Duality in nonconvex optimization

Duality Theory: Biduality in Nonconvex Optimization Duality Theory: Biduality in Nonconvex Optimization

It is known that in convex optimization, the Lagrangian associated with a constrained problem is usually a saddle function, which leads to the classical saddle Lagrange duality (i. e. the monoduality) theory. In nonconvex optimization, a so-called superLagrangian was introduced in [1], which leads to a nice biduality theory in convex Hamiltonian systems and in the so-called d.c. programming.

Duality gaps in nonconvex stochastic optimization

We consider multistage stochastic optimization models containing nonconvex constraints, e.g., due to logical or integrality requirements. We study three variants of Lagrangian relaxations and of the corresponding decomposition schemes, namely, scenario, nodal and geographical decomposition. Based on convex equivalents for the Lagrangian duals, we compare the duality gaps for these decomposition...

Subgradient Optimization, Generalized Programming, and Nonconvex Duality

Spectral Approach to Duality in Nonconvex Global Optimization∗

A nonconvex problem of constrained optimization is analyzed in terms of its ordinary Lagrangian function. New sufficient conditions are obtained for the duality gap to vanish. Among them, the main condition is that the objective and constraint functions be the sums of convex functionals and nonconvex quadratic forms with certain specific spectral properties. The proofs are related to extensions...

Augmented Lagrangian Duality and Nondifferentiable Optimization Methods in Nonconvex Programming

Abstract. In this paper we present augmented Lagrangians for nonconvex minimization problems with equality constraints. We construct a dual problem with respect to the presented here Lagrangian, give the saddle point optimality conditions and obtain strong duality results. We use these results and modify the subgradient and cutting plane methods for solving the dual problem constructed. Algorit...