Geometry of optimality conditions and constraint qualifications

Certain types of necessary optimality conditions for mathematical programming problems are equivalent to corresponding regularity conditions on the constraint set. For any problem, a certain natural optimality condition, dependent upon the particular constraint set, is always satisfied. This condition can be strengthened in numerous ways by invoking appropriate regularity assumptions on the constraint set. Results are presented for Euclidean spaces and some extensions to Banach spaces are given.This work was supported in part by the Office of Naval Research, Contract No. N00014-67-A-0321-0003 (NR-047-095).     

A note on convex cones and constraint qualifications in infinite-dimensional vector spaces

In connection with mathematical programming in infinite-dimensional vector spaces, Zowe has studied the relationship between the Slater constraint qualification and a formally weaker qualification used by Kurcyusz. The attractive feature of the latter is that it involves only active constraints. Zowe has proved that, in barreled spaces, the two qualifications are equivalent and has asked whether the assumption of barreledness is superfluous. By studying cores and interiors of convex cones, we show that the two constraint qualifications are equivalent in a given topological vector spaceE iff every barrel inE is a neighborhood of the origin. Thus, whenE is locally convex, the two constraint qualifications are equivalent iffE is barreled. Other questions of Zowe are also answered.This research was supported in part by the Office of Naval Research, and in part by the Sonderforschungsbereich 21, Institut für Operations Research, Bonn, Federal Republic of Germany. The author is indebted to Professor J. Zowe for some helpful comments.     

An extension of the Basic Constraint Qualification to nonconvex vector optimization problems

In this paper a Basic Constraint Qualification is introduced for a nonconvex infinite-dimensional vector optimization problem extending the usual one from convex programming assuming the Hadamard differentiability of the maps. Corresponding KKT conditions are established by considering a decoupling of the constraint cone into half-spaces. This extension leads to generalized KKT conditions which are finer than the usual abstract multiplier rule. A second constraint qualification expressed directly in terms of the data is also introduced, which allows us to compute the contingent cone to the feasible set and, as a consequence, it is proven that this condition is a particular case of the first one. Relationship with other constraint qualifications in infinite-dimensional vector optimization, specially with the Kurcyuscz-Robinson-Zowe constraint qualification, are also given.     

Lineare Optimierung in halbgeordneten Vektorräumen

Summary The problem of linear programming in partially ordered vector spaces is formulated as an immediate generalization of the same problem in Euclidean spaces. Sufficient conditions for the existence of solutions of the problem and its dual are obtained. In the special case of function spaces the sufficient conditions for the solvability of the dual problem are satisfied if a certain regularity condition is assumed.

---

---

This research was supported by the Air Force Office of Scientific Research under grant AF-AFO SR-93 7-6 7     

Regularity and stability for the mathematical programming problem in Banach spaces

This paper deals with a regularity assumption for the mathematical programming problem in Banach spaces. The attractive feature of our constraint qualification is the fact that it can be considered as a condition on the active part only of the constraint, and that it is preserved under small perturbations. Moreover, we show that our condition is almost equivalent to the existence of a non-empty and weakly compact set of Lagrange multipliers. The main step in the proof of our results is a generalization of the open mapping theorem.The early parts of this article result from fruitful correspondence with S. Kurcyusz, who died tragically in 1978. This paper is dedicated to his memory.     

Optimality conditions and duality in continuous programming II. The linear problem revisited

This is Part II of a two-part paper; the purpose of this two-part paper is (a) to develop new concepts and techniques in the theory of infinite-dimensional programming, and (b) to obtain fruitful applications in continuous time programming. In Part II the continuous time version of Farkas' theorem developed in Part I serves as the foundation for the duality theory for a broad class of linear continuous time programming problems distinct from those previously examined. In particular, we establish duality under analytic conditions, e.g., whether the given functions are measurable or continuous, that are weaker, and algebraic conditions that are more general, than those previously imposed. The new class of problems arising from these conditions allows for several important resource allocation problems previously excluded from consideration. In addition, an assumption needed to prove the Kuhn-Tucker theorem for the nonlinear problem of Part I is shown in the linear case to be completely analogous to the well-known Slater condition utilized in finite-dimensional programming theory. An example is given that exhibits the essential role of the constraint qualification in linear continuous time programming, a result at variance with the theory in finite dimensions but consistent with other results concerning linear programs in infinite-dimensional spaces.     

