Metric regularity of the feasible set mapping in semi-infinite optimization

For parametric systems of (finitely many) equations and (infinitely many) inequalities the well-known concept of metric regularity is shown to be equivalent to the so-called extended Mangasarian-Fromovitz constraint qualification. By this, a corresponding result obtained by Robinson for finite optimization problems my be transferred to semi-infinite optimization. For the proof a local epigraph representation of the constraint set is mainly used.     

On feasible sets defined through Chebyshev approximation

LetZ be a compact set of the real space with at leastn + 2 points;f,h1,h2:Z continuous functions,h1,h2 strictly positive andP(x,z),x(x ₀,...,x _n ) ⁿ⁺¹,z , a polynomial of degree at mostn. Consider a feasible setM {x ⁿ⁺¹z Z, –h ₂(z) P(x, z)–f(z)h ₁(z)}. Here it is proved the null vector 0 of ⁿ⁺¹ belongs to the compact convex hull of the gradients ± (1,z,...,z ⁿ ), wherez Z are the index points in which the constraint functions are active for a givenx* M, if and only ifM is a singleton.This work was partially supported by CONACYT-MEXICO.     

Unified steerable phase I-phase II method of feasible directions for semi-infinite optimization

In this paper, we complete a cycle in the construction of methods of feasible directions for solving semi-infinite constrained optimization problems. Earlier phase I-phase II methods of feasible directions used one search direction rule in all of ⁿ with two stepsize rules, one for feasible points and one for infeasible points. The algorithm presented in this paper uses both a single search direction rule and a single stepsize rule in all of ⁿ . In addition, the new algorithm incorporates a steering parameter which can be used to control the speed with which feasibility is achieved. The new algorithm is simpler to analyze and performs somewhat better than existing, first order, phase I-phase II methods. The new algorithm is globally convergent, with linear rate.The research reported herein was sponsored in part by the National Science Foundation Grant ECS-8713334, the Air Force Office of Scientific Research Contract AFOSR-86-0116, and the State of California MICRO Program Grant 532410-19900.The authors would like to thank Dr. J. Higgins for providing the C-code of Algorithm 3.1.     

Semi-infinite optimization: Structure and stability of the feasible set

The problem of the minimization of a functionf: ⁿ under finitely many equality constraints and perhaps infinitely many inequality constraints gives rise to a structural analysis of the feasible setM[H, G]={xⁿ¦H(x)=0,G(x, y)0,yY} with compactY^r. An extension of the well-known Mangasarian-Fromovitz constraint qualification (EMFCQ) is introduced. The main result for compactM[H, G] is the equivalence of the topological stability of the feasible setM[H, G] and the validity of EMFCQ. As a byproduct, we obtain under EMFCQ that the feasible set admits local linearizations and also thatM[H, G] depends continuously on the pair (H, G). Moreover, EMFCQ is shown to be satisfied generically.The authors would like to thank Rainer Hettich and Doug Ward for fruitful discussions. Moreover, the authors are indebted to the anonymous referees for their valuable comments.     

One-Parameter Families of Feasible Sets in Semi-infinite Optimization

Feasible sets in semi-infinite optimization are basically defined by means of infinitely many inequality constraints. We consider one-parameter families of such sets. In particular, all defin-ing functions - including those defining the index set of the inequality constraints - will depend on a parameter. We note that a semi-infinite problem is a two-level problem in the sense that a point is feasible if and only if all global minimizers of a corresponding marginal function are nonnegative. For a quite natural class of mappings we characterize changes in the global topological structure of the corresponding feasible set as the parameter varies. As long as the index set (-mapping) of the inequality constraints is lower semicontinuous, all changes in topology are those which generically appear in one-parameter sets defined by finitely many constraints. In the case, however, that some component of the mentioned index set is born (or vanishes), the topological change is of global nature and is not controllable. In fact, the change might be as drastic as that when adding or deleting an (arbitrary) inequality constraint.     

On duality in multiobjective semi-infinite optimization

Abstract

In this paper, we consider multiobjective semi-infinite optimization problems which are defined in a finite-dimensional space by finitely many objective functions and infinitely many inequality constraints. We present duality results both for the convex and nonconvex case. In particular, we show weak, strong and converse duality with respect to both efficiency and weak efficiency. Moreover, the property of being a locally properly efficient point plays a crucial role in the nonconvex case.     

On the Classical Necessary Second-Order Optimality Conditions

In this paper, the property of a necessary second-order optimality condition to hold with the same Lagrange multiplier for all critical vectors is investigated. It is limited to nonconvex optimization Problems in ⁿ with equality and inequality constraints; the Mangasarian-Fromovitz constraint qualification is assumed to be hold. A counterexample was given recently by Anitescu. We give some sufficient conditions and we prove that this property holds if n 2 or if the number of active inequality constraints is at most two. For three active inequality constraints and n =3, a counterexample is given.     

A norm-relaxed method of feasible directions for finely discretized problems from semi-infinite programming

In this paper, a class of finely discretized Semi-Infinite Programming (SIP) problems is discussed. Combining the idea of the norm-relaxed Method of Feasible Directions (MFD) and the technique of updating discretization index set, we present a new algorithm for solving the Discretized Semi-Infinite (DSI) problems from SIP. At each iteration, the iteration point is feasible for the discretized problem and an improved search direction is computed by solving only one direction finding subproblem, i.e., a quadratic program, and some appropriate constraints are chosen to reduce the computational cost. A high-order correction direction can be obtained by solving another quadratic programming subproblem with only equality constraints. Under weak conditions such as Mangasarian–Fromovitz Constraint Qualification (MFCQ), the proposed algorithm possesses weak global convergence. Moreover, the superlinear convergence is obtained under Linearly Independent Constraint Qualification (LICQ) and other assumptions. In the end, some elementary numerical experiments are reported.     

