DepartmentUniversity of MinnesotaMinneapolisUSA