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

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## 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* ∈*R*^{ n }, while the linear part is in terms of*y* ∈*R*^{k}. 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.

## Key words

Global Minimization Separable Programming Quadratic Programming Large-Scale## Preview

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