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.     

\epsilon -Henig proper efficiency of set-valued optimization problems in real ordered linear spaces

The aim of this paper is to investigate \(\epsilon \) -Henig proper efficiency of set-valued optimization problems in linear spaces. Firstly, a new notion of \(\epsilon \) -Henig properly efficient point is introduced in linear spaces. Secondly, scalarization theorems of set-valued optimization problems are established in the sense of \(\epsilon \) -Henig proper efficiency. Finally, under the assumption of generalized cone subconvexlikeness, Lagrange multiplier theorems are obtained. Our results generalize some known results in the literature from topological spaces to linear spaces.     

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.     

The semi-<Emphasis Type="Italic">E</Emphasis> cone convex set-valued map and its applications

In this paper, firstly, a new notion of the semi-E cone convex set-valued map is introduced in locally convex spaces. Secondly, without any convexity assumption, we investigate the existence conditions of the weakly efficient element of the set-valued optimization problem. Finally, under the assumption of the semi-E cone convexity of set-valued maps, we obtain that the local weakly efficient element of the set-valued optimization problem is the weakly efficient element. We also give some examples to illustrate our results.     

Benson Proper Efficiency in the Vector Optimization of Set-Valued Maps

This paper extends the concept of cone subconvexlikeness of single-valued maps to set-valued maps and presents several equivalent characterizations and an alternative theorem for cone-subconvexlike set-valued maps. The concept and results are then applied to study the Benson proper efficiency for a vector optimization problem with set-valued maps in topological vector spaces. Two scalarization theorems and two Lagrange multiplier theorems are established. After introducing the new concept of proper saddle point for an appropriate set-valued Lagrange map, we use it to characterize the Benson proper efficiency. Lagrange duality theorems are also obtained     

A Midpoint Method for Generalized Equations Under Mild Differentiability Condition

The aim of this study is the approximation of a solution x ^? of the generalized equation 0∈f(x)+F(x) in Banach spaces, where f is a single function whose second order Fréchet derivative ?² f verifies an Hölder condition, and F stands for a set-valued map with closed graph. Using a fixed point theorem and proceeding by induction under the pseudo-Lipschitz property of F, we obtain a sequence defined by a midpoint formula whose convergence to x ^? is superquadratic. Taking a weaker condition, we present the result obtained when ?² f satisfies a center-Hölder conditioning.     

Operator inclusions and quasi-variational inequalities

The operator inclusion 0 ∈ A(x) + N(x) is studied. Themain results are concerned with the case where A is a bounded monotone-type operator from a reflexive space to its dual and N is a cone-valued operator. A criterion for this inclusion to have no solutions is obtained. Additive and homotopy-invariant integer characteristics of set-valued maps are introduced. Applications to the theory of quasi-variational inequalities with set-valued operators are given.     

Spaces of lower semicontinuous set-valued maps I

We introduce a lower semicontinuous analog, L ^?(X), of the well-studied space of upper semicontinuous set-valued maps with nonempty compact interval images. Because the elements of L ^?(X) contain continuous selections, the space C(X) of real-valued continuous functions on X can be used to establish properties of L ^?(X), such as the two interrelated main theorems. The first of these theorems, the Extension Theorem, is proved in this Part I. The Extension Theorem says that for binormal spaces X and Y, every bimonotone homeomorphism between C(X) and C(Y) can be extended to an ordered homeomorphism between L ^?(X) and L ^?(Y). The second main theorem, the Factorization Theorem, is proved in Part II. The Factorization Theorem says that for binormal spaces X and Y, every ordered homeomorphism between L ^?(X) and L ^?(Y) can be characterized by a unique factorization.     

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.     

Quasiregular mappings to generalized manifolds

We establish the basic analytic and geometric properties of quasiregular maps f: ω → X, where ω ? ? ⁿ is a domain and X is a generalized n-manifold with a suitably controlled geometry. Generalizing the classical Väisälä and Poletsky inequalities, our main theorem shows that the path family method applies to these maps.     

Subdifferential and optimality conditions for the difference of set-valued mappings

In this paper, an existence theorem of the subgradients for set-valued mappings, which introduced by Borwein (Math Scand 48:189?C204, 1981), and relations between this subdifferential and the subdifferential introduced by Baier and Jahn (J Optim Theory Appl 100:233?C240, 1999), are obtained. By using the concept of this subdifferential, the sufficient optimality conditions for generalized D.C. multiobjective optimization problems are established. And the necessary optimality conditions, which are the generalizations of that in Gadhi (Positivity 9:687?C703, 2005), are also established. Moreover, by using a special scalarization function, a real set-valued optimization problem is introduced and the equivalent relations between the solutions are proved for the real set-valued optimization problem and a generalized D.C. multiobjective optimization problem.     

A saddle point characterization of efficient solutions for interval optimization problems

In this article, we attempt to characterize efficient solutions of constrained interval optimization problems. Towards this aim, at first, we study a scalarization characterization to capture efficient solutions. Then, with the help of saddle point of a newly introduced Lagrangian function, we investigate efficient solutions of an interval optimization problem. Several parts of the results are supported with numerical and pictorial illustration.     

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.     

