Constraint qualifications in a class of nondifferentiable mathematical programming problems

The Kuhn-Tucker type necessary optimality conditions are given for the problem of minimizing the sum of a differentiable function and a convex function subject to a set of differentiable nonlinear inequalities on a convex subset C of , under the conditions similar to the Kuhn-Tucker constraint qualification or the Arrow-Hurwicz-Uzawa constraint qualification. The case when the set C is open (not necessarily convex) is shown to be a special one of our results, which helps us to improve some of the existing results in the literature.     

Constraint qualifications in multiobjective optimization problems: Differentiable case

In this paper, we are concerned with a multiobjective optimization problem with inequality constraints. We introduce a constraint qualification and derive the Kuhn-Tucker type necessary conditions for efficiency. Moreover, we give conditions which ensure the constraint qualification.This work was done while the author was visiting the University of California, Berkeley, California.     

Lagrange multipliers and infinite-dimensional equilibrium problems

We prove the existence of the Lagrange multipliers for a constrained optimization problem, being the constraint set given by the convex set which characterizes the most important equilibrium problems. In order to obtain our result, we’ll make use of the new concept of quasi relative interior.     

Lagrange multipliers theorem and saddle point optimality criteria in mathematical programming

We prove a version of Lagrange multipliers theorem for nonsmooth functionals defined on normed spaces. Applying these results, we extend some results about saddle point optimality criteria in mathematical programming.     

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.     

Pseudonormality and a Lagrange Multiplier Theory for Constrained Optimization

We consider optimization problems with equality, inequality, and abstract set constraints, and we explore various characteristics of the constraint set that imply the existence of Lagrange multipliers. We prove a generalized version of the Fritz–John theorem, and we introduce new and general conditions that extend and unify the major constraint qualifications. Among these conditions, two new properties, pseudonormality and quasinormality, emerge as central within the taxonomy of interesting constraint characteristics. In the case where there is no abstract set constraint, these properties provide the connecting link between the classical constraint qualifications and two distinct pathways to the existence of Lagrange multipliers: one involving the notion of quasiregularity and the Farkas lemma, and the other involving the use of exact penalty functions. The second pathway also applies in the general case where there is an abstract set constraint.     

Lagrange multiplers for pareto nonsmooth programming problems in banach spaces

We establish the existence of Lagrange multipliers for general Pareto multiobjective mathematical programming problems in Banach spaces. Here the data are general nonsmooth strongly compactly ipschitzian mappings     

Existence of optimal Lagrange multipliers for certain nonconvex allocation problems

The use of Lagrange multipliers for decentralization of large resource allocation problems is well known. However, these dual techniques may suffer from the drawback ofduality gaps, to guarantee the absence of which various functions are required to be convex. This limits greatly the applicability of the decentralized approach. We show that less restrictive conditions can be formulated for a certain class of allocation problems, which we call resource management problems, which typically occur in large operational systems. We present a theorem for the existence of optimal multipliers, while placing almost no restrictions on the forms of the resource usage functions or the domains of the decision variables. Efficient solution algorithms, with provable convergence properties, have been given in a companion paper. Our results justify the application of dual methods to this class ofreal-world problems.The author is indebted to Mr. G. Karady and Professor Y. C. Ho of Harvard University for their valuable comments, and also to the referees for their helpful suggestions. This research was partially supported by the Office of Naval Research, under the Joint Services Electronic Program, Contract No. N0001475-C-0648, and by the National Science Foundation, Grant No. ENG-78-15231.     

Separation of Sets,Lagrange Multipliers,and Totally Regular Extremum Problems

We introduce the concept of total regularity for the separation of sets, and we give a characterization of it. Also, we prove the equivalence between total regularity and boundedness of the generalized multipliers associated to the separation, and we compute the value of the bound. Then, we give a theorem concerning the uniqueness of such multipliers. Afterward, the previous results are applied to the study of the impossibility of generalized systems; particular attention is devoted to systems arising from the optimality conditions of constrained extremum problems.     

Coupled FEM-BEM for elastoplastic contact problems using Lagrange multipliers

The coupling of the elastoplastic finite element and elastic boundary element methods for two-dimensional frictionless contact stress analysis is presented. Interface traction matching (boundary element approach), which involves the force terms in the finite element analysis being transformed to tractions, is chosen for the coupling method. The analysis at the contact region is performed by the finite element method, and the Lagrange multiplier approach is used to apply the contact constraints. Since the analyses of elastoplastic problems are non-linear and involve iterative solution, the reduced size of the final system of equations introduced by combining the two methods is very advantageous, especially for contact problems where the nature of the problem also involves an iterative scheme.     

