Unifying local–global type properties in vector optimization

It is well-known that all local minimum points of a semistrictly quasiconvex real-valued function are global minimum points. Also, any local maximum point of an explicitly quasiconvex real-valued function is a global minimum point, provided that it belongs to the intrinsic core of the function’s domain. The aim of this paper is to show that these “local min–global min” and “local max–global min” type properties can be extended and unified by a single general local–global extremality principle for certain generalized convex vector-valued functions with respect to two proper subsets of the outcome space. For particular choices of these two sets, we recover and refine several local–global properties known in the literature, concerning unified vector optimization (where optimality is defined with respect to an arbitrary set, not necessarily a convex cone) and, in particular, classical vector/multicriteria optimization.     

On the functions with pseudoconvex sublevel sets and optimality conditions

A new class of generalized convex functions, called the functions with pseudoconvex sublevel sets, is defined. They include quasiconvex ones. A complete characterization of these functions is derived. Further, it is shown that a continuous function admits pseudoconvex sublevel sets if and only if it is quasiconvex. Optimality conditions for a minimum of the nonsmooth nonlinear programming problem with inequality, equality and a set constraints are obtained in terms of the lower Hadamard directional derivative. In particular sufficient conditions for a strict global minimum are given where the functions have pseudoconvex sublevel sets.     

Generalized Monotone Operators on Dense Sets

In the present work we show that the local generalized monotonicity of a lower semicontinuous set-valued operator on some certain type of dense sets ensures the global generalized monotonicity of that operator. We achieve this goal gradually by showing at first that the lower semicontinuous set-valued functions of one real variable, which are locally generalized monotone on a dense subsets of their domain are globally generalized monotone. Then, these results are extended to the case of set-valued operators on arbitrary Banach spaces. We close this work with a section on the global generalized convexity of a real valued function, which is obtained out of its local counterpart on some dense sets.     

拟凸函数判别准则的一个注记

我们在上半连续的条件下,给出了拟凸函数的一个新的判别准则,即：凸集上的一个上半连续函数是拟凸的充分必要条件是这个函数是中间拟凸的。     

Generalized convexities of lower semicontinuous functions

In this paper a general theorem on the replacement of the condition “for all λ” in the definition of generalized convexity properties of lower semicontinuous functions by the condition “there exists a λ” is shown. This result can be applied to a number of special kinds of convexity and completes, for instance, studies of Behbikgeb concerning (explicitly) quasiconvex functions.     

Adaptive subgradient method for the split quasi-convex feasibility problems

In this paper, we consider a type of the celebrated convex feasibility problem, named as split quasi-convex feasibility problem (SQFP). The SQFP is to find a point in a sublevel set of a quasi-convex function in one space and its image under a bounded linear operator is contained in a sublevel set of another quasi-convex function in the image space. We propose a new adaptive subgradient algorithm for solving SQFP problem. We also discuss the convergence analyses for two cases: the first case where the functions are upper semicontinuous in the setting of finite dimensional, and the second case where the functions are weakly continuous in the infinite-dimensional settings. Finally some numerical examples in order to support the convergence results are given.     

Minimizing Irregular Convex Functions: Ulam Stability for Approximate Minima

The main concern of this article is to study Ulam stability of the set of ε-approximate minima of a proper lower semicontinuous convex function bounded below on a real normed space X, when the objective function is subjected to small perturbations (in the sense of Attouch & Wets). More precisely, we characterize the class all proper lower semicontinuous convex functions bounded below such that the set-valued application which assigns to each function the set of its ε-approximate minima is Hausdorff upper semi-continuous for the Attouch–Wets topology when the set $\mathcal{C}(X)$ of all the closed and nonempty convex subsets of X is equipped with the Hausdorff topology. We prove that a proper lower semicontinuous convex function bounded below has Ulam-stable ε-approximate minima if and only if the boundary of any of its sublevel sets is bounded.     

