Solutions of evolution inclusions generated by a difference of subdifferentials

An evolution inclusion with the right-hand side containing the difference of subdifferentials of proper convex lower semicontinuous functions and a multivalued perturbation whose values are nonconvex closed sets is considered in a separable Hilbert space. In addition to the original inclusion, we consider an inclusion with convexified perturbation and a perturbation whose values are extremal points of the convexified perturbation that also belong to the values of the original perturbation. Questions of the existence of solutions under various perturbations are studied and relations between solutions are established. The primary focus is on the weakening of assumptions on the perturbation as compared to the known assumptions under which existence and relaxation theorems are valid. All our assumptions, in contrast to the known assumptions, concern the convexified rather than original perturbation.     

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.     

Random evolution inclusions of the subdifferential type

In this paper, we consider random evolution inclusion of the subdifferential type with a convex valued perturbation and we establish the existence of a random strong solution. Two examples, the first a nonlinear random parabolic partial differential inclusion and the second a random differential variational inequality, are also worked out in detail     

Properties of Attainable Sets of Evolution Inclusions and Control Systems of Subdifferential Type

In a separable Hilbert space we consider an evolution inclusion with a multivalued perturbation and the evolution operators that are the compositions of a linear operator and the subdifferentials of a time-dependent proper convex lower semicontinuous function. Alongside the initial inclusion, we consider a sequence of approximating evolution inclusions with the same perturbation and the evolution operators that are the compositions of the same linear operator and the subdifferentials of the Moreau–Yosida regularizations of the initial function. We demonstrate that the attainable set of the initial inclusion as a multivalued function of time is the time uniform limit of a sequence of the attainable sets of the approximating inclusions in the Hausdorff metric. We obtain similar results for evolution control systems of subdifferential type with mixed constraints on control. As application we consider an example of a control system with discontinuous nonlinearities containing some linear functions of the state variables of the system.     

Extremality in Solving General Quasilinear Parabolic Inclusions

The paper deals with an initial boundary-value problem for a parabolic inclusion whose multivalued term has the structure of a difference between the Clarke generalized gradient of some locally Lipschitz function verifying a unilateral growth condition and the subdifferential of a convex function, and where the elliptic part is expressed by a general quasilinear operator of the Leray-Lions type. Our results address not only the existence of solutions, but also the extremality inside an order interval determined by appropriately defined upper and lower solutions as well as the compactness of the solution set in suitable spaces.     

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.     

Optimal Control Problems Governed by Time Dependent Subdifferential Operators with Delay

This article is devoted to the study, in the infinite dimensional setting, of a Bolza-type problem governed by a class of functional evolution inclusion which involves a time-dependent subdifferential operator with a time-delay perturbation. We present a relaxation result associated with such equations where the controls are Young measures in order to show the existence of an optimal solution under a suitable convexity assumption.     

Density of the Extremal Solutions for a Class of Second Order Boundary Value Problem

In this paper, we prove a theorem on the existence of extremal solutions to a second-order differential inclusion with boundary conditions, governed by the subdifferential of a convex function. We also show that the extremal solutions set is dense in the solutions set of the original problem.     

Relaxation Results for Hybrid Inclusions

The Filippov–Wa?ewski relaxation theorem describes when the set of solutions to a differential inclusion is dense in the set of solutions to the relaxed (convexified) differential inclusion. This paper establishes relaxation results for a broad range of hybrid systems which combine differential inclusions, difference inclusions, and constraints on the continuous and discrete motions induced by these inclusions. The relaxation results are used to deduce continuous dependence on initial conditions of the sets of solutions to hybrid systems.     

Epi-convergence Properties of Smoothing by Infimal Convolution

This paper concerns smoothing by infimal convolution for two large classes of functions: convex, proper and lower semicontinous as well as for (the nonconvex class of) convex-composite functions. The smooth approximations are constructed so that they epi-converge (to the underlying nonsmooth function) and fulfill a desirable property with respect to graph convergence of the gradient mappings to the subdifferential of the original function under reasonable assumptions. The close connection between epi-convergence of the smoothing functions and coercivity properties of the smoothing kernel is established.     

Quasilinear elliptic inclusions of hemivariational type: Extremality and compactness of the solution set

We consider the Dirichlet boundary value problem for an elliptic inclusion governed by a quasilinear elliptic operator of Leray-Lions type and a multivalued term which is given by the difference of Clarke's generalized gradient of some locally Lipschitz function and the subdifferential of some convex function. Problems of this kind arise, e.g., in mechanical models described by nonconvex and nonsmooth energy functionals that result from nonmonotone, multivalued constitutive laws. Our main goal is to characterize the solution set of the problem under consideration. In particular we are going to prove that the solution set possesses extremal elements with respect to the underlying natural partial ordering of functions, and that the solution set is compact. The main tools used in the proofs are abstract results on pseudomonotone operators, truncation, and special test function techniques, Zorn's lemma as well as tools from nonsmooth analysis.     