Lagrange Multipliers and saddle points in multiobjective programming

In this paper, we present several conditions for the existence of a Lagrange multiplier or a weak saddle point in multiobjective optimization. Relations between a Lagrange multiplier and a weak saddle point are established. A sufficient condition is also given for the equivalence of the Benson proper efficiency and the Borwein proper efficiency.This research was supported by NSFC under Grant No. 78900011 and by BMADIS. The authors are grateful to two referees for supplying valuable comments and pointing out detailed corrections to the draft paper. The authors also wish to thank Dr. P. L. Yu for valuable comments and suggestions.The revised version of this paper was completed while the second author visited the Faculty of Technical Mathematics and Informatics, Delft University of Technology, Delft, The Netherlands.     

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.     

On the existence and nonexistence of Lagrange multipliers in Banach spaces

The paper deals with the existence of Lagrange multipliers for a general nonlinear programming problem. Some regularity conditions are formulated which are, in a sense, the weakest to assure the existence of multipliers. A number of related conditions are discussed. The connection between the choice of suitable function spaces and the existence of multipliers is analyzed.This work was partly supported by the National Science Foundation, Grant No. GF-37298, to the Institute of Automatic Control, Technical University of Warsaw, Warsaw, Poland, and the Department of Computer and Control Sciences, University of Minnesota, Minneapolis, Minnesota.The author wishes to thank Professor A. P. Wierzbicki for many important remarks concerning the subject of this paper.     

Lagrange multipliers for set-valued optimization problems associated with coderivatives

In this paper we investigate a vector optimization problem (P) where objective and constraints are given by set-valued maps. We show that by mean of marginal functions and suitable scalarizing functions one can characterize certain solutions of (P) as solutions of a scalar optimization problem (SP) with single-valued objective and constraint functions. Then applying some classical or recent results in optimization theory to (SP) and using estimates of subdifferentials of marginal functions, we obtain optimality conditions for (P) expressed in terms of Lagrange or sequential Lagrange multipliers associated with various coderivatives of the set-valued data.     

Constraint qualifications for nonsmooth programming

ABSTRACT

By using an implicit function theorem and a result of error bound, we provide new constraint qualifications ensuring the Karush–Kuhn–Tuker necessary optimality conditions for both smooth and nonsmooth optimization problems in normed spaces or Banach spaces.     

Regularity conditions for constrained extremum problems via image space

Exploiting the image-space approach, we give an overview of regularity conditions. A notion of regularity for the image of a constrained extremum problem is given. The relationship between image regularity and other concepts is also discussed. It turns out that image regularity is among the weakest conditions for the existence of normal Lagrange multipliers.     

Duality and existence theory for nondifferentiable programming

Necessary and sufficient conditions of optimality are given for a nonlinear nondifferentiable program, where the constraints are defined via closed convex cones and their polars. These results are then used to obtain an existence theorem for the corresponding stationary point problem, under some convexity and regularity conditions on the functions involved, which also guarantee an optimal solution to the programming problem. Furthermore, a dual problem is defined, and a strong duality theorem is obtained under the assumption that the constraint sets of the primal and dual problems are nonempty.     

Lagrange multipliers for functions derivable along directions in a linear subspace

We prove a Lagrange multipliers theorem for a class of functions that are derivable along directions in a linear subspace of a Banach space where they are defined. Our result is available for topological linear vector spaces and is stronger than the classical one even for two-dimensional spaces, because we only require the differentiablity of functions at critical points. Applying these results we generalize the Lax-Milgram theorem. Some applications in variational inequalities and quasilinear elliptic equations are given.

     

On generalized Nash equilibrium in infinite dimension: the Lagrange multipliers approach

In this note we study a class of generalized Nash equilibrium problems and characterize the solutions which have the property that all players share the same Lagrange multipliers. Nash equilibria of this kind were introduced by Rosen in 1965, in finite-dimensional spaces. In order to obtain the same property in infinite dimension, we use very recent developments of a new duality theory. In view of its usefulness in the study of time-dependent or stochastic equilibrium problems, an application in Lebesgue spaces is given.     

On the mixed finite element method with Lagrange multipliers

In this note we analyze a modified mixed finite element method for second‐order elliptic equations in divergence form. As a model we consider the Poisson problem with mixed boundary conditions in a polygonal domain of R ². The Neumann (essential) condition is imposed here in a weak sense, which yields the introduction of a Lagrange multiplier given by the trace of the solution on the corresponding boundary. This approach allows to handle nonhomogeneous Neumann boundary conditions, theoretically and computationally, in an alternative and usually easier way. Then we utilize the classical Babu?ka‐Brezzi theory to show that the resulting mixed variational formulation is well posed. In addition, we use Raviart‐Thomas spaces to define the associated finite element method and, applying some elliptic regularity results, we prove the stability, unique solvability, and convergence of this discrete scheme, under appropriate assumptions on the mesh sizes. Finally, we provide numerical results illustrating the performance of the algorithm for smooth and singular problems. © 2003 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 19: 192–210, 2003