A New Constraint Qualification and Sharp Optimality Conditions for Nonsmooth Mathematical Programming Problems in Terms of Quasidifferentials

In the research of optimization problems, optimality conditions play an important role. By using some derivatives, various types of necessary and/or sufficient optimality conditions have been introduced by many researchers. Especially, in convex programming, necessary and sufficient optimality conditions in terms of the subdifferential have been studied extensively. Recently, necessary and sufficient optimality conditions for quasiconvex programming have been investigated by the authors. However, there are not so many results concerned with Karush–Kuhn–Tucker type optimality conditions for non-differentiable quasiconvex programming. In this paper, we study a Karush–Kuhn–Tucker type optimality condