Partly convex programming and zermelo's navigation problems

Abstract

Abstract. 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 Zerrnelo's navigation problems and establish global optimality of the numerical solutions. (1991). 90C25, 90C30, 90C90. Mathematics Subject Classifications

Stable parametric programming*

“…The results given here are essentially different from those obtained in, e.g. [39,41,431, because they do not involve a particular subset of the index set known as the "minimal index set of active constraints". In Section 4 we find new conditions when a well-known marginal value formula, see, e.g.…”

“…(One such program describes an optimal pricing problem in a cheese shop at the end of Christmas week, see [5, Problems 1.10 and 1.111.) Recently, some advances have been made in Zermelo's navigation problems (see [41,43]) and in bilevel optimization (see [13]) using SPP. In particular, duality theories have been formulated and local and global optimality have been characterized for this class of programs.…”

“…In a classical navigational situation (recall Zermelo's problem, e.g. [41]), the input data are the speed of the river, relative speed of the boat and the steering angle of the boat, while the output is the time required to cross the river or reach a target. For linear and convex optimization models, continuity of some important output, such as the optimal value function and the optimal solution (assumed unique), is implied by continuity of the feasible set point-to-set mapping (more precisely : its lower semicontinuity), see, e.g.…”

Sensitivity driven experimental design to facilitate control of dynamical systems

“…In it, we let g denote our nominal model, (x, u, T ) denote the reference solution generated using g = g, g denote our improved model after acquiring new data to improve the model, and ( x, ũ, T ) denote the updated reference solution generated using g = g. Our goal is that the reference trajectory x is easier to track than x since the additional data we collect reduces error in the directions which place the greatest strain on the feedback controller. The Zermelo problem [25], a classical navigation problem in the optimal control literature, considers control of a boat being driven by a current. Mathematically, it is modeled by an ODE system with states x = (x 1 , x 2 ) corresponding to the position of a boat along a river (with the river running parallel to the x 1 -axis).…”