PARTLY CONVEX-PROGRAMMING AND ZERMELOS NAVIGATION PROBLEMS

Mathematical programs, that become convex programs after ''freezing'' some variables, are termed partly convex. For such programs we give saddle-point conditions that are both necessary and sufficient that a feasible point be globally optimal. The conditions require ''cooperation'' of the feasible point tested for optimality, an assumption implied by lower semicontinuity of the feasible set mapping. The characterizations are simplified if certain point-to-set mappings satisfy a ''sandwich condition''. The tools of parametric optimization and basic point-to-set topology are used in formulating both optimality conditions and numerical methods. In particular, we solve a large class of Zermelo's navigation problems and establish global optimality of the numerical solutions.