Convex Set-valued Maps Research Articles

On metric subregularity for convex constraint systems by primal equivalent conditions

In this paper, we mainly study metric subregularity for a convex constraint system defined by a convex set-valued mapping and a convex constraint subset. The main work is to provide several primal equivalent conditions for metric subregularity by contingent cone and graphical derivative. Further it is proved that these primal equivalent conditions can characterize strong basic constraint qualification of convex constraint system given by Zheng and Ng [SIAM J. Optim., 18(2007), pp. 437-460].

Fractional differential equations and inclusions with semiperiodic and three-point boundary conditions

In this article, we investigate the existence of solutions for boundary value problems of fractional differential equations and inclusions with semiperiodic and three-point boundary conditions. The existence results for equations are obtained by applying Banach’s contraction mapping principle, Schaefer-type fixed point theorem, Leray-Schauder degree theory, Krasnoselskii’s fixed point theorem, and Leray-Schauder nonlinear alternative, whereas the existence of solutions for convex and nonconvex set-valued maps (inclusion case) is shown via nonlinear alternative of Leray-Schauder type for multivalued maps and Wegrzyk’s fixed point theorem for generalized contractions, respectively. We emphasize that a variety of fixed point theorems are used to obtain different existence criteria for the problems at hand. Several examples are discussed for illustration of the obtained results. Moreover, an interesting observation related to symmetric second-order three-point boundary value problems is presented.

Global proper efficiency and vector optimization with cone-arcwise connected set-valued maps

This paper deals withthe characteristics of global proper efficient points and theoptimality conditions of vector optimization problems involvinggeneralized convex set-valued maps. Several equivalent properties ofglobal proper efficient points are proposed. Utilizing cone-directedcontingent derivative, it presents the unified necessary andsufficient optimality conditions for global proper efficient elementin vector optimization problem with cone-arcwise connectedset-valued mapping.

Prederivatives of convex set-valued maps and applications to set optimization problems

We investigate the existence of several kinds of prederivatives for set-valued mappings enjoying convex properties. Then, we established both necessary and sufficient optimality conditions, involvi...

A derivative for complex Lipschitz maps with generalised Cauchy–Riemann equations

We introduce the Lipschitz derivative or the L-derivative of a locally Lipschitz complex map: it is a Scott continuous, compact and convex set-valued map that extends the classical derivative to the bigger class of locally Lipschitz maps and allows an extension of the fundamental theorem of calculus and a new generalisation of Cauchy–Riemann equations to these maps, which form a continuous Scott domain. We show that a complex Lipschitz map is analytic in an open set if and only if its L-derivative is a singleton at all points in the open set. The calculus of the L-derivative for sum, product and composition of maps is derived. The notion of contour integration is extended to Scott continuous, non-empty compact, convex valued functions on the complex plane, and by using the L-derivative, the fundamental theorem of contour integration is extended to these functions.

CONVEXITY AND GLOBAL WELL-POSEDNESS IN SET-OPTIMIZATION

Well-posedness for vector optimization problems has been extensively studied. More recently, some attempts to extend thee results to set-valued optimization have been proposed, mainly applying some scalarization. In this paper we propose a new definition of global well-posedness for set-optimization problems. Using an embedding technique proposed by Kuroiwa and Nuriya (2006), we prove well-posedness property of a class of generalized convex set-valued maps.

Split Algorithms for New Implicit Feasibility Null-Point Problems

Inspired by the very recent work by Noor and Noor (9) and given a closed convex set-valued mapping C, we propose a split algorithm for solving the problem of finding an element xwhich is a zero of a given maximal monotone operator T such that its image, Ax � , under a linear operator, A, is in a closed convex set C(x � ). Then, we present two strong convergence results and state some examples as applications. The ideas and techniques of this paper may motivate the readers to discover some novel and innovative applications of the implicit split feasibility problems in various branches of pu re and applied sciences.

On selections of generalized convex set-valued maps

We obtain selections of generalized convex set-valued maps satisfying certain functional equations. As a consequence there follow some results on stability for functional equations on bisymmetric grupoids.

Strongly convex set-valued maps

We introduce the notion of strongly \(t\)-convex set-valued maps and present some properties of it. In particular, a Bernstein–Doetsch and Sierpinski-type theorems for strongly midconvex set-valued maps, as well as a Kuhn-type result are obtained. A representation of strongly \(t\)-convex set-valued maps in inner product spaces and a characterization of inner product spaces involving this representation is given. Finally, a connection between strongly convex set-valued maps and strongly convex sets is presented.

The Relationship between Two Kinds of Generalized Convex Set-Valued Maps in Real Ordered Linear Spaces

A new notion of the ic-cone convexlike set-valued map characterized by the algebraic interior and the vector closure is introduced in real ordered linear spaces. The relationship between the ic-cone convexlike set-valued map and the nearly cone subconvexlike set-valued map is established. The results in this paper generalize some known results in the literature from locally convex spaces to linear spaces.

