Existence and properties of solutions of a control system with hysteresis effect

We consider a control system described by two ordinary nonlinear differential equations subject to a control constraint given by a multivalued mapping with closed nonconvex values, which depends on the phase variables. One of the equations contains the subdifferential of the indicator function of a closed convex set depending on the unknown phase variable. The equation containing the subdifferential describes an input-output relation of hysteresis type.Along with the original control constraint, we also consider the convexified control constraint and the constraint consisting of the extremal points of the convexified control constraint.We prove the existence of solutions of our control system with various control constraints and establish certain relationships between corresponding solution sets.     

Continuity in the parameter of the minimum value of an integral functional over the solutions of an evolution control system

The problem of minimization of an integral functional with an integrand that is nonconvex with respect to the control is considered. We minimize our functional over the solution set of a nonlinear evolution control system with a time-dependent subdifferential operator in a Hilbert space. The control constraint is given by a nonconvex closed bounded set. The integrand, the control constraint, the initial conditions and the operators in the equation describing the control system all depend on a parameter. We consider, along with the original problem, the problem of minimizing an integral functional with an integrand convexified with respect to the control over the solution set of the same system, but now subject to the convexified control constraint. By a solution of the control system we mean a “trajectory–control” pair. We prove that for each value of the parameter the convexified problem has a solution, which is the limit of a minimizing sequence of the original problem, and the minimum value of the functional of the convexified problem is a continuous function of the parameter.     

Properties of the set of “trajectory-control” pairs of a control systemwith subdifferential operators

We consider a control system described by an evolution equation with control constraint which is a multivalued mapping of a phase variable with closed nonconvex values. One of the evolution operators of the system is the subdifferential of a time-dependent proper, convex, and lower semicontinuous function. The other operator, acting on the derivative of the required functions, is the subdifferential of a convex continuous function. We also consider systems with the following control constraints: multivalued mappings whose values are the closed convex hulls of the values of the original constraint and multivalued mapping whose values are the extreme points of the convexified constraint that belong to the original one. We study topological properties of the sets of admissible “trajectory–control” pairs of the system with various control constraints and clarify the relations between them. An example of a parabolic system with hysteresis and diffusion phenomena is considered in detail. Bibliography: 19 titles.     

Relaxation in nonconvex optimal control problems with subdifferential operators

We consider the minimization problem of an integral functional in a separable Hilbert space with integrand not convex in the control defined on solutions of the control system described by nonlinear evolutionary equations with mixed nonconvex constraints. The evolutionary operator of the system is the subdifferential of a proper, convex, lower semicontinuous function depending on time. Along with the initial problem, the author considers the relaxed problem with the convexicated control constraint and the integrand convexicated with respect to the control. Under sufficiently general assumptions, it is proved that the relaxed problem has an optimal solution, and for any optimal solution, there exists a minimizing sequence of the initial problem converging to the optimal solution with respect to trajectories and the functional. An example of a controlled parabolic variational inequality with obstacle is considered in detail. __________ Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 26, Nonlinear Dynamics, 2005.     

Properties of Solutions to Second Order Evolution Control Systems. I

We consider a control system described by a nonlinear second order evolution equation defined on an evolution triple of Banach spaces (Gelfand triple) with a mixed multivalued control constraint whose values are nonconvex closed sets. Alongside the original system we consider a system with the following control constraints: a constraint whose values are the closed convex hull of the values of the original constraint and a constraint whose values are extreme points of the constraint which belong simultaneously to the original constraint. By a solution to the system we mean an admissible trajectory-control pair. In this part of the article we study existence questions for solutions to the control system with various constraints and density of the solution set with nonconvex constraints in the solution set with convexified constraints.     

Another characterization of convexity for set-valued maps

We give a new necessary and sufficient condition for convexity of a set-valued map F between Banach spaces. It is established for a closed map F having nonconvex values. The main tool in this paper is the coderivative of F which is constructed with the help of an abstract subdifferential notion of Penot . A detailed discussion is devoted to special cases when the contingent, the Fréchet and the Clarke-Rockafellar subdifFerentials Sixe used as this abstract subdifferential.     

Duality Principles for Optimization Problems Dealing with the Difference of Vector-Valued Convex Mappings

Using the concept of a subdifferential of a vector-valued convex mapping, we provide duality formulas for the minimization of nonconvex composite functions and related optimization problems such as the minimization of a convex function over a vectorial DC constraint.     

Stability and augmented Lagrangian duality in nonconvex semi-infinite programming

In this paper the pseudo-Lipschitz property of the constraint set mapping and the Lipschitz property of the optimal value function of parametric nonconvex semi-infinite optimization problems are obtained under suitable conditions on the limiting subdifferential and the limiting normal cone. Then we derive sufficient conditions for the strong duality of nonconvex semi-infinite optimality problems and a criterion for exact penalty representations via an augmented Lagrangian approach. Examples are given to illustrate the obtained results.     

Subgradients of the Value Function to a Parametric Optimal Control Problem

This paper studies the first-order behavior of the value function of a parametric optimal control problem with linear constraints and nonconvex cost functions. By establishing an abstract result on the Fréchet subdifferential of the value functions of a parametric mathematical programming problem, a new formula for computing the Fréchet subdifferential of the value function to a parametric optimal control problem is obtained.     