Finite perturbation of convex programs

This paper concerns a characterization of the finite-perturbation property of a convex program. When this property holds, finite perturbation of the objective function of a convex program leads to a solution of the original problem which minimizes the perturbation function over the set of solutions of the original problem. This generalizes a finite-termination property of the proximal point algorithm for linear programs and characterizes finite Tikhonov regularization of convex programs.This material is based on research supported by National Science Foundation Grants DCR-8521228 and CCR-8723091 and Air Force Office of Scientific Research Grants AFOSR-86-0172 and AFOSR-89-0410.     

Subdifferential set of the supremum of lower semi-continuous convex functions and the conical hull intersection property

In this paper we give some characterizations for the subdifferential set of the supremum of an arbitrary (possibly infinite) family of proper lower semi-continuous convex functions. This is achieved by means of formulae depending exclusively on the (exact) subdifferential sets and the normal cones to the domains of the involved functions. Our approach makes use of the concept of conical hull intersection property (CHIP, for short). It allows us to establish sufficient conditions guarantying explicit representations for this subdifferential set at any point of the effective domain of the supremum function. Research supported by grant SB2003-0344 of SEUI (MEC), Spain.     

Characterizations of the Solution Sets of Convex Programs and Variational Inequality Problems

For a convex program in a normed vector space with the objective function admitting the Gateaux derivative at an optimal solution, we show that the solution set consists of the feasible points lying in the hyperplane whose normal vector equals the Gateaux derivative. For a general continuous convex program, a feasible point is an optimal solution iff it lies in a hyperplane with a normal vector belonging to the subdifferential of the objective function at this point. In several cases, the solution set of a variational inequality problem is shown to coincide with the solution set of a convex program with its dual gap function as objective function, while the mapping involved can be used to express the above normal vectors.The research was supported by the National Science Council of the Republic of China. The authors are grateful to the referees for valuable comments and constructive suggestions.     

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...     

序扰动多目标规划的锥次微分稳定性

对于局部凸拓扑向量空间的多目标规划问题,本文研究并得到当确定空间序的控制锥受扰动,它们的锥有效点（解）集和锥弱有效点（解）集分别在锥次微分和锥弱次微分意义下的稳定性结果.     

Continuous Gradient Projection Method in Hilbert Spaces

This paper is concerned with the asymptotic analysis of the trajectories of some dynamical systems built upon the gradient projection method in Hilbert spaces. For a convex function with locally Lipschitz gradient, it is proved that the orbits converge weakly to a constrained minimizer whenever it exists. This result remains valid even if the initial condition is chosen out of the feasible set and it can be extended in some sense to quasiconvex functions. An asymptotic control result, involving a Tykhonov-like regularization, shows that the orbits can be forced to converge strongly toward a well-specified minimizer. In the finite-dimensional framework, we study the differential inclusion obtained by replacing the classical gradient by the subdifferential of a continuous convex function. We prove the existence of a solution whose asymptotic properties are the same as in the smooth case.     

Characterizations of the solution set for non-essentially quasiconvex programming

Characterizations of the solution set in terms of subdifferentials play an important role in research of mathematical programming. Previous characterizations are based on necessary and sufficient optimality conditions and invariance properties of subdifferentials. Recently, characterizations of the solution set for essentially quasiconvex programming in terms of Greenberg–Pierskalla subdifferential are studied by the authors. Unfortunately, there are some examples such that these characterizations do not hold for non-essentially quasiconvex programming. As far as we know, characterizations of the solution set for non-essentially quasiconvex programming have not been studied yet. In this paper, we study characterizations of the solution set in terms of subdifferentials for non-essentially quasiconvex programming. For this purpose, we use Martínez–Legaz subdifferential which is introduced by Martínez–Legaz as a special case of c-subdifferential by Moreau. We derive necessary and sufficient optimality conditions for quasiconvex programming by means of Martínez–Legaz subdifferential, and, as a consequence, investigate characterizations of the solution set in terms of Martínez–Legaz subdifferential. In addition, we compare our results with previous ones. We show an invariance property of Greenberg–Pierskalla subdifferential as a consequence of an invariance property of Martínez–Legaz subdifferential. We give characterizations of the solution set for essentially quasiconvex programming in terms of Martínez–Legaz subdifferential.     

Subdifferential estimate of the directional derivative,optimality criterion and separation principles

We provide an inequality relating the radial directional derivative and the subdifferential of proper lower semicontinuous functions, which extends the known formula for convex functions. We show that this property is equivalent to other subdifferential properties of Banach spaces, such as controlled dense subdifferentiability, optimality criterion, mean value inequality and separation principles. As an application, we obtain a first-order sufficient condition for optimality, which extends the known condition for differentiable functions in finite-dimensional spaces and which amounts to the maximal monotonicity of the subdifferential for convex lower semicontinuous functions. Finally, we establish a formula describing the subdifferential of the sum of a convex lower semicontinuous function with a convex inf-compact function in terms of the sum of their approximate ?-subdifferentials. Such a formula directly leads to the known formula relating the directional derivative of a convex lower semicontinuous function to its approximate ?-subdifferential.