Mathematical programs with vanishing constraints: optimality conditions and constraint qualifications

We consider a difficult class of optimization problems that we call a mathematical program with vanishing constraints. Problems of this kind arise in various applications including optimal topology design problems of mechanical structures. We show that some standard constraint qualifications like LICQ and MFCQ usually do not hold at a local minimum of our program, whereas the Abadie constraint qualification is sometimes satisfied. We also introduce a suitable modification of the standard Abadie constraint qualification as well as a corresponding optimality condition, and show that this modified constraint qualification holds under fairly mild assumptions. We also discuss the relation between our class of optimization problems with vanishing constraints and a mathematical program with equilibrium constraints.     

Necessary optimality conditions for constrained optimization problems under relaxed constraint qualifications

We derive first- and second-order necessary optimality conditions for set-constrained optimization problems under the constraint qualification-type conditions significantly weaker than Robinson’s constraint qualification. Our development relies on the so-called 2-regularity concept, and unifies and extends the previous studies based on this concept. Specifically, in our setting constraints are given by an inclusion, with an arbitrary closed convex set on the right-hand side. Thus, for the second-order analysis, some curvature characterizations of this set near the reference point must be taken into account.      

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).     

Optimality conditions for nondifferentiable convex semi-infinite programming

This paper gives characterizations of optimal solutions to the nondifferentiable convex semi-infinite programming problem, which involve the notion of Lagrangian saddlepoint. With the aim of giving the necessary conditions for optimality, local and global constraint qualifications are established. These constraint qualifications are based on the property of Farkas-Minkowski, which plays an important role in relation to certain systems obtained by linearizing the feasible set. It is proved that Slater's qualification implies those qualifications.     

On constraint qualifications for second-order optimality conditions depending on a single Lagrange multiplier

We present new constraint qualifications (CQs) to ensure the validity of some well-known second-order optimality conditions. Our main interest is on second-order conditions that can be associated with numerical methods for solving constrained optimization problems. Such conditions depend on a single Lagrange multiplier, instead of the whole set of Lagrange multipliers. For each condition, we characterize the weakest CQ that guarantees its fulfillment at local minimizers, while proposing new weak conditions implying them. Relations with other CQs are discussed.     

The bilevel programming problem: reformulations, constraint qualifications and optimality conditions

We consider the bilevel programming problem and its optimal value and KKT one level reformulations. The two reformulations are studied in a unified manner and compared in terms of optimal solutions, constraint qualifications and optimality conditions. We also show that any bilevel programming problem where the lower level problem is linear with respect to the lower level variable, is partially calm without any restrictive assumption. Finally, we consider the bilevel demand adjustment problem in transportation, and show how KKT type optimality conditions can be obtained under the partial calmness, using the differential calculus of Mordukhovich.     

Optimality conditions and duality in multiple objective programming involving semilocally convex and related functions

A nonlinear multiple objective programming problem is considered where the functions involved are nondifferentiable. By considering the concept of weak minima, the Fritz John type and Karush-Kuhn- Tucker type necessary optimality conditions and Wolfe and Mond-Weir type duality results are given in terms of the right differentials of the functions. The duality results are stated by using the concepts of generalized semilocally convex functions     

Characterization of optimality in convex programming without a constraint qualification

Necessary and sufficient conditions of optimality are given for convex programming problems with no constraint qualification. The optimality conditions are stated in terms of consistency or inconsistency of a family of systems of linear inequalities and cone relations.This research was supported by Project No. NR-047-021, ONR Contract No. N00014-67-A-0126-0009 with the Center for Cybernetics Studies, The University of Texas; by NSF Grant No. ENG-76-10260 at Northwestern University; and by the National Research Council of Canada.     

Necessary and sufficient conditions for some constraint qualifications in quasiconvex programming

In this paper, we investigate relations between constraint qualifications in quasiconvex programming. At first, we show a necessary and sufficient condition for the closed cone constraint qualification for quasiconvex programming (Q-CCCQ), and investigate some sufficient conditions for the Q-CCCQ. Also, we consider a relation between the Q-CCCQ and the basic constraint qualification for quasiconvex programming (Q-BCQ) and we compare the Q-BCQ with some constraint qualifications.     