DC Optimization Approach to Metric Regularity of Convex Multifunctions with Applications to Infinite Systems

The paper develops a new approach to the study of metric regularity and related well-posedness properties of convex set-valued mappings between general Banach spaces by reducing them to unconstrained minimization problems with objectives given as the difference of convex (DC) functions. In this way, we establish new formulas for calculating the exact regularity bound of closed and convex multifunctions and apply them to deriving explicit conditions ensuring well-posedness of infinite convex systems described by inequality and equality constraints.

On subdifferential calculus for set-valued mappings and optimality conditions

The aim of this paper is to establish formulas for the subdifferentials of the sum and the composition of convex set-valued mappings under the Attouch–Brézis qualification condition. An application to a general set-valued optimization problem is considered.

Stationary Set Analysis for PD Controlled Mechanical Systems

This brief presents a comprehensive method to analyze the stationary set of the high-dimensional mechanical systems with proportional-derivative (PD) control. The system behavior is characterized by Filippov's differential inclusion, in which the discontinuous disturbance mainly caused by static friction is modeled by a convex set-valued map with respect to the velocity. The stationary set of the closed-loop system is obtained by calculating the static solution in the sense of Filippov. The stationary set is also proved to be asymptotically stable, while the asymptotic stability cannot be obtained for proportional-integral-derivative (PID) controlled systems. Finally, the experiments on a three-axis rotation table are carried out to illustrate our analysis results.

On linear selections of convex set-valued maps

We study continuous subadditive set-valued maps taking points of a linear space X to convex compact subsets of a linear space Y. The subadditivity means that φ(x 1 + x 2) ⊂ φ(x 1) + φ(x 2). We characterize all pairs of locally convex spaces (X, Y) for which any such map has a linear selection, i.e., there exists a linear operator A: X → Y such that Ax ∈ φ(x), x ∈ X. The existence of linear selections for a class of subadditive maps generated by differences of a continuous function is proved. This result is applied to the Lipschitz stability problem for linear operators in Banach spaces.

On $\epsilon$-OPTIMALITY CONDITIONS FOR CONVEX SET-VALUED OPTIMIZATION PROBLEMS

In this paper, $\epsilon$-subgradients for convex set-valued maps are defined. We prove an existence theorem for $\epsilon$-subgradients of convex set-valued maps. Also, we give necessary $\epsilon$- optimality conditions for an $\epsilon$-solution of a convex set-valued optimization problem (CSP). Moreover, using the single-valued function induced from the set-valued map, we obtain theorems describing the $\epsilon$-subgradient sum formula for two convex set-valued maps, and then give necessary and sufficient $\epsilon$-optimality conditions for the problem (CSP).

On -OPTIMALITY CONDITIONS FOR CONVEX SET-VALUED OPTIMIZATION PROBLEMS

In this paper, $\epsilon$-subgradients for convex set-valued maps are defined. We prove an existence theorem for $\epsilon$-subgradients of convex set-valued maps. Also, we give necessary $\epsilon$- optimality conditions for an $\epsilon$-solution of a convex set-valued optimization problem (CSP). Moreover, using the single-valued function induced from the set-valued map, we obtain theorems describing the $\epsilon$-subgradient sum formula for two convex set-valued maps, and then give necessary and sufficient $\epsilon$-optimality conditions for the problem (CSP).

A metric on the space of finite measures with an application to fixed point theory

In this work, we give a metric function on the space of finite measures on a compact metric space, which has some interesting metric properties. As an application, we establish the existence of essential components of the set of fixed points for every upper semicontinuous convex set-valued mapping of a compact convex subset of the space of finite measures into itself.

Optimality Conditions for the Difference of Convex Set-Valued Mappings

Using the concept of subdifferential of cone-convex set valued mappings recently introduced by Baier and Jahn J. Optimiz. Theory Appl. 100 (1999), 233–240, we give necessary optimality conditions for nonconvex multiobjective optimization problems. An example illustrating the usefulness of our results is also given.

Optimization with set relations: conjugate duality

The aim of this article is to develop a conjugate duality theory, based on set relation approach, for convex set-valued maps. The basic idea is to understand a convex set-valued map as a function with values in the space of closed convex subsets of . The usual inclusion of sets provides a natural ordering relation in this space. Infimum and supremum with respect to this ordering relation can be expressed with the aid of union and intersection. Our main result is a strong duality assertion formulated along the lines of classical duality theorems for extended real-valued convex functions.

Proximal proper efficiency in set-valued optimization

In this paper, we introduce the concept of cone semilocal convex and cone semilocal convexlike set-valued maps and obtain characterization of these maps in terms of locally star-shaped sets. We derive an alternative theorem involving cone semilocal convexlike set-valued maps under the assumption of closedness of the translation of the image set of the map by the cone under consideration. We introduce proximal proper efficiency for a set-valued optimization problem in finite-dimensional spaces and obtain certain scalarization and Lagrange multiplier theorems. In the end, we consider a Lagrange form of dual and establish weak and strong duality theorems.