On Directional Metric Regularity, Subregularity and Optimality Conditions for Nonsmooth Mathematical Programs

This paper mainly deals with the study of directional versions of metric regularity and metric subregularity for general set-valued mappings between infinite-dimensional spaces. Using advanced techniques of variational analysis and generalized differentiation, we derive necessary and sufficient conditions, which extend even the known results for the conventional metric regularity. Finally, these results are applied to non-smooth optimization problems. We show that that at a locally optimal solution M-stationarity conditions are fulfilled if the constraint mapping is subregular with respect to one critical direction and that for every critical direction a M-stationarity condition, possibly with different multipliers, is fulfilled.     

Necessary optimality conditions for a special class of bilevel programming problems with unique lower level solution

We consider a bilevel programming problem in Banach spaces whose lower level solution is unique for any choice of the upper level variable. A condition is presented which ensures that the lower level solution mapping is directionally differentiable, and a formula is constructed which can be used to compute this directional derivative. Afterwards, we apply these results in order to obtain first-order necessary optimality conditions for the bilevel programming problem. It is shown that these optimality conditions imply that a certain mathematical program with complementarity constraints in Banach spaces has the optimal solution zero. We state the weak and strong stationarity conditions of this problem as well as corresponding constraint qualifications in order to derive applicable necessary optimality conditions for the original bilevel programming problem. Finally, we use the theory to state new necessary optimality conditions for certain classes of semidefinite bilevel programming problems and present an example in terms of bilevel optimal control.     

Multivalued differential equations in Banach space. An application to control theory

This paper is concerned with the study of existence theorems for multivalued differential systems in infinite-dimensional Banach space: the method used is based on techniques (extension theorem for linear operator, compactly convergent sequences) developed earlier by the authors for multivalued differential systems defined inn-dimensional vector spaces. As an application, the authors consider a distributed-parameter control problem arising in mathematical physics, more specifically, in the study of heat transfer in solids.This work was performed under the auspices of the National Research Council of Italy (CNR).     

Affine Variational Inequalities on Normed Spaces

This paper studies infinite-dimensional affine variational inequalities on normed spaces. It is shown that infinite-dimensional quadratic programming problems and infinite-dimensional linear fractional vector optimization problems can be studied by using affine variational inequalities. We present two basic facts about infinite-dimensional affine variational inequalities: the Lagrange multiplier rule and the solution set decomposition.     

Proper solutions of vector optimization problems

We define proper solutions in the Kuhn-Tucker sense for multiobjective mathematical programming problems with parameters in infinite-dimensional spaces and compare them with other definitions via suitable representatives: the Benson, Geoffrion, and Hurwicz properness. Necessary and/or sufficient conditions for proper solutions are proved. Problems with and without constraint qualifications are considered under relaxed convexity and differentiability assumptions.The author is grateful to Prof. W. Stadler and two referees for valuable remarks and suggestions concerning a previous draft of this paper.     

Scalarizing vector optimization problems

A scalarization of vector optimization problems is proposed, where optimality is defined through convex cones. By varying the parameters of the scalar problem, it is possible to find all vector optima from the scalar ones. Moreover, it is shown that, under mild assumptions, the dependence is differentiable for smooth objective maps defined over reflexive Banach spaces. A sufficiency condition of optimality for a general mathematical programming problem is also given in the Appendix.     

Constrained Extremum Problems,Regularity Conditions and Image Space Analysis. Part I: The Scalar Finite-Dimensional Case

Image space analysis has proved to be instrumental in unifying several theories, apparently disjoint from each other. With reference to constraint qualifications/regularity conditions in optimization, such an analysis has been recently introduced by Moldovan and Pellegrini. Based on this result, the present paper is a preliminary part of a work, which aims at exploiting the image space analysis to establish a general regularity condition for constrained extremum problems. The present part deals with scalar constrained extremum problems in a Euclidean space. The vector case as well as the case of infinite-dimensional image will be the subject of a subsequent part.     

弧连通锥-凸数学规划的最优性条件

在弧连通锥-凸假设下讨论Hausdorff局部凸空间中的一类数学规划的最优性条件问题.首先,利用择一定理得到了锥约束标量优化问题的一个必要最优性条件.其次,利用凸集分离定理证明了无约束向量优化问题关于弱极小元的标量化定理和一个一致的充分必要条件.所得结果深化和丰富了最优化理论及其应用的内容.     