An extension of proximal methods for quasiconvex minimization on the nonnegative orthant

In this paper we propose an extension of proximal methods to solve minimization problems with quasiconvex objective functions on the nonnegative orthant. Assuming that the function is bounded from below and lower semicontinuous and using a general proximal distance, it is proved that the iterations given by our algorithm are well defined and stay in the positive orthant. If the objective function is quasiconvex we obtain the convergence of the iterates to a certain set which contains the set of optimal solutions and convergence to a KKT point if the function is continuously differentiable and the proximal parameters are bounded. Furthermore, we introduce a sufficient condition on the proximal distance such that the sequence converges to an optimal solution of the problem.     

On extremal points of quasiconvex functions

In this paper we examine the linear sectionwise relative minimums of a quasiconvex function and give a sufficient condition for quasiconvex functions to have a strict global minimum on an open convex set.     

Characterization of R-Evenly Quasiconvex Functions

A function defined on a locally convex space is called evenly quasiconvex if its level sets are intersections of families of open half-spaces. Furthermore, if the closures of these open halfspaces do not contain the origin, then the function is called R-evenly quasiconvex. In this note, R-evenly quasiconvex functions are characterized as those evenly-quasiconvex functions that satisfy a certain simple relation with their lower semicontinuous hulls.     

关于局部极小值点为整体极小值点的函数的一点注记

本引进了一种广义严格拟凸函数的定义,给出了它的一个刻画,证明 了局部极小值点为整体极小值点的函数与本定义的广义严格拟凸函数的等价性。     

Normal Characterization of the Main Classes of Quasiconvex Functions

In this article we explore the concept of the normal cone to the sublevel sets (or strict sublevel sets) of a function. By slightly modifying the original definition of Borde and Crouzeix, we obtain here a new (but strongly related to the already existent) notion of a normal operator. This technique turns out to be appropriate in Quasiconvex Analysis since it allows us to reveal characterizations of the various classes of quasiconvex functions in terms of the generalized quasimonotonicity of their `normal' multifunctions.     

Subdifferential Properties of Quasiconvex and Pseudoconvex Functions: Unified Approach

In this paper, we are mainly concerned with the characterization of quasiconvex or pseudoconvex nondifferentiable functions and the relationship between those two concepts. In particular, we characterize the quasiconvexity and pseudoconvexity of a function by mixed properties combining properties of the function and properties of its subdifferential. We also prove that a lower semicontinuous and radially continuous function is pseudoconvex if it is quasiconvex and satisfies the following optimality condition: 0f(x)f has a global minimum at x. The results are proved using the abstract subdifferential introduced in Ref. 1, a concept which allows one to recover almost all the subdifferentials used in nonsmooth analysis.     

锥拟凸集值映射多目标优化的解集的连通性

在一般拓扑向量空间中,本文引进了一类锥进凸（锥严格拟凸）集值映射。并在目标映射是锥拟凸（锥严格拟凸）和上半连续的条件下,利用Minkowski泛函,证明了弱有效解（有效解）集是连通的。文[9]中的结论是本文所得结果大目标映射为单值和拓扑空间为格的情况下的特例。     

Optimality Conditions for Isolated Local Minimum of Nonsmooth Multiobjective Problems

This article is devoted to the study of Fritz John and strong Kuhn-Tucker necessary conditions for properly efficient solutions, efficient solutions and isolated efficient solutions of a nonsmooth multiobjective optimization problem involving inequality and equality constraints and a set constraints in terms of the lower Hadamard directional derivative. Sufficient conditions for the existence of such solutions are also provided where the involved functions have pseudoconvex sublevel sets. Our results are based on the concept of pseudoconvex sublevel sets. The functions with pseudoconvex sublevel sets are a class of generalized convex functions that include quasiconvex functions.     

