ε-Optimality Conditions for Vector Optimization Problems with Set-Valued Maps

In this paper, I introduce two kinds of ε-subgradients—ε-Pareto subgradients and ε-Benson proper subgradients—for set-valued maps. I give sufficient conditions for the existence of such ε-subgradients. A generalized ε-Moreau–Rockafellar type theorem for ε-Benson proper subdifferentials of set-valued maps is formulated and applied to establish ε-optimality conditions for ε-Benson proper efficient solutions of vector optimization problems with set-valued maps. Scalarization theorems for such a problem are also obtained.     

Existence of ∈-Minima for Vector Optimization Problems

The -generalized minima for vector optimization problems are defined and a sufficient condition for the existence of -generalized minima for vector optimization problems is established.     

ε-super Effcient Solutions of Optimization Problems with Set-valued Maps in Locally Convex Space

In this paper,the ε-super effcient solution for set-valued map vector optimization in locally convex space is introduced.And under the assumption of the nearly generalized cone-subconvexlikeness for set-valued maps,used new methods,the scalarization theorem and Lagrange multiplier theorem for ε-super effcient solution are established.     

Existence of Solutions to Generalized Vector Quasi-Equilibrium Problems with Discontinuous Mappings

Let X, Y be two finite-dimensional topological vector spaces, Z a Hausdorff topological vector space, K C X and D C Z be two nonempty sets, C be a pointed, closed, and convex cone in Y with int C ≠θ Let S ： K → 2^K and T ： K → 2^D be two multivalued mappings, and φ ： K × D × K → Y be a trifunction. In this paper, we consider the generalized vector quasi-equilibrium problem, which is formulated by finding X∈ K and y∈ T（x） such that x∈ E S（x） and φ（x,y, u） （∈/） -int C for all u ∈ S（x）. We establish an existence result in which T is not supposed to have any continuity property. Our results extend and improve the corresponding results of Cubiotti, Yao and Guo.     

Generalized Clarke Epiderivatives of Parametric Vector Optimization Problems

This paper deals with the generalized Clarke epiderivative of the extremum (or efficient point) multifunction in parametric vector optimization problems. The formulas for computing and/or estimating the generalized Clarke epiderivative of this extremum multifunction are given in terms of the Clarke tangent cone to the graph of a multifunction or the constraint mapping and/or the Fréchet derivative of the objective function. An application to semi-infinite programming is given.     

Hölder Continuity of Solutions to Parametric Vector Equilibrium Problems with Nonlinear Scalarization

In this article, using the nonlinear scalarization approach by virtue of the nonlinear scalarization function, commonly known as the Gerstewitz function in the theory of vector optimization, Hölder continuity of solution mappings for both set-valued and single-valued cases to parametric vector equilibrium problems is studied. The nonlinear scalarization function is a powerful tool that plays a key role in the proofs, and its main properties (such as sublinearity, continuity, convexity) are fully employed. Especially, its locally and globally Lipschitz properties are provided and the Lipschitz property is first exploited to investigate the Hölder continuity of solutions.     

Convergence in Distribution of Optimal Solutions to Stochastic Optimization Problems

1.IntroductionDistributionproblemsareofgreatimportanceinstochasticoptimizationandstatis-tics.Usuallythiskindofproblemscanbedescribedinthefollowingform:wheref(x,w)isafuncti0ndefinedonR"xflandSisasubsetin'R".Because0fc0mplexityoftheproblems,ingeneral,onecangetonlytheirapproximatesolutions.Thefollowingtypeofapproximationis0ftenused:Letuscall(2)thefirsttype0fapproximation.DenotebyZ(w),A(w)theoptima1valueandoptimalsolutionsetofproblem(1)respectivelyandbyZk(w),Ak(w)thecorrespondingonesofproblem(…     

Nonsmooth Vector Optimization Problems and Minty Vector Variational Inequalities

The vector optimization problem may have a nonsmooth objective function. Therefore, we introduce the Minty vector variational inequality (Minty VVI) and the Stampacchia vector variational inequality (Stampacchia VVI) defined by means of upper Dini derivative. By using the Minty VVI, we provide a necessary and sufficient condition for a vector minimal point (v.m.p.) of a vector optimization problem for pseudoconvex functions involving Dini derivatives. We establish the relationship between the Minty VVI and the Stampacchia VVI under upper sign continuity. Some relationships among v.m.p., weak v.m.p., solutions of the Stampacchia VVI and solutions of the Minty VVI are discussed. We present also an existence result for the solutions of the weak Minty VVI and the weak Stampacchia VVI.     