Optimality conditions and the basic constraint qualification for quasiconvex programming

In this paper, we consider optimality conditions and a constraint qualification for quasiconvex programming. For this purpose, we introduce a generator and a new subdifferential for quasiconvex functions by using Penot and Volle’s theorem.     

First and second-order necessary and sufficient optimality conditions for infinite-dimensional programming problems

First-order and second-order necessary and sufficient optimality conditions are given for infinite-dimensional programming problems with constraints defined by arbitrary closed convex cones. The necessary conditions are immediate generalizations of those known for the finite-dimensional case. However, this does not hold for the sufficient conditions as illustrated by a counterexample. Here, to go from finite to infinite dimensions, causes an essential change in the proof-techniques and the results. We present modified sufficient conditions of first-order and of second-order which are based on a strengthening of the usual assumptions on the derivative of the objective function and on the second derivative of the Lagrangian.     

KKT conditions for weak compact convex sets,theorems of the alternative,and optimality conditions

We present several equivalent conditions for the Karush–Kuhn–Tucker conditions for weak^? compact convex sets. Using them, we extend several existing theorems of the alternative in terms of weak^? compact convex sets. Such extensions allow us to express the KKT conditions and hence necessary optimality conditions for more general nonsmooth optimization problems with inequality and equality constraints. Furthermore, several new equivalent optimality conditions for optimization problems with inequality constraints are obtained.     

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.     

A note on strong duality in convex semidefinite optimization: necessary and sufficient conditions

A strong duality which states that the optimal values of the primal convex problem and its Lagrangian dual problem are equal (i.e. zero duality gap) and the dual problem attains its maximum is a corner stone in convex optimization. In particular it plays a major role in the numerical solution as well as the application of convex semidefinite optimization. The strong duality requires a technical condition known as a constraint qualification (CQ). Several CQs which are sufficient for strong duality have been given in the literature. In this note we present new necessary and sufficient CQs for the strong duality in convex semidefinite optimization. These CQs are shown to be sharper forms of the strong conical hull intersection property (CHIP) of the intersecting sets of constraints which has played a critical role in other areas of convex optimization such as constrained approximation and error bounds. Research was partially supported by the Australian Research Council. The author is grateful to the referees for their helpful comments     

Constraint qualifications for optimality conditions and total Lagrange dualities in convex infinite programming

For an inequality system defined by an infinite family of proper convex functions (not necessarily lower semicontinuous), we introduce some new notions of constraint qualifications. Under the new constraint qualifications, we provide necessary and/or sufficient conditions for the KKT rules to hold. Similarly, we provide characterizations for constrained minimization problems to have total Lagrangian dualities. Several known results in the conic programming problem are extended and improved.     

First and second order sensitivity analysis of nonlinear programs under directional constraint qualification conditions

An extension and unification is presented, of the recent results of Shapiro [16] and Gatjvin & Janin [8] about second order differentiability of the optimal value function and directional differentiability of optimal solutions of perturbed mathematical programs, under a relaxed directional version of the Managasarian-Fromowitz constraint qualification condition introduced by Gollan [9]     

Necessary and sufficient conditions for regularity of constraints in convex programming

This paper derives some necessary and sufficient conditions for (Lagrangian) regularity of the nondifferentiable convex programming problem. Furthermore, some weakest constraint qualifications are presented using the supporting functions and their derivatives, the outer normal cones, the single constraint function and its directional derivatives and epigraph and the projections of the outer normal cones     

On solvability and regularity of a parametrized version of optimality conditions

We investigate a linear homotopyF(·,t) connecting an appropriate smooth equationG=0 with Kojima's (nonsmooth) systemK=0 describing critical points (primal —dual) of a nonlinear optimization problem (NLP) in finite dimension.Fort=0, our system may be seen e.g. as a starting system for an embedding procedure to determine a critical point to NLP. Fort1, it may be regarded as a regularization ofK.Conditions for regularity (necessary and sufficient) and solvability (sufficient) are studied. Though, formally, they can be given in a unified way, we show that their meaning differs fort < 1 andt=1. Particularily, no MFCQ-like condition must be imposed in order to ensure regularity fort < 1.     

Theorems of the alternative and optimality conditions

Several corrections to Ref. 1 are pointed out.