Lower Semicontinuity Property Research Articles

Locating network trees by a bilevel scheme

AbstractIn this paper we investigate how to choose an optimal position of a specific facility that is constrained to a network tree connecting some given demand points in a given area. A bilevel formulation is provided and existence results are given together with some properties when a density describes the construction cost of the networks in the area. This includes the presence of an obstacle or of free regions. To prove existence of a solution of the bilevel problem, that is framed in Euclidean spaces, a lower semicontinuity property is required. This is obtained proving an extension of Goła̧b’s theorem in the general setting of metric spaces, which allows for considering a density function.

Spectral Stability of the {overline{partial }}-Neumann Laplacian: Domain Perturbations

We study spectral stability of the {bar{partial }}-Neumann Laplacian on a bounded domain in {mathbb {C}}^n when the underlying domain is perturbed. In particular, we establish upper semi-continuity properties for the variational eigenvalues of the {bar{partial }}-Neumann Laplacian on bounded pseudoconvex domains in {mathbb {C}}^n, lower semi-continuity properties on pseudoconvex domains that satisfy property (P), and quantitative estimates on smooth bounded pseudoconvex domains of finite D’Angelo type in {mathbb {C}}^n.

Coepi-analysis of lower semicontinuity property of vector-valued mappings

Topological properties of coepigraphs of vector-valued mappings are investigated. Lower semicontinuous regularization of such mappings and its variational representation are received with the help of coepigraphs.

On the Existence of Weak Efficient Solutions of Nonconvex Vector Optimization Problems

We study vector optimization problems with solid non-polyhedral convex ordering cones, without assuming any convexity or quasiconvexity assumption. We state a Weierstrass-type theorem and existence results for weak efficient solutions for coercive and noncoercive problems. Our approach is based on a new coercivity notion for vector-valued functions, two realizations of the Gerstewitz scalarization function, asymptotic analysis and a regularization of the objective function. We define new boundedness and lower semicontinuity properties for vector-valued functions and study their properties. These new tools rely heavily on the solidness of the ordering cone through the notion of colevel and level sets. As a consequence of this approach, we improve various existence results from the literature, since weaker assumptions are required.

Common fixed point theorems for nonexpansive mappings using the lower semicontinuity property

Suppose that $E$ is a Banach space, $\tau $ a topology under which the norm of $E$ becomes $\tau $-lower semicontinuous and $\mathcal {S}$ a commuting family of $\tau $-continuous nonexpansive mappings defined on a $\tau $-compact convex subset $C$ of $E.

Lower Semicontinuity Results in Parametric Multivalued Weak Vector Equilibrium Problems and Applications

ABSTRACTThis article gives new sufficient conditions for the lower semicontinuity of the solution mapping of a parametric multivalued weak vector equilibrium problem with moving cones. A scalarizing approach, based on the signed distance function of Hiriart Urruty is used to discuss this lower semicontinuity property. The main results of the article are obtained under some assumptions different from those introduced earlier by previous linear and nonlinear scalarizing approaches. Some applications to the study of connectedness of weak solution sets of multivalued vector equilibrium problems are given.

Gamma Convergence of a Family of Surface–Director Bending Energies with Small Tilt

We prove a Gamma-convergence result for a family of bending energies defined on smooth surfaces in $\mathbb{R}^3$ equipped with a director field. The energies strongly penalize the deviation of the director from the surface unit normal and control the derivatives of the director. Such type of energies for example arise in a model for bilayer membranes introduced by Peletier and R\"oger [Arch. Ration. Mech. Anal. 193 (2009)]. Here we prove in three space dimensions in the vanishing-tilt limit a Gamma-liminf estimate with respect to a specific curvature energy. In order to obtain appropriate compactness and lower semi-continuity properties we use tools from geometric measure theory, in particular the concept of generalized Gauss graphs and curvature varifolds.

On Caristi-type mappings without lower semicontinuity assumptions

In this paper, we establish some new fixed point theorems for generalized distances and Caristi-type mappings without assuming that the dominated functions possess lower semicontinuity property. As applications of these results, we obtain some new fixed point theorems which generalize and improve the famous Banach contraction principle.

On $$\Gamma $$ Γ -convergence of vector-valued mappings

This paper deals with a new concept of limit for sequences of vector-valued mappings in normed spaces. We generalize the well-known concept of \(\Gamma \)-convergence to the case of vector-valued mappings and specify notion of \(\Gamma ^{\Lambda ,\mu }\)-convergence similar to the one previously introduced in Dovzhenko et al. (Far East J Appl Math 60:1–39, 2011). In particular, we show that \(\Gamma ^{\Lambda ,\mu }\)-convergence concept introduced in this paper possesses a compactness property whereas this property was failed in Dovzhenko et al. (Far East J Appl Math 60:1–39, 2011). In spite of the fact this paper contains another definition of \(\Gamma ^{\Lambda ,\mu }\)-limits for vector-valued mapping we prove that the \(\Gamma ^{\Lambda ,\mu }\)-lower limit in the new version coincides with the previous one, whereas the \(\Gamma ^{\Lambda ,\mu }\)-upper limit leads to a different mapping in general. Using the link between the lower semicontinuity property of vector-valued mappings and the topological properties of their coepigraphs, we establish the connection between \(\Gamma ^{\Lambda ,\mu }\)-convergence of the sequences of mappings and \(K\)-convergence of their epigraphs and coepigraphs in the sense of Kuratowski and study the main topological properties of \(\Gamma ^{\Lambda ,\mu }\)-limits. The main results are illustrated by numerous examples.

