Boundedness and Nonemptiness of the Efficient Solution Sets in Multiobjective Optimization

For a given multiobjective optimization problem, we study recession properties of the sets of efficient solutions and properly efficient solutions. We work out various consequences based on the obtained recession properties, including a characterization for the boundedness and nonemptiness of the set of (properly) efficient solutions when the problem is a convex problem. We also show that the boundedness and nonemptiness of the set of efficient solutions is equivalent to that of the set of properly efficient solutions under an additional mild condition. Finally, we provide some new verifiable necessary conditions for the nonemptiness of the set of efficient solutions in terms of the associated recession functions and recession cones.     

A hybrid iterative scheme for mixed equilibrium problems and fixed point problems

The purpose of this paper is to investigate the problem of finding a common element of the set of solutions of a mixed equilibrium problem (MEP) and the set of common fixed points of finitely many nonexpansive mappings in a real Hilbert space. First, by using the well-known KKM technique we derive the existence and uniqueness of solutions of the auxiliary problems for the MEP. Second, by virtue of this result we introduce a hybrid iterative scheme for finding a common element of the set of solutions of MEP and the set of common fixed points of finitely many nonexpansive mappings. Furthermore, we prove that the sequences generated by the hybrid iterative scheme converge strongly to a common element of the set of solutions of MEP and the set of common fixed points of finitely many nonexpansive mappings.     

Approximate solutions and scalarization in set-valued optimization

Abstract

Certain notions of approximate weak efficient solutions are considered for a set-valued optimization problem based on vector and set criteria approaches. For approximate solutions based on the vector approach, a characterization is provided in terms of an extended Gerstewitz’s function. For the set approach case, two notions of approximate weak efficient solutions are introduced using a lower and an upper quasi order relations for sets and further compactness and stability aspects are discussed for these approximate solutions. Existence and scalarization using a generalized Gerstewitz’s function are also established for approximate solutions, based on the lower set order relation.     

Hybrid extragradient method for general equilibrium problems and fixed point problems in Hilbert space

In this paper, we introduce an iterative scheme by the hybrid methods for finding a common element of the set of fixed points of nonexpansive mappings, the set of solutions of an equilibrium problem and the set of solutions of a variational inequality problem in a Hilbert space. Then, we prove the strongly convergent theorem by a hybrid extragradient method to the common element of the set of fixed points of nonexpansive mappings, the set of solutions of an equilibrium problem and the set of solutions of a variational inequality problem. Our results extend and improve the results of Bnouhachem et al. [A. Bnouhachem, M. Aslam Noor, Z. Hao, Some new extragradient iterative methods for variational inequalities, Nonlinear Analysis (2008) doi:10.1016/j.na.2008.02.014] and many others.     

Proper efficiency in vector infinite programming problems

Using the notion of invexity, we give sufficient conditions of optimality for properly efficient solutions of a vector infinite programming problem and show that the set of properly efficient solutions coincides with the set of optimal solutions of a related scalar problem.     

On essential sets and essential components of efficient solutions for vector optimization problems

In this paper, we first introduce the notions of an essential set and an essential component of the set of efficient solutions for continuous vector optimizations on a nonempty compact subset of a metric space. Then we show that for each of these vector optimizations, each set of all efficient solutions corresponding to the same optimal values is essential. Basing on this result, we give full characterizations of an essential point, an essential set and an essential component, respectively. As an application, we prove that for continuous quasiconvex vector optimization problems on a nonempty compact subset of a metric vector space, each component of the set of efficient solutions is essential even though the efficient solution set is not connected.     

Levitin–Polyak well-posedness for constrained quasiconvex vector optimization problems

In this paper, a notion of Levitin–Polyak (LP in short) well-posedness is introduced for a vector optimization problem in terms of minimizing sequences and efficient solutions. Sufficient conditions for the LP well-posedness are studied under the assumptions of compactness of the feasible set, closedness of the set of minimal solutions and continuity of the objective function. The continuity assumption is then weakened to cone lower semicontinuity for vector-valued functions. A notion of LP minimizing sequence of sets is studied to establish another set of sufficient conditions for the LP well-posedness of the vector problem. For a quasiconvex vector optimization problem, sufficient conditions are obtained by weakening the compactness of the feasible set to a certain level-boundedness condition. This in turn leads to the equivalence of LP well-posedness and compactness of the set of efficient solutions. Some characterizations of LP well-posedness are given in terms of the upper Hausdorff convergence of the sequence of sets of approximate efficient solutions and the upper semicontinuity of an approximate efficient map by assuming the compactness of the set of efficient solutions, even when the objective function is not necessarily quasiconvex. Finally, a characterization of LP well-posedness in terms of the closedness of the approximate efficient map is provided by assuming the compactness of the feasible set.     

