On Characterizing the Solution Sets of Pseudoinvex Extremum Problems

In this paper, we study the minimization of a pseudoinvex function over an invex subset and provide several new and simple characterizations of the solution set of pseudoinvex extremum problems. By means of the basic properties of pseudoinvex functions, the solution set of a pseudoinvex program is characterized, for instance, by the equality , for each feasible point x, where is in the solution set. Our study improves naturally and extends some previously known results in Mangasarian (Oper. Res. Lett. 7: 21–26, 1988) and Jeyakumar and Yang (J. Opt. Theory Appl. 87: 747–755, 1995). This research was partially supported by National Natural Science Foundation of China Grants No. 10771228 and 10831009.     

Extremum Problems of Laplacian Eigenvalues and Generalized Polya Conjecture

In this survey on extremum problems of Laplacian-Dirichlet eigenvalues of Euclidian domains,the author briefly presents some relevant classical results and recent progress.The main goal is to describe the well-known conjecture due to Polya,its connections to Weyl's asymptotic formula for eigenvalues and shape optimizations.Many related open problems and some preliminary results are also discussed.     

On the Solution of Inverse Problems for Generalized Oxygen Consumption

We present the solution of some inverse problems for one-dimensional free boundary problems of oxygen consumption type, with a semilinear convection-diffusion-reaction parabolic equation. Using a fixed domain transformation (Landau's transformation) the direct problem is reduced to a system of ODEs. To minimize the objective functionals in the inverse problems, we approximate the data by a finite number of parameters with respect to which automatic differentiation is applied.     

On the Use of Augmented Lagrangians in the Solution of Generalized Semi-Infinite Min-Max Problems

We present an approach for the solution of a class of generalized semi-infinite optimization problems. Our approach uses augmented Lagrangians to transform generalized semi-infinite min-max problems into ordinary semi-infinite min-max problems, with the same set of local and global solutions as well as the same stationary points. Once the transformation is effected, the generalized semi-infinite min-max problems can be solved using any available semi-infinite optimization algorithm. We illustrate our approach with two numerical examples, one of which deals with structural design subject to reliability constraints.     

On the Solution of the Generalized Wave and Generalized Sine-Gordon Equations

The generalized wave equation and generalized sine-Gordon equations are known to be natural multidimensional differential geometric generalizations of the classical two-dimensional versions. In this paper we associate a system of linear differential equations with these equations and show how the direct and inverse problems can be solved for appropriately decaying data on suitable lines. An initial-boundary-value problem is solved for these equations.     

Generalized Multiplicative Inequalities for Ideal Spaces

We study the problem of completely describing the domains that enjoy the generalized multiplicative inequalities of the embedding theorem type. We transfer the assertions for the Sobolev spaces L _p ¹() to the function classes that result from the replacement of L _p () with an ideal space of vector-functions. We prove equivalence of the functional and geometric inequalities between the norms of indicators and the capacities of closed subsets of . The most comprehensible results relate to the case of the rearrangement invariant ideal spaces.     

On the Solution of Min-sum-min Problems

One class of min-sum-min problems is discussed in the paper. Min-sum-min problems appear in a natural way in many applications (e.g., in cluster analysis, pattern recognition, classification theory etc.). Like min-max-min problems, min-sum-min problems represent a very important family of nonsmooth problems. Problems of this type can be treated by means of the existing tools of Nonsmooth Analysis. However, most of algorithms available provide a local minimizer only, since they are based on necessary conditions which are of local nature. In the paper it is proved that the original problem can be reduced to the problem of minimizing a finite number of sum-functions. A necessary condition for a global minimum and a sufficient condition for a local minimum are stated. The necessary condition is of nonlocal nature. An algorithm (so-called Exchange algorithm) for finding points, satisfying necessary conditions, is described. An ɛ-Exchange algorithm is formulated, allowing, in principle, to escape from a ‘shallow’ local minimizer. An example is presented to illustrate the results and algorithms. An application of the proposed algorithms to solving one clustering problem is also given. Numerical results are provided.AMS Subject Classification:90C30, 49J40.     

On Generalized Vector Equilibrium Problems

In this paper we prove the existence of solutions of the generalized vector equilibrium problem in the setting of Hausdorff topological vector spaces. As applications, we present some relevant particular cases: a generalized vector variational-like inequality in Hausdorff topological vector spaces, and equilibrium problem in the case of pseudomonotone real functions, and a generalized weak Pareto optima problem.     

