Lagrange Duality in Partly Convex Programming

Abstract

A Lagrangian duality theory is formulated for a large class of nonconvex optimization problems. The theoiy is based on the recent characterizations of global and local optima for partly convex programs. It uses basic notions from convex and parametric programming, such as the minimal index set of active constraints and continuity of the feasible set point-to-set mapping. The results are applied to the classic navigation problem of Zermelo. The primal problem consists of finding a steering angle that minimizes the sailing time to a target. Its dual solution is a function that associates, with every steering angle, the sensitivity of the sailing time relative to small perturbations of the target. The unconstrained minima of the dual solution yield the most “robust” steering angles.