BDD-based heuristics for binary optimization

In this paper we introduce a new method for generating heuristic solutions to binary optimization problems. We develop a technique based on binary decision diagrams. We use these structures to provide an under-approximation to the set of feasible solutions. We show that the proposed algorithm delivers comparable solutions to a state-of-the-art general-purpose optimization solver on randomly generated set covering and set packing problems.     

Fixed point solutions of generalized equilibrium problems for nonexpansive mappings

In this paper, we introduce an iterative scheme for finding a common element of the set of fixed points of a nonexpansive mapping and the set of solutions of a generalized equilibrium problem in a real Hilbert space. Then, strong convergence of the scheme to a common element of the two sets is proved. As an application, problem of finding a common element of the set of fixed points of a nonexpansive mapping and the set of solutions of an equilibrium problem is solved. Moreover, solution is given to the problem of finding a common element of fixed points set of nonexpansive mappings and the set of solutions of a variational inequality problem.     

Strong convergence theorem by a new hybrid method for equilibrium problems and variational inequality problems

In this paper, we prove a strong convergence theorem for finding a common element of the set of solutions of an equilibrium problem and the set of the solutions of the variational inequality problem by using a new hybrid method. We obtain a new result for finding a solution of an equilibrium problem and the solutions of the variational inequality problem.     

平衡问题变分包含问题及不动点问题的二次极小化

借助预解式技巧,寻求二次极小化问题minx∈Ω‖x‖2的解,其中Ω是Hilbert空间中某一广义平衡问题的解集,与一无穷族非扩张映像的公共不动点的集合,以及某一变分包含的解集的交集．在适当的条件下,逼近上述极小化问题的解的一新的强收敛定理被证明．     

Sobolev type embedding and weak solutions with a prescribed singular set

New Sobolev type embeddings for some weighted Banach spaces are established. Using such embeddings and the singular positive radial entire solutions, we construct singular positive weak solutions with a prescribed singular set for a weighted elliptic equation. Our main results in this paper also provide positive weak solutions with a prescribed singular set to an equation with Hardy potential.     

Weighting factor results in vector optimization

This paper deals with the study, in a convex vector optimization problem, of the set of efficient solutions and the set of properly efficient solutions, the latter being obtained by a weighting factor technique. Relationships between these two sets are discussed; they are shown to be nonempty when the objective functions have no common direction of recession and to be closed and equal when, moreover, the objective functions are locally polyhedral. An example is provided where the set of efficient solutions is not included in the closure of the nonempty set of properly efficient solutions.The author wishes to thank the unknown referee for the helpful comments that improved the quality of this paper.     

Viscosity approximation methods for generalized equilibrium problems and fixed point problems

The purpose of this paper is to investigate the problem of finding a common element of the set of solutions of a generalized equilibrium problem (for short, GEP) and the set of fixed points of a nonexpansive mapping in the setting of Hilbert spaces. By using well-known Fan-KKM lemma, we derive the existence and uniqueness of a solution of the auxiliary problem for GEP. On account of this result and Nadler’s theorem, we propose an iterative scheme by the viscosity approximation method for finding a common element of the set of solutions of GEP and the set of fixed points of a nonexpansive mapping. Furthermore, it is proven that the sequences generated by this iterative scheme converge strongly to a common element of the set of solutions of GEP and the set of fixed points of a nonexpansive mapping.     

Cone convexity,cone extreme points,and nondominated solutions in decision problems with multiobjectives

Although there is no universally accepted solution concept for decision problems with multiple noncommensurable objectives, one would agree that agood solution must not be dominated by the other feasible alternatives. Here, we propose a structure of domination over the objective space and explore the geometry of the set of all nondominated solutions. Two methods for locating the set of all nondominated solutions through ordinary mathematical programming are introduced. In order to achieve our main results, we have introduced the new concepts of cone convexity and cone extreme point, and we have explored their main properties. Some relevant results on polar cones and polyhedral cones are also derived. Throughout the paper, we also pay attention to an important special case of nondominated solutions, that is, Pareto-optimal solutions. The geometry of the set of all Pareto solutions and methods for locating it are also studied. At the end, we provide an example to show how we can locate the set of all nondominated solutions through a derived decomposition theorem.     