Exact relaxations for parametric robust linear optimization problems

We first show that the closedness of the characteristic cone of the constraint system of a parametric robust linear optimization problem is a necessary and sufficient condition for each robust linear program with the finite optimal value to admit exact semidefinite linear programming relaxations. We then provide the weakest regularity condition that guarantees exact second-order cone programming relaxations for parametric robust linear programs.     

An application of Neustadt's abstract maximum principle to probability kinematics

An important class of problems in philosophy can be formulated as mathematical programming problems in an infinite-dimensional vector space. One such problem is that of probability kinematics: the study of how an individual ought to adjust his degree-of-belief function in response to new information. Much work has recently been done to establish maximum principles for these generalized programming problems (Refs. 3–4). Perhaps, the most general treatment of the problem presented to date is that by Neustadt (Ref. 1). In this paper, the problem of probability kinematics is formulated as a generalized mathematical programming problem and necessary conditions for the optimal revised degree-of-belief function are derived from an abstract maximum principle contained in Neustadt's paper.This work was supported by the National Research Council of Canada.The author is grateful to G. J. Lastman and J. A. Baker of the University of Waterloo for numerous suggestions made for improvement of this paper. The problem of probability kinematics was brought to the author's attention by W. L. Harper of the University of Western Ontario.     

OPTIMALITY CONDITIONS AND APPROXIMATE OPTIMALITY CONDITIONS IN LOCALLY LIPSCHITZ VECTOR OPTIMIZATION

Abstract

In this paper, we study constrained locally Lipschitz vector optimization problems in which the objective and constraint spaces are Hilbert spaces, the decision space is a Banach space, the dominating cone and the constraint cone may be with empty interior. Necessary optimality conditions for this type of optimization problems are derived. A sufficient condition for the existence of approximate efficient solutions to a general vector optimization problem is presented. Necessary conditions for approximate efficient solutions to a constrained locally Lipschitz optimization problem is obtained.     

A second order <Emphasis Type="Italic">η</Emphasis>-approximation method for constrained optimization problems involving second order invex functions

A new approach for obtaining the second order sufficient conditions for non-linear mathematical programming problems which makes use of second order derivative is presented. In the so-called second order η-approximation method, an optimization problem associated with the original nonlinear programming problem is constructed that involves a second order η-approximation of both the objective function and the constraint function constituting the original problem. The equivalence between the nonlinear original mathematical programming problem and its associated second orderη-approximated optimization problem is established under second order invexity assumption imposed on the functions constituting the original optimization problem.     

On optimality conditions for multiobjective optimization problems in topological vector space

In this paper, we are concerned with a differentiable multiobjective programming problem in topological vector spaces. An alternative theorem for generalized K subconvexlike mappings is given. This permits the establishment of optimality conditions in this context: several generalized Fritz John conditions, in line to those in Hu and Ling [Y. Hu, C. Ling, The generalized optimality conditions of multiobjective programming problem in topological vector space, J. Math. Anal. Appl. 290 (2004) 363-372] are obtained and, in the presence of the generalized Slater's constraint qualification, the Karush-Kuhn-Tucker necessary optimality conditions.     

Multiobjective optimization problem with variational inequality constraints

 We study a general multiobjective optimization problem with variational inequality, equality, inequality and abstract constraints. Fritz John type necessary optimality conditions involving Mordukhovich coderivatives are derived. They lead to Kuhn-Tucker type necessary optimality conditions under additional constraint qualifications including the calmness condition, the error bound constraint qualification, the no nonzero abnormal multiplier constraint qualification, the generalized Mangasarian-Fromovitz constraint qualification, the strong regularity constraint qualification and the linear constraint qualification. We then apply these results to the multiobjective optimization problem with complementarity constraints and the multiobjective bilevel programming problem. Received: November 2000 / Accepted: October 2001 Published online: December 19, 2002 Key Words. Multiobjective optimization – Variational inequality – Complementarity constraint – Constraint qualification – Bilevel programming problem – Preference – Utility function – Subdifferential calculus – Variational principle Research of this paper was supported by NSERC and a University of Victoria Internal Research Grant Research was supported by the National Science Foundation under grants DMS-9704203 and DMS-0102496 Mathematics Subject Classification (2000): Sub49K24, 90C29