Maximal Monotone Multifunctions of Brøndsted–Rockafellar Type

We consider whether the “inequality-splitting” property established in the Brøndsted–Rockafellar theorem for the subdifferential of a proper convex lower semicontinuous function on a Banach space has an analog for arbitrary maximal monotone multifunctions. We introduce the maximal monotone multifunctions of type (ED), for which an “inequality-splitting” property does hold. These multifunctions form a subclass of Gossez"s maximal monotone multifunctions of type (D); however, in every case where it has been proved that a multifunction is maximal monotone of type (D) then it is also of type (ED). Specifically, the following maximal monotone multifunctions are of type (ED): ? ultramaximal monotone multifunctions, which occur in the study of certain nonlinear elliptic functional equations; ? single-valued linear operators that are maximal monotone of type (D); ? subdifferentials of proper convex lower semicontinuous functions; ? “subdifferentials” of certain saddle-functions. We discuss the negative alignment set of a maximal monotone multifunction of type (ED) with respect to a point not in its graph – a mysterious continuous curve without end-points lying in the interior of the first quadrant of the plane. We deduce new inequality-splitting properties of subdifferentials, almost giving a substantial generalization of the original Brøndsted–Rockafellar theorem. We develop some mathematical infrastructure, some specific to multifunctions, some with possible applications to other areas of nonlinear analysis: ? the formula for the biconjugate of the pointwise maximum of a finite set of convex functions – in a situation where the “obvious” formula for the conjugate fails; ? a new topology on the bidual of a Banach space – in some respects, quite well behaved, but in other respects, quite pathological; ? an existence theorem for bounded linear functionals – unusual in that it does not assume the existence of any a priori bound; ? the 'big convexification" of a multifunction.

     

Stable generalization of convex functions

A kind of generalized convex functions is said to be stable with respect to some property (P) if this property is maintaincd during an arbitrary function from this class is disturbed by a linear functional with sufficiently small norm. Unfortunately. known generallzed convexities iike quasicunvexity, explicit quasiconvexity. and pseudoconvexity are not stable with respect to such optimization properties which are expected to be true by these generalizations, even if the domain ol the functions is compact. Therefore, we introduce the notion of s-quasiconvex functions. These functions are quasiconvex, explicitly quasicon vex. and pseudoconvex if they are continuously differentiable. Especially, the s-quasiconvexity is stable with respect to the following important properties: (Pl) all lower level sets are convex, (P2) each local minimum is a global minimum. and (P3) each stationary point is a global minimizer. In this paper, different aspects. of s–quasiconvexity and its stability are investigated.     

Subdifferential test for optimality

We provide a first-order necessary and sufficient condition for optimality of lower semicontinuous functions on Banach spaces using the concept of subdifferential. From the sufficient condition we derive that any subdifferential operator is monotone absorbing, hence maximal monotone when the function is convex.     

Optimality Conditions for Nonsmooth Equilibrium Problems via Hadamard Directional Derivative

In this paper, we consider a mathematical program with equilibrium constraints (MPEC) formulated as a mathematical program with complementarity constraints. We obtain necessary conditions of Fritz John (FJ) and Karush-Kuhn-Tucker (KKT) types for a nonsmooth (MPEC) problem in terms of the lower Hadamard directional derivative. In particular sufficient conditions for MPECs are given where the involved functions have pseudoconvex sublevel sets. The functions with pseudoconvex sublevel sets is a class of generalized convex functions that include quasiconvex functions.     

Extended radial epiderivatives of non-convex vector-valued maps and parametric quasiconvex programming

In this paper, we consider extended vector-valued mappings defined on a normed linear space. Based on the recent semicontinuous regularizations related to hypographical and/or epigraphical profile mappings of the considered function introduced, we define semicontinuous radial epiderivatives. We, then, demonstrate that the properties of these epiderivatives amount to properties of hypographical and/or epigraphical profile mappings of the corresponding difference quotient of the underlying function, which simplify fairly well the proofs in the radial epiderivative formulaes. In particular, we stress the impact of semicontinuity, hence, we characterize with new arguments the radial epiderivatives in terms of the suprema and/or infima of the interiorly radial cone of the hypograph and/or epigraph of the considered function. Finally, we obtain optimality conditions for general non-convex constrained vector optimization problems. We apply thereafter the obtained pattern to a parametric quasiconvex programming problem for which we derive necessary and sufficient optimality conditions that are not sensitive to perturbation at the nominal level, yielding henceforth more – and strong at least under asymptotically regular constraints – information than the recent stability results obtained under additional conditions on the regularity of the normal cone to the adjusted sublevel sets of the underlying function.