On Optimality Condition for Vector Optimization Based on Convex Set-Valued Mapping

1.IntroductionTheoptimalityconditionsofmathematicalprogrammingisaveryimportantsubjectbecausetLeyprovideausefulanalyticaltoolforstudingthedualitytheoryandnonlinearprogrammingalgoirthms.Inrecelltyears,someauthorshavebeguntostudytheoptimalityconditionsforvectoroptimizationproblemofset-valuedmapping,suchas[4][51.Inthispaperlwedefinetheconceptofcone--weaklyefficientsubdifferentialofset-valuedmappinginthecaseofgeneralpartiallyorderedlocallyconvextopologicalvectorspaces.Thecone-weaklysubdifferential…     

The quadratic convergence of the topological epsilon algorithm for systems of nonlinear equations

In this work we consider the topological epsilon algorithm for solving systems of nonlinear equations. In section 2, a sufficient condition for its quadratic convergence is given. In section 3, some geometrical remarks about this condition are made.     

Finite Coverings by Cones and an Application in Multi-Objective Programming

This paper considers analogues of statements concerning compactness and finite coverings, in which the roles of spheres are replaced by cones. Furthermore, one of the finite covering results provides an application in Multi-Objective Programming; infinite sets of alternatives are reduced to finite sets.     

Lorentz Manifolds with the Three Largest Degrees of Symmetry

We show that if a Lorentz manifold (M, g) has a sufficiently large group of isometries and if, in addition, this group has no null orbits, then (M, g) is homogeneous. A list of Lorentz manifolds with the three largest groups of isometries is given.     

Nonlinear Lagrange Duality Theorems and Penalty Function Methods In Continuous Optimization

We propose a general dual program for a constrained optimization problem via generalized nonlinear Lagrangian functions. Our dual program includes a class of general dual programs with explicit structures as special cases. Duality theorems with the zero duality gap are proved under very general assumptions and several important corollaries which include some known results are given. Using dual functions as penalty functions, we also establish that a sequence of approximate optimal solutions of the penalty function converges to the optimal solution of the original optimization problem.     

Start‐up class models in multiple‐class queues with N‐policy

In this paper, we consider multipleclass queueing systems with Npolicy in which the idle server starts service as soon as the number of customers in the startup class reaches threshold N. We consider the cases of FCFS and nonpreemptive priority. We obtain the Laplace–Stieltjes transform of the waiting times of each class of customers. We also show some results for the general behavior of such systems.     

Parallel Algorithms for Solving the Convex Minimum Cost Flow Problem

In this paper we deal with the solution of the separable convex cost network flow problem. In particular, we propose a parallel asynchronous version of the -relaxation method and we prove theoretically its correctness.We present two implementations of the parallel method for a shared memory multiprocessor system, and we empirically analyze their numerical performance on different test problems. The preliminary numerical results show a good reduction of the execution time of the parallel algorithm with the respect to the sequential counterpart.     

A large deviation principle for (r,p)-capacities on the Wiener space

Summary We formulate and prove a large deviation principle for the (r, p)-capacity on an abstract Wiener space. As an application, we obtain a sharpening of Strassen's law of the iterated logarithm in terms of the capacity.     

On a Theorem of Holický and Zelený Concerning Borel Maps Without σ-Compact Fibers

The paper is concerned with a recent very interesting theorem obtained by Holický and Zelený. We provide an alternative proof avoiding games used by Holický and Zelený and give some generalizations to the case of set-valued mappings.     

Primal-relaxed dual global optimization approach

A deterministic global optimization approach is proposed for nonconvex constrained nonlinear programming problems. Partitioning of the variables, along with the introduction of transformation variables, if necessary, converts the original problem into primal and relaxed dual subproblems that provide valid upper and lower bounds respectively on the global optimum. Theoretical properties are presented which allow for a rigorous solution of the relaxed dual problem. Proofs of -finite convergence and -global optimality are provided. The approach is shown to be particularly suited to (a) quadratic programming problems, (b) quadratically constrained problems, and (c) unconstrained and constrained optimization of polynomial and rational polynomial functions. The theoretical approach is illustrated through a few example problems. Finally, some further developments in the approach are briefly discussed.The authors gratefully acknowledge financial support from National Science Foundation Presidential Young Investigator Award CBT-88-57013. The authors are also grateful to Drs. F. A. Al-Khayyal, B. Jaumard, P. M. Pardalos, and H. D. Sherali for helpful comments on an earlier draft of this paper.