Exact solutions of one-dimensional nonlinear shallow water equations over even and sloping bottoms

We establish an equivalence of two systems of equations of one-dimensional shallow water models describing the propagation of surface waves over even and sloping bottoms. For each of these systems, we obtain formulas for the general form of their nondegenerate solutions, which are expressible in terms of solutions of the Darboux equation. The invariant solutions of the Darboux equation that we find are simplest representatives of its essentially different exact solutions (those not related by invertible point transformations). They depend on 21 arbitrary real constants; after “proliferation” formulas derived by methods of group theory analysis are applied, they generate a 27-parameter family of essentially different exact solutions. Subsequently using the derived infinitesimal “proliferation” formulas for the solutions in this family generates a denumerable set of exact solutions, whose linear span constitutes an infinite-dimensional vector space of solutions of the Darboux equation. This vector space of solutions of the Darboux equation and the general formulas for nondegenerate solutions of systems of shallow water equations with even and sloping bottoms give an infinite set of their solutions. The “proliferation” formulas for these systems determine their additional nondegenerate solutions. We also find all degenerate solutions of these systems and thus construct a database of an infinite set of exact solutions of systems of equations of the one-dimensional nonlinear shallow water model with even and sloping bottoms.     

Compromise solutions and estimation of the noninferior set

A general framework is presented in which the relation of the set of noninferior points and the set of compromise solutions is studied. It is shown that the set of compromise solutions is dense in the set of noninferior points and that each compromise solution is properly noninferior. Also, under convexity of the criteria space, a characterization of the properly noninferior points in terms of the compromise solutions is presented. In this characterization, the compromise solutions depend continuously on the weights. Use of the maximum norm is studied also. It is shown that a subset of these max-norm solutions, obtained by taking certain limits of compromise solutions, is dense and contained in the closure of the set of noninferior points.     

The set of all nondominated solutions in linear cases and a multicriteria simplex method

In this note we are interested in the properties of, and methods for locating the set of all nondominated solutions of multiple linear criteria defined over a polyhedron. We first show that the set of all dominated solutions is convex and that the set of all nondominated solutions is a subset of the convex hull of the nondominated extreme points. When the domination cone is polyhedral, we derive a necessary and sufficient condition for a point to be nondominated. The condition is stronger than that of Ref. [1] and enables us to give a simple proof that the set of all nondominated extreme points indeed is connected. In order to locate the entire set of all nondominated extreme points, we derive a generalized version of simplex method—multicriteria simplex method. In addition to some useful results, a necessary and sufficient condition for an extreme point to be nondominated is derived. Examples and computer experience are also given. Finally, we focus on how to generate the entire set of all nondominated solutions through the set of all nondominated extreme points. A decomposition theorem and some necessary and sufficient conditions for a face to be nondominated are derived. We then describe a systematic way to identify the entire set of all nondominated solutions. Through examples, we show that in fact our procedure is quite efficient.     

Stationary points of the Yang-Mills action

We examine the structure of a recently discovered set of non-self-dual solutions of the Yang-Mills equations. These solutions have a symmetry that reduces the YM equations to a set of ODE's. The distinct solutions are indexed by two postive odd integers. We develop a scheme to approximate on a computer the solutions for small values of the indexing integers and present some numerical results. We then analyze the asymptotic behavior of the solutions as the indexing integers become large.     

强向量均衡问题与不动点问题的粘性逼近算法

讨论了强向量均衡问题与非扩张映射不动点问题的公共解.首先,给出了强向量均衡问题的辅助问题,并在适当的条件下,证明了其解的存在性和唯一性结果.然后,利用这些结果,提出了强向量均衡问题与非扩张映射不动点问题公共解的粘性逼近算法,并进一步证明了,在适当的条件下,由该算法产生的迭代序列强收敛于强向量均衡问题和非扩张映射不动点问题的公共解.