Solutions to quadratic minimization problems with box and integer constraints

  • Authors:

  • David Yang Gao;Ning Ruan

  • Affiliations:

  • Department of Mathematics, Virginia Tech, Blacksburg, USA 24061;Department of Mathematics, Virginia Tech, Blacksburg, USA 24061

  • Venue:
  • Journal of Global Optimization
  • Year:
  • 2010

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Abstract

This paper presents a canonical duality theory for solving quadratic minimization problems subjected to either box or integer constraints. Results show that under Gao and Strang's general global optimality condition, these well-known nonconvex and discrete problems can be converted into smooth concave maximization dual problems over closed convex feasible spaces without duality gap, and can be solved by well-developed optimization methods. Both existence and uniqueness of these canonical dual solutions are presented. Based on a second-order canonical dual perturbation, the discrete integer programming problem is equivalent to a continuous unconstrained Lipschitzian optimization problem, which can be solved by certain deterministic technique. Particularly, an analytical solution is obtained under certain condition. A fourth-order canonical dual perturbation algorithm is presented and applications are illustrated. Finally, implication of the canonical duality theory for the popular semi-definite programming method is revealed.