On equivalent stability properties in semi-infinite optimization

This paper studies noncompact feasible sets of a semi-infinite optimization problem which are defined by finitely many equality constraints and infinitely many inequality constraints. The main result is the equivalence of the overall validity of the Extended Mangasarian Fromovitz Constraint Qualification with certain (topological) stability conditions. Furthermore, two perturbation theorems being of independent interest are presented.This work was supported by the Deutsche Forschungsgemeinschaft under grant Gu 304/1-2.     

On the closure of the feasible set in generalized semi-infinite programming

In generalized semi-infinite programming the feasible set is known to be not closed in general. In this paper, under natural and generic assumptions, the closure of the feasible set is described in explicit terms. Oliver Stein gratefully acknowledges support through a Heisenberg grant of the Deutsche Forschungsgemeinschaft.     

On existence and uniqueness of stationary points in semi-infinite optimization

The paper deals with semi-infinite optimization problems which are defined by finitely many equality constraints and infinitely many inequality constraints. We generalize the concept of strongly stable stationary points which was introduced by Kojima for finite problems; it refers to the local existence and uniqueness of a stationary point for each sufficiently small perturbed problem, where perturbations up to second order are allowed. Under the extended Mangasarian-Fromovitz constraint qualification we present equivalent conditions for the strong stability of a considered stationary point in terms of first and second derivatives of the involved functions. In particular, we discuss the case where the reduction approach is not satisfied. Received June 30, 1995 / Revised version received October 9, 1998? Published online June 11, 1999     

Bounded sets of Lagrange multipliers for vector optimization problems in infinite dimension

The aim of this paper is to point out some sufficient constraint qualification conditions ensuring the boundedness of a set of Lagrange multipliers for vectorial optimization problems in infinite dimension. In some (smooth) cases these conditions turn out to be necessary for the existence of multipliers as well.     

轨道逼近时间集的密度

任意给定0pq1,证明了在符号系统中(进而在帐篷映射中)存在Mycielski集C,使得C中任意两个互异的点的轨道按照下密度p,上密度q的"速率"逼近.构造了线段上的连续映射,使其具有一个满Lebesgue测度的Mycielski集S,使得S中任意两个互异的点的轨道按照下密度p,上密度q的"速率"逼近.     

On generalized semi-infinite programming

This paper surveys some basic properties of the class of generalized semi-infinite programming problems (GSIP) where the infinite index set of inequality constraints depends on the state variables and all emerging functions are assumed to be continuously differentiable. There exists a wide range of applications which can be modelled as a (GSIP). The paper discusses extensions of the Mangasarian-Fromovitz, Kuhn-Tucker and Abadie constraint qualification to (GSIP) and presents related first order optimality conditions of Fritz-John and Karush-Kuhn-Tucker type. By using directional differentiability properties of the optimal value function of the lower level problem, first and second order necessary and sufficient optimality conditions are discussed. Several examples illustrate the results presented. The work of this author was supported by CONACYT (México) under grant 44003.     

Characterization of (weakly/properly/robust) efficient solutions in nonsmooth semi-infinite multiobjective optimization using convexificators

The main aim of this paper is to investigate weakly/properly/robust efficient solutions of a nonsmooth semi-infinite multiobjective programming problem, in terms of convexificators. In some of the results, we assume the feasible set to be locally star-shaped. The appearing functions are not necessarily smooth/locally Lipschitz/convex. First, constraint qualifications and the normal cone to the feasible set are studied. Then, as a major part of the paper, various necessary and sufficient optimality conditions for solutions of the problem under consideration are presented. The paper is closed by a linear approximation problem to detect the solutions and by studying a gap function.     

An improved feasible QP-free algorithm for inequality constrained optimization

In this paper, an improved feasible QP-free method is proposed to solve nonlinear inequality constrained optimization problems. Here, a new modified method is presented to obtain the revised feasible descent direction. In view of the computational cost, the most attractive feature of the new algorithm is that only one system of linear equations is required to obtain the revised feasible descent direction. Thereby, per single iteration, it is only necessary to solve three systems of linear equations with the same coefficient matrix. In particular, without the positive definiteness assumption on the Hessian estimate, the proposed algorithm is still global convergence. Under some suitable conditions, the superlinear convergence rate is obtained.     

Use of Pτ-nets for the approximation of the Edgeworth-Pareto set in multicriteria optimization

Engineering optimization problems, design problems among them, are multicriteria in their essence. When designing machines, mechanisms, and structures, one has to deal with numerous contradictory criteria.Experience gained in solving engineering optimization and optimal design problems shows that the designer cannot formulate them correctly. Unfortunately, the known optimization methods offer little assistance. To assure a correct formulation and solution of engineering optimization problems, the parameter space investigation (PSI) method and methods of approximation have been developed and widely integrated into different fields of industry, science, and technology. The PSI method has become one of the basic working tools for choosing optimal parameters in many fields of the national economy in Russia.     

Isolated calmness of solution mappings in convex semi-infinite optimization

This paper is concerned with isolated calmness of the solution mapping of a parameterized convex semi-infinite optimization problem subject to canonical perturbations. We provide a sufficient condition for isolated calmness of this mapping. This sufficient condition characterizes the strong uniqueness of minimizers, under the Slater constraint qualification. Moreover, on the assumption that the objective function and the constraints are linear, we show that this condition is also necessary for isolated calmness.