The Lipschitzianity of convex vector and set-valued functions

It is well known that every scalar convex function is locally Lipschitz on the interior of its domain in finite dimensional spaces. The aim of this paper is to extend this result for both vector functions and set-valued mappings acting between infinite dimensional spaces with an order generated by a proper convex cone C. Under the additional assumption that the ordering cone C is normal, we prove that a locally C-bounded C-convex vector function is Lipschitz on the interior of its domain by two different ways. Moreover, we derive necessary conditions for Pareto minimal points of vector-valued optimization problems where the objective function is C-convex and C-bounded. Corresponding results are derived for set-valued optimization problems.     

Scalarization of Set-Valued Optimization Problems with Generalized Cone Subconvexlikeness in Real Ordered Linear Spaces

In real ordered linear spaces, an equivalent characterization of generalized cone subconvexlikeness of set-valued maps is firstly established. Secondly, under the assumption of generalized cone subconvexlikeness of set-valued maps, a scalarization theorem of set-valued optimization problems in the sense of ?-weak efficiency is obtained. Finally, by a scalarization approach, an existence theorem of ?-global properly efficient element of set-valued optimization problems is obtained. The results in this paper generalize and improve some known results in the literature.     

Nonlinear separation concerning <Emphasis Type="Italic">E</Emphasis>-optimal solution of constrained multi-objective optimization problems

This paper aims at investigating optimality conditions in terms of E-optimal solution for constrained multi-objective optimization problems in a general scheme, where E is an improvement set with respect to a nontrivial closed convex point cone with apex at the origin. In the case where E is not convex, nonlinear vector regular weak separation functions and scalar weak separation functions are introduced respectively to realize the separation between the two sets in the image space, and Lagrangian-type optimality conditions are established. These results extend and improve the convex ones in the literature.     

On convex hulls of compact sets of probability measures with countable supports

E. Michael and I. Namioka proved the following theorem. Let Y be a convex G _δ -subset of a Banach space E such that if K ? Y is a compact space, then its closed (in Y) convex hull is also compact. Then every lower semicontinuous set-valued mapping of a paracompact space X to Y with closed (in Y) convex values has a continuous selection. E. Michael asked the question: Is the assumption that Y is G _δ essential? In this note we give an affirmative answer to this question of Michael.     

Mappings of preserving <Emphasis Type="Italic">n</Emphasis>-distance one in <Emphasis Type="Italic">n</Emphasis>-normed spaces

We give a positive answer to the Aleksandrov problem in n-normed spaces under the surjectivity assumption. Namely, we show that every surjective mapping preserving n-distance one is affine, and thus is an n-isometry. This is the first time the Aleksandrov problem is solved in n-normed spaces with only the surjectivity assumption even in the usual case \(n=2\). Finally, when the target space is n-strictly convex, we prove that every mapping preserving two n-distances with an integer ratio is an affine n-isometry.     

The <Emphasis Type="Italic">p</Emphasis>-Harmonic Measure of Small Axially Symmetric Sets

Using the first eigenvalue/eigenvector pair of a singular eigenvalue problem (motivated by the Dirichlet eigenvalue problem for the Laplace-Beltrami operator on a spherical cap), we define certain nonnegative p-superharmonic and p-subharmonic functions on a convex cone which are singular at the vertex and vanish on the rest of the boundary. We use these functions to give upper and lower estimates of the p-harmonic measure near the vertex of the cone as well as the p-harmonic measure of a small spherical cap.     

On proximinality of convex sets in superspaces

In this paper, we show that a closed convex subset C of a Banach space is strongly proximinal (proximinal, resp.) in every Banach space isometrically containing it if and only if C is locally (weakly, resp.) compact. As a consequence, it is proved that local compactness of C is also equivalent to that for every Banach space Y isometrically containing it, the metric projection from Y to C is nonempty set-valued and upper semi-continuous.     