The convolution method for fourier expansions. The case of generalized transmission conditions of crack (screen) type in piecewise inhomogeneous media

We consider boundary value problems for the equation ? _x (K ? _x ?) + KL[?] = 0 in the space R ⁿ with generalized transmission conditions of the type of a strongly permeable crack or a weakly permeable screen on the surface x = 0. (Here L is an arbitrary linear differential operator with respect to the variables y ₁, …, y _n?1.) The coefficients K(x) > 0 are monotone functions of certain classes in the regions separated by the screen x = 0. The desired solution has arbitrary given singular points and satisfies a sufficiently weak condition at infinity.We derive formulas expressing the solutions of the above-mentioned problems in the form of simple quadratures via the solutions of the considered equation with a constant coefficient K for given singular points in the absence of a crack or a screen. In particular, the obtained formulas permit one to solve boundary value problems with generalized transmission conditions for equations with functional piecewise continuous coefficients in the framework of the theory of harmonic functions.     

On generalized Heawood inequalities for manifolds: a van Kampen–Flores-type nonembeddability result

The fact that the complete graph K₅ does not embed in the plane has been generalized in two independent directions. On the one hand, the solution of the classical Heawood problem for graphs on surfaces established that the complete graph K_n embeds in a closed surface M (other than the Klein bottle) if and only if (n?3)(n?4) ≤ 6b₁(M), where b₁(M) is the first Z₂-Betti number of M. On the other hand, van Kampen and Flores proved that the k-skeleton of the n-dimensional simplex (the higher-dimensional analogue of K_n+1) embeds in R^2k if and only if n ≤ 2k + 1.Two decades ago, Kühnel conjectured that the k-skeleton of the n-simplex embeds in a compact, (k ? 1)-connected 2k-manifold with kth Z₂-Betti number b_k only if the following generalized Heawood inequality holds: ( _k+1 ^n?k?1 ) ≤ ( _k+1 ^2k+1 )b_k. This is a common generalization of the case of graphs on surfaces as well as the van Kampen–Flores theorem.In the spirit of Kühnel’s conjecture, we prove that if the k-skeleton of the n-simplex embeds in a compact 2k-manifold with kth Z₂-Betti number bk, then n ≤ 2b_k( _k ^2k+2 )+2k+4. This bound is weaker than the generalized Heawood inequality, but does not require the assumption that M is (k?1)-connected. Our results generalize to maps without q-covered points, in the spirit of Tverberg’s theorem, for q a prime power. Our proof uses a result of Volovikov about maps that satisfy a certain homological triviality condition.     

Hadamard well-posedness for a set-valued optimization problem

In this paper, we introduce a kind of Hadamard well-posedness for a set-valued optimization problem. By virtue of a scalarization function, we obtain some relationships between weak ${(\varepsilon, e)}$ -minimizers of the set-valued optimization problem and ${\varepsilon}$ -approximate solutions of a scalar optimization problem. Then, we establish a scalarization theorem of P.K. convergence for sequences of set-valued mappings. Based on these results, we also derive a sufficient condition of Hadamard well-posedness for the set-valued optimization problem.     

Characterizations of improvement sets via quasi interior and applications in vector optimization

In this paper, we first give some characterizations of improvement sets via quasi interior. Furthermore, as applications of these characterizations, we establish an alternative theorem via improvement sets and quasi interior, and then obtain a scalarization result of weak \(E\)-efficient solutions defined by improvement sets and quasi interior for vector optimization problems with set-valued maps. Moreover, we also present some examples to illustrate the main conditions and results.     

On implicit function theorems at abnormal points

We consider the equation F(x, σ) = 0, x ∈ K, in which σ is a parameter and x is an unknown variable taking values in a specified convex cone K lying in a Banach space X. This equation is investigated in a neighborhood of a given solution (x*, σ*), where Robinson’s constraint qualification may be violated. We introduce the 2-regularity condition, which is considerably weaker than Robinson’s constraint qualification; assuming that it is satisfied, we obtain an implicit function theorem for this equation. The theorem is a generalization of the known implicit function theorems even in the case when the cone K coincides with the whole space X.     

Mixed problem for the wave equation with integrable potential in the case of two-point boundary conditions of distinct orders

We study a mixed problem for the wave equation with integrable potential and with two-point boundary conditions of distinct orders for the case in which the corresponding spectral problem may have multiple spectrum. Based on the resolvent approach in the Fourier method and the Krylov convergence acceleration trick for Fourier series, we obtain a classical solution u(x, t) of this problem under minimal constraints on the initial condition u(x, 0) = ?(x). We use the Carleson–Hunt theorem to prove the convergence almost everywhere of the formal solution series in the limit case of ?(x) ∈ L _p[0, 1], p > 1, and show that the formal solution is a generalized solution of the problem.     

On the divergence of trigonometric Fourier series in classes <Emphasis Type="Italic">ϕ</Emphasis>(<Emphasis Type="Italic">L</Emphasis>) close to <Emphasis Type="Italic">L</Emphasis>

We show the unimprovability of a theorem on sufficient convergence conditions for the trigonometric Fourier series of a function in classes ?(L) in the case when the class ?(L) is “close” to L.