Directional Hölder Metric Regularity

This paper studies the first-order behavior of the value function of a parametric optimal control problem with nonconvex cost functions and control constraints. By establishing an abstract result on the Fréchet subdifferential of the value function of a parametric minimization problem, we derive a formula for computing the Fréchet subdifferential of the value function to a parametric optimal control problem. The obtained results improve and extend some previous results.     

Uniform subsmoothness and linear regularity for a collection of infinitely many closed sets

Motivated by the subsmoothness of a closed set introduced by Aussel et al. (2005) [8], we introduce and study the uniform subsmoothness of a collection of infinitely many closed subsets in a Banach space. Under the uniform subsmoothness assumption, we provide an interesting subdifferential formula on distance functions and consider uniform metric regularity for a kind of multifunctions frequently appearing in optimization and variational analysis. Different from the existing works, without the restriction of convexity, we consider several fundamental notions in optimization such as the linear regularity, CHIP, strong CHIP and property (G) for a collection of infinitely many closed sets. We establish relationships among these fundamental notions for an arbitrary collection of uniformly subsmooth closed sets. In particular, we extend duality characterizations of the linear regularity for a collection of closed convex sets to the nonconvex setting.     

Optimality conditions for minimizing the difference of nonconvex vector-valued mappings

In this paper, we consider vector optimization problems involving the difference of nonconvex vector-valued mappings. By a nonconvex scalarization function, we establish necessary optimality conditions in terms of the Mordukhovich subdifferential, strong subdifferential and Ioffe subdifferential without any convexity assumption. As an application, we discuss the optimality condition on a nonconvex multiobjective fractional programming problem.     

Optimality Conditions for Nonconvex Constrained Optimization Problems

Abstract

We present optimality conditions for a class of nonsmooth and nonconvex constrained optimization problems. To achieve this aim, various well-known constraint qualifications are extended based on the concept of tangential subdifferential and the relations between them are investigated. Moreover, local and global necessary and sufficient optimality conditions are derived in the absence of convexity of the feasible set. In addition to the theoretical results, several examples are provided to illustrate the advantage of our outcomes.     

Mordukhovich subgradients of the value function to a parametric discrete optimal control problem

This paper is devoted to the study of the first-order behavior of the value function of a parametric discrete optimal control problem with nonconvex cost functions and control constraints. By establishing an abstract result on the Mordukhovich subdifferential of the value function of a parametric mathematical programming problem, we derive a formula for computing the Mordukhovich subdifferential of the value function to a parametric discrete optimal control problem.     

Subgradients of value functions in parametric dynamic programming

We study in this paper the first-order behavior of value functions in parametric dynamic programming with linear constraints and nonconvex cost functions. By establishing an abstract result on the Fréchet subdifferential of value functions of parametric mathematical programming problems, some new formulas on the Fréchet subdifferential of value functions in parametric dynamic programming are obtained.     

On global subdifferentials with applications in nonsmooth optimization

The notions of global subdifferentials associated with the global directional derivatives are introduced in the following paper. Most common used properties, a set of calculus rules along with a mean value theorem are presented as well. In addition, a diversity of comparisons with well-known subdifferentials such as Fréchet, Dini, Clarke, Michel–Penot, and Mordukhovich subdifferential and convexificator notion are provided. Furthermore, the lower global subdifferential is in fact proved to be an abstract subdifferential. Therefore, the lower global subdifferential satisfies standard properties for subdifferential operators. Finally, two applications in nonconvex nonsmooth optimization are given: necessary and sufficient optimality conditions for a point to be local minima with and without constraints, and a revisited characterization for nonsmooth quasiconvex functions.

     

Conditioning and Regularization of Nonsymmetric Operators

In this paper, we study conditioning problems for convex and nonconvex functions defined on normed linear spaces. We extend the notion of upper Lipschitzness for multivalued functions introduced by Robinson, and show that this concept ensures local conditioning in the nonconvex case via an abstract subdifferential; in the convex case, we obtain complete characterizations of global conditioning in terms of an extension of the upper-Lipschitz property.     

Existence and relaxation for subdifferential inclusions with unbounded perturbation

Mathematical Programming - We consider a differential inclusion of subdifferential type with a nonconvex and unbounded valued perturbation. Existence and relaxation results are obtained for this...     

Nonconvex Separation Theorem for Multifunctions,Subdifferential Calculus and Applications

We derive a nonconvex separation theorem for multifunctions that generalizes an early result of Borwein and Jofré and show that this result is equivalent to several other subdifferential calculus results in smooth Banach spaces. Then we apply this nonconvex separation theorem to improve a second welfare theorem in economics and a necessary optimality condition for a multi-objective optimization problem.     

Subdifferential of the supremum function: moving back and forth between continuous and non-continuous settings

In this paper we establish general formulas for the subdifferential of the pointwise supremum of convex functions, which cover and unify both the compact continuous and the non-compact non-continuous settings. From the non-continuous to the continuous setting, we proceed by a compactification-based approach which leads us to problems having compact index sets and upper semi-continuously indexed mappings, giving rise to new characterizations of the subdifferential of the supremum by means of upper semicontinuous regularized functions and an enlarged compact index set. In the opposite sense, we rewrite the subdifferential of these new regularized functions by using the original data, also leading us to new results on the subdifferential of the supremum. We give two applications in the last section, the first one concerning the nonconvex Fenchel duality, and the second one establishing Fritz-John and KKT conditions in convex semi-infinite programming.