Branch and Bound Method to Resolve the Non-convex Quadratic Problems

Abstract

In this paper, we present a new rectangle Branch and Bound approach for solving non convex quadratic programming problems in which we construct a new lower approximate convex quadratic function of the objective quadratic function over an n-rectangle $S^{k}=\left[ a^{k},b^{k}\right]$ or $S^{k}= \left[ L^{k},U^{k}\right]$. This quadratic function (the approximate one) is given to determine a lower bound of the global optimal value of the original problem (NQP) over each rectangle. In the other side, we apply a simple two-partition technique on rectangle, as well as, the tactics on reducing and deleting subrectangles are used to accelerate the convergence of the proposed algorithm. This proposed algorithm is proved to be convergent and shown to be effective with some examples.