On quadratic scalarization of vector optimization problems in Banach spaces

We study vector optimization problems in partially ordered Banach spaces and suppose that the objective mapping possesses a weakened property of lower semicontinuity and make no assumptions on the interior of the ordering cone. We discuss the so-called adaptive scalarization of such problems and show that the corresponding scalar non-linear optimization problems can be by-turn approximated by quadratic minimization problems.

Epi and coepi-analysis of one class of vector-valued mappings

This article deals with a new characterization of lower semicontinuity of vector-valued mappings in normed spaces. We study the link between the lower semicontinuity property of vector-valued mappings and the topological properties of their epigraphs and coepigraphs, respectively. We show that if the objective space is partially ordered by a pointed cone with nonempty interior, then coepigraphs are stable with respect to the procedure of their closure and, moreover, the locally semicompact vector-valued mappings with closed coepigraphs are lower semicontinuous. Using these results we propose some regularization schemes for vector-valued functions. In the case when there are no assumptions on the topological interior of the ordering cone, we introduce a new concept of lower semicontinuity for vector-valued mappings, the so-called epi-lower semicontinuity, which is closely related to the closedness of epigraphs of such mappings, and study their main properties. All principal notions and assertions are illustrated by numerous examples.

Про квадратичну скаляризацію одного класу задач векторної оптимізації в банахових просторах

We study vector optimization problems in partially ordered Banach Spaces. We sup­pose that the objective mapping possesses a weakened property of lower semicontinuity and make no assumptions on the interior of the ordering cone. We discuss the ”clas­sical” scalarization of vector optimization problems in the form of weighted sum and also we propose other type of scalarization for vector optimization problem, the socalled adaptive scalarization, which inherits some ideas of Pascoletti-Serafini approach. As a result, we show that the scalar nonlinear optimization problems can byturn approxi­mated by the quadratic minimization problems. The advantage of such regularization is especially interesting from a numerical point of view because it gives a possibility to apply rather simple computational methods for the approximation of the whole set of efficient solutions.

The efficient hedging problem for American options

In this paper, we prove the existence of efficient partial hedging strategies for a trader unable to commit the initial minimal amount of money needed to implement a hedging strategy for an American option. The attitude towards the shortfall is modeled in terms of a decreasing and convex risk functional satisfying a lower semicontinuity property with respect to the Fatou convergence of stochastic processes. Some relevant examples of risk functionals are analyzed. Numerical computations in a discrete-time market model are provided. In a Levy market, an approximating solution is given assuming discrete-time trading.

Continuity Properties of Optimal Multiple Stopping Value

This article is concerned with the optimal multiple stopping problem for discrete time finite stage stochastic processes. We study lower semicontinuity and continuity properties of optimal stopping values with respect to the topology of convergence in distribution. Also, we formulate the multiple stopping version of the prophet inequality for the optimal stopping problem and apply the lower semicontinuity property of optimal stopping values to the prophet inequality for the optimal multiple stopping problem.

Lower Semicontinuity Properties of Functionals with Free Discontinuities

This paper is concerned with functionals with free discontinuities, considered as depending on the set variable. A splitting operation divides the sets in which the functional takes low values in the union of a noise part and a macroscopic part. Then we show lower semicontinuity results on the limit of the macroscopic parts by Hausdorff distance, along minimizing sequences.

On the Stability of the Feasible Set in Linear Optimization

This paper deals with the stability of two families of linear optimization problems, each one formed by the dual problems to the members of the other family. We characterize the problems of these families that are stable in the sense that they remain consistent (inconsistent) under sufficiently small arbitrary perturbations of all the data. This characterization is established in terms of the lower semicontinuity property of the feasible set mapping and the boundedness of the optimal set of the corresponding coupled problem. Other continuity properties of the feasible set mapping are also derived. This stability theory extends some well-known theorems of Williams and Robinson on the stability of ordinary linear programming problems to linear optimization problems with infinitely many variables or constraints.

Stability results for Ekeland's ε variational principle for vector valued functions

In this paper, under the assumption that the nonconvex vector valued function f satisfies some lower semicontinuity property and bounded below, the nonconvex vector valued function sequence fn satisfies the same lower semicontinuity property and uniformly bounded below, and fn converges to f in the generalized sense of Mosco, we obtain the relation: \(\), when \(\), where \(\)when\(\), C is the pointed closed convex dominating cone with nonempty interior int C, e∈int C. Under some conditions, we also prove the same result when fn converges to f in the generalized sense of Painleve'-Kuratowski.

Lower Semi-Continuity in Parametric Variational Problems, the Area Formula and Related Results

In the investigation of variational problems associated with functionals of the form (*), lower semi-continuity properties of the functionals play an important role. These properties are strongly related to certain convexity properties of the integrand f. In many variational problems of non-parametric type, the integrand f is convex with respect to px. Such problems were investigated by Tonelli, Cesari, Morrey, Serrin and others. The assumption that f is convex with respect to 0,, makes it possible to obtain very strong lower semi-continuity theorems for If. Results of this type, under very general conditions, were obtained by Serrin [11].

Coninuous Glimm-Type Functionals and Spreading of Rarefaction Waves

Several Glimm-type functionals for (piecewise smooth) approximate solutions of nonlinear hyperbolic systems have been introduced in recent years. In this paper, following a work by Baiti and Bressan on genuinely nonlinear systems we provide a framework to prove that such functionals can be extended to general functions with bounded variation and we investigate their lower semi-continuity properties with respect to the strong L1topology. In particular, our result applies to the functionals introduced by Iguchi-LeFloch and Liu-Yang for systems with general flux-functions, as well as the functional introduced by Baiti-LeFloch-Piccoli for nonclassical entropy solutions. As an illustration of the use of continuous Glimm-type functionals, we also extend a result by Bressan and Colombo for genuinely nonlinear systems, and establish an estimate on the spreading of rarefaction waves in solutions of hyperbolic systems with general flux-function.