We show that it is fruitful to dualize the integrality constraints in a combinatorial optimization problem. First, this reproduces the known SDP relaxations of the max-cut and max-stable problems. Then we apply the approach to general combinatorial problems. We show that the resulting duality gap is smaller than with the classical Lagrangian relaxation; we also show that linear constraints need a special treatment.

Abstract not found

We study convergence properties of a modified subgradient algorithm, applied to the dual problem defined by the sharp augmented Lagrangian. The primal problem we consider is nonconvex and nondifferentiable, with equality constraints. We obtain primal and dual convergence results, as well as a condition for existence of a dual solution. Using a practical selection of the step-size parameters, we demonstrate the algorithm and its advantages on test problems, including an integer programming and an optimal control problem. Key words: Nonconvex programming; nonsmooth optimization; augmented Lagrangian; sharp Lagrangian; subgradient optimization.