扰动广义向量变分不等式解的半连续性

本文通过标量化的方法在Hausdorff拓扑向量空间中讨论了扰动广义向量变分不等式解的下半连续性.     

椭圆型方程奇摄动问题的广义解

讨论了一类奇摄动椭圆型方程边值问题.在适当的条件下,研究了问题广义解的存在、唯一性及其渐近性态.     

On Generalized Vector Equilibrium Problems

A new generalized vector equilibrium problem involving set-valued mappings and the proper quasi concavity of set-valued mappings in topological vector spaces are introduced; its existence theorems and the convexity of the solution sets are established.     

Solution of Linear Generalized Eigenvalue Problems Containing Singular Matrices

A general method is presented for the solution of the lineargeneralized eigenvalue problem Ax = Bx, where matrices A andB may both be singular but ||A — B|| 0. The procedurecan easily be implemented on a digital computer using a Gaussianelimination algorithm with pivoting together with a QR eigensolutionpackage. Results are given for two problems with sensitive eigenvalues.     

On the Equivalence of Extended Generalized Complementarity and Generalized Least-Element Problems

In this paper, we derive some equivalences of generalized nonlinear programs, generalized least-element problems, and extended generalized complementarity problems under certain regularity and growth conditions. We also generalize the notion of a Z-map for point-to-set maps. Our results extend recent results by Schaible and Yao (Ref. 1).     

结构动力学逆特征值问题的同伦解法

将结构动力学反问题视为拟乘法逆特征值问题,利用求解非线性方程组的同伦方法来解决结构动力学逆特征值问题,这种方法由于沿同伦路径求解,对初值的选取没有本质的要求,算例说明了这种方法是可行的．     

On the Question of Generalized Solution to Algebro-Differential Systems

We study the possibility of constructing a Sobolev–Schwartz generalized solution to the problem A(t) x(t) + B (t) x (t) = f (t), t T = [0 , + ) , x ( 0 ) = a, whose coefficient (n × n)-matrix of derivatives is degenerate for every tT in the situation when there is no classical solution x(t)C ¹(T) (the initial data do not satisfy the agreement conditions and the right-hand side is not a sufficiently smooth vector-function). We prove that the generalized solution is the limit of a sequence of classical solutions of the Cauchy problem for a system with constant coefficients, obtained by the perturbation method.     

Generalized Convex Multiplicative Programming via Quasiconcave Minimization

We present a new method for minimizing the sum of a convex function and aproduct of k nonnegative convex functions over a convex set. This problem isreduced to a k-dimensional quasiconcave minimization problem which is solvedby a conical branch-and-bound algorithm. Comparative computational results areprovided on test problems from the literature.     

Semicontinuity of Solution Mappings of Parametric Generalized Vector Equilibrium Problems

In this paper, we study the ε-generalized vector equilibrium problem (ε-GVEP) and the ε-extended vector equilibrium problem (ε-EVEP), which can be regarded as approximate problems to the generalized vector equilibrium problems (GVEP). Existence results for ε-GVEP and ε-EVEP are established. We investigate also the continuity of the solution mappings of ε-GVEP and ε-EVEP. In particular, two results concerning the lower semicontinuity of the solution mappings of ε-GVEP and ε-EVEP are presented. This research was partially supported by Grant NSC 95-2811-M-110-010.     

Quasi-Multiplier Rules for Quasidifferentiable Extremum Problems

The purpose of the present article is to contribute to clarify the role of the Lagrange multipliers within the theory of the first order necessary optimality conditions for nonsmooth constrained optimization, when the directional derivatives of functions involved in the extremum problems are not sublinear. This task is accomplished in the particular case of quasidifferentiable problems with side constraints. In such setting, making use of the image-space approach, it is possible to establish a generalized (nonlinear) separation result by means of which a new Lagrange principle is obtained. According to this principle, which seems to fit better quasidifferentiable extremum problems than the classic one, the concept of linear multiplier is to be replaced with that of quasi-multiplier, a sublinear and continuous functional whose existence can be guaranteed under mild assumptions, even when classic multipliers fail to exist. Such as extension allows to formulate in terms of Lagrange function the known optimality necessary condition for unconstrained quasidifferentiable optimization expressed in form of quasidifferential inclusion. Along with this, other multiplier rules are established.     

Semicontinuity of Solution Mappings to Parametric Generalized Vector Equilibrium Problems

In this article, stability results concerning the lower semicontinuity and the Hausdorff upper semicontinuity of the solution mappings to parametric generalized vector equilibrium problems with neither the monotonicity of mappings nor any information of the solution mappings are established by using scalarization methods and a new density result.