Title: Fully convex Bolza problems with state constraints and impulses

Abstract: In this talk, we shall review the Hamilton-Jacobi theory for A Fully Convex Bolza (FCB) problems when the data has no implicit state constraints and is coercive, in which case the minimizing class of arcs are Absolutely Continuous (AC). When a state constraint x(t) in X is added to the problem formulation, the dual variable may exhibit an impulse or “jump” when the constraint is active. The two properties of a state constraint and noncoercive data (which induce impulsive behavior) are in fact dual to each other, and the minimizing class becomes those of bounded variation. We shall describe Rockafellar’s optimality conditions for these problems and a new technique for approximating them by AC problems that utilizes Goebel’s self-dual envelope. The approximating AC problems maintain duality and the existing theory can be applied to them. It is proposed that an HJ theory can be developed for BV problems as an appropriate limit of the approximating AC problems. An explicit example will illustrate this.

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Abstract: his talk concerns the study of criticality of Lagrange multipliers in variational systems that have been recognized in both theoretical and numerical aspects of optimization and variational analysis. In contrast to the previous developments dealing with polyhedral KKT systems and the like, we now focus on general nonpolyhedral systems that are associated, in particular, with problems of conic programming. Developing a novel approach, which is mainly based on advanced techniques and tools of second-order variational analysis and generalized differentiation, allows us to overcome principal challenges of nonpolyhedrality and to establish complete characterizations on noncritical multipliers in such settings. We present applications of noncritical multipliers to deriving efficient conditions of the sequential quadratic programming method for conic programs.

Based on joint work with Ebrahim Sarabi (Miami University, Oxford, OH, USA)