Strongly convex weakly complex Berwald metrics and real Landsberg metrics

Under the assumption that' is a strongly convex weakly Khler Finsler metric on a complex manifold M, we prove that F is a weakly complex Berwald metric if and only if F is a real Landsberg metric.This result together with Zhong(2011) implies that among the strongly convex weakly Kahler Finsler metrics there does not exist unicorn metric in the sense of Bao(2007). We also give an explicit example of strongly convex Kahler Finsler metric which is simultaneously a complex Berwald metric, a complex Landsberg metric,a real Berwald metric, and a real Landsberg metric.     

On vector variational-like inequalities and vector optimization problems with (<Emphasis Type="Italic">G</Emphasis>, <Emphasis Type="Italic">α</Emphasis>)-invexity

The aim of this paper is to study the relationship among Minty vector variational-like inequality problem, Stampacchia vector variational-like inequality problem and vector optimization problem involving (G, α)-invex functions. Furthermore, we establish equivalence among the solutions of weak formulations of Minty vector variational-like inequality problem, Stampacchia vector variational-like inequality problem and weak efficient solution of vector optimization problem under the assumption of (G, α)-invex functions. Examples are provided to elucidate our results.     

Uniform Convexity and Associate Spaces

We prove that the associate space of a generalized Orlicz space L^?(·) is given by the conjugate modular ?* even without the assumption that simple functions belong to the space. Second, we show that every weakly doubling Φ-function is equivalent to a doubling Φ-function. As a consequence, we conclude that L^?(·) is uniformly convex if ? and ?* are weakly doubling.     

Bernstein–Doetsch-type criteria for the continuity and Lipschitz continuity of convex set-valued mappings

The Bernstein–Doetsch criterion (for convex and midconvex functionals) has been repeatedly generalized to convex and midconvex set-valued mappings F: X → 2 ^Y ; continuity and local Lipschitz continuity were understood in the sense of the Hausdorff distance. However, all such results imposed restrictive additional boundedness-type conditions on the images F(x). In this paper, the Bernstein–Doetsch criterion is generalized to arbitrary convex and midconvex set-valued mappings acting on normed linear spaces X,Y.     

Preservation of <Emphasis Type="Italic">ω</Emphasis>-Categoricity In Expanding The Models of Weakly <Emphasis Type="Italic">o</Emphasis>-Minimal Theories

Studying the model-theoretic properties that are preserved under expansion of the models of countably categorical weakly o-minimal theories of finite convexity rank with convex unary predicates, we show that countable categoricity and convexity rank are among these properties.     

Numerical approximation of the solution in infinite dimensional global optimization using a representation formula

A non convex optimization problem, involving a regular functional J, on a closed and bounded subset S of a separable Hilbert space V is here considered. No convexity assumption is introduced. The solutions are represented by using a closed formula involving means of convenient random variables, analogous to Pincus (Oper Res 16(3):690–694, 1968). The representation suggests a numerical method based on the generation of samples in order to estimate the means. Three strategies for the implementation are examined, with the originality that they do not involve a priori finite dimensional approximation of the solution and consider a hilbertian basis or enumerable dense family of V. The results may be improved on a finite-dimensional subspace by an optimization procedure, in order to get higher-quality solutions. Numerical examples involving both classical situation and an engineering application issued from calculus of variations are presented and establish that the method is effective to calculate.     

Scalarization of \epsilon -Super Efficient Solutions of Set-Valued Optimization Problems in Real Ordered Linear Spaces

In this paper, we investigate the scalarization of \(\epsilon \) -super efficient solutions of set-valued optimization problems in real ordered linear spaces. First, in real ordered linear spaces, under the assumption of generalized cone subconvexlikeness of set-valued maps, a dual decomposition theorem is established in the sense of \(\epsilon \) -super efficiency. Second, as an application of the dual decomposition theorem, a linear scalarization theorem is given. Finally, without any convexity assumption, a nonlinear scalarization theorem characterized by the seminorm is obtained.     

A revised pre-order principle and set-valued Ekeland variational principles with generalized distances

In my former paper "A pre-order principle and set-valued Ekeland variational principle"(see [J. Math. Anal. Appl., 419, 904–937(2014)]), we established a general pre-order principle.From the pre-order principle, we deduced most of the known set-valued Ekeland variational principles(denoted by EVPs) in set containing forms and their improvements. But the pre-order principle could not imply Khanh and Quy's EVP in [On generalized Ekeland's variational principle and equivalent formulations for set-valued mappings, J. Glob. Optim., 49, 381–396(2011)], where the perturbation contains a weak τ-function, a certain type of generalized distances. In this paper, we give a revised version of the pre-order principle. This revised version not only implies the original pre-order principle,but also can be applied to obtain the above Khanh and Quy's EVP. In particular, we give several new set-valued EVPs, where the perturbations contain convex subsets of the ordering cone and various types of generalized distances.     

Complex convexity and monotonicity in quasi-Banach lattices

In this paper we study the monotonicity and convexity properties in quasi-Banach lattices. We establish relationship between uniform monotonicity, uniform ?-convexity, H-and PL-convexity. We show that if the quasi-Banach lattice E has α-convexity constant one for some 0 < α < ∞, then the following are equivalent: (i) E is uniformly PL-convex; (ii) E is uniformly monotone; and (iii) E is uniformly ?-convex. In particular, it is shown that if E has α-convexity constant one for some 0 < α < ∞ and if E is uniformly ?-convex of power type then it is uniformly H-convex of power type. The relations between concavity, convexity and monotonicity are also shown so that the Maurey-Pisier type theorem in a quasi-Banach lattice is proved.Finally we study the lifting property of uniform PL-convexity: if E is a quasi-Köthe function space with α-convexity constant one and X is a continuously quasi-normed space, then it is shown that the quasi-normed Köthe-Bochner function space E(X) is uniformly PL-convex if and only if both E and X are uniformly PL-convex.     

Duality between Fréchet differentiability and strong convexity

This paper revisits the duality between differentiability and strict or strong convexity under the Legendre–Fenchel transform \({f\mapsto f^*}\). Functions f defined on a Banach space X are considered. For a lower semicontinuous but not necessarily convex function f we relate essential Fréchet differentiability of the conjugate function f* to essential strong convexity of f.     

Revisiting <Emphasis Type="Italic">k</Emphasis>-sum optimization

In this paper, we address continuous, integer and combinatorial k-sum optimization problems. We analyze different formulations of this problem that allow to solve it through the minimization of a relatively small number of minisum optimization problems. This approach provides a general tool for solving a variety of k-sum optimization problems and at the same time, improves the complexity bounds of many ad-hoc algorithms previously reported in the literature for particular versions of this problem. Moreover, the results developed for k-sum optimization have been extended to the more general case of the convex ordered median problem, improving upon existing solution approaches.     

On representations of the feasible set in convex optimization

We consider the convex optimization problem \({\min_{\mathbf{x}} \{f(\mathbf{x}): g_j(\mathbf{x})\leq 0, j=1,\ldots,m\}}\) where f is convex, the feasible set \({\mathbf{K}}\) is convex and Slater’s condition holds, but the functions g _j ’s are not necessarily convex. We show that for any representation of \({\mathbf{K}}\) 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 ’s are convex. So in convex optimization, and as far as one is concerned with KKT points, what really matters is the geometry of \({\mathbf{K}}\) and not so much its representation.     

On merit functions for <Emphasis Type="Italic">p</Emphasis>-order cone complementarity problem

Merit function approach is a popular method to deal with complementarity problems, in which the complementarity problem is recast as an unconstrained minimization via merit function or complementarity function. In this paper, for the complementarity problem associated with p-order cone, which is a type of nonsymmetric cone complementarity problem, we show the readers how to construct merit functions for solving p-order cone complementarity problem. In addition, we study the conditions under which the level sets of the corresponding merit functions are bounded, and we also assert that these merit functions provide an error bound for the p-order cone complementarity problem. These results build up a theoretical basis for the merit method for solving p-order cone complementarity problem.