Article

# Hidden convexity in some nonconvex quadratically constrained quadratic programming.

Mathematical Programming (Impact Factor: 2.09). 01/1996; 72:51-63. DOI: 10.1007/BF02592331

Source: DBLP

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**ABSTRACT:**The paper identifies classes of nonconvex optimization problems whose convex relaxations have optimal solutions which at the same time are global optimal solutions of the original nonconvex problems. Such a hidden convexity property was so far limited to quadratically constrained quadratic problems with one or two constraints. We extend it here to problems with some partial separable structure. Among other things, the new hidden convexity results open up the possibility to solve multi-stage robust optimization problems using certain nonlinear decision rules.06/2011; - [Show abstract] [Hide abstract]

**ABSTRACT:**We present in this paper new sufficient conditions for verifying zero duality gap in nonconvex quadratically/linearly constrained quadratic programs (QP). Based on saddle point condition and conic duality theorem, we first derive a sufficient condition for the zero duality gap between a quadratically constrained QP and its Lagrangian dual or SDP relaxation. We then use a distance measure to characterize the duality gap for nonconvex QP with linear constraints. We show that this distance can be computed via cell enumeration technique in discrete geometry. Finally, we revisit two sufficient optimality conditions in the literature for two classes of nonconvex QPs and show that these conditions actually imply zero duality gap.Journal of Global Optimization 01/2012; 52:229-242. · 1.31 Impact Factor - [Show abstract] [Hide abstract]

**ABSTRACT:**We propose a branch-and-bound algorithm for minimizing a not necessarily convex quadratic function over integer variables. The algorithm is based on lower bounds computed as continuous minima of the objective function over appropriate ellipsoids. In the nonconvex case, we use ellipsoids enclosing the feasible region of the problem. In spite of the nonconvexity, these minima can be computed quickly. We present several ideas that allow to accelerate the solution of the continuous relaxation within a branch-and-bound scheme and examine the performance of the overall algorithm by computational experiments.01/2012;

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