On representations of the feasible set in convex optimization

We consider the convex optimization problem $\min \{f(x) : g_j(x)\leq 0, j=1,...,m\}$ where $f$ is convex, the feasible set K is convex and Slater's condition holds, but the functions $g_j$ are not necessarily convex. We show that for any representation of K that satisfies a mild nondegeneracy assumption, every minimizer is a Karush-Kuhn-Tucker (KKT) point and conversely every KKT point is a minimizer. That is, the KKT optimality conditions are necessary and sufficient as in convex programming where one assumes that the $g_j$ are convex. So in convex optimization, and as far as one is concerned with KKT points, what really matters is the geometry of K and not so much its representation.Comment: to appear in Optimization Letter

On the Maximum Principle for Impulsive Hybrid Systems

Hybrid systems consist of dynamical systems where both continuous and discrete event dynamics are interacting. These are widely accepted as realistic models of diverse technical systems and processes, for instance, industrial electronics, power systems, maneuvering aircrafts, automotive control systems, chemical processes and communication networks. Recently optimization problems and variants of the Maximum Principle (MP) for hybrid systems have attracted a great deal of attention. Both theoretical results and computational techniques were developed see e.g.,[1]-[14]. These results are extended here to a general class of hybrid systems with state jumps and the corresponding constrained optimal control problems. The family of hybrid optimization problems under consideration (called Impulsive Hybrid Optimal Control Problems IHOCP) include dynamical systems with internally forced switchings and continuous control signals. The discrete state transitions are triggered by the continuous state and are accompanied by a discontinuous change in the latter variable. This class captures phenomena arising e.g., in cyclically operated batch processes and certain epidemic propagation models. Using the mathematical techniques of distributional derivatives and impulsive differential equations, we extend the necessary optimality conditions to the above class of problems (IHOCPs). We obtain specific elements of the Impulsive Hybrid MP (IHMP), namely, the corresponding boundary-value problem and some additional relations. As in the classical case, the proposed IHMP provides a basis for diverse computational algorithms for the treatment of IHOCPs

Team optimization problems with Lipschitz continuous strategies

Sufficient conditions for the existence and Lipschitz continuity of optimal strategies for static team optimization problems are studied. Revised statements and proofs of some results appeared in the literature are presented. Their extensions are discussed. As an example of application, optimal production in a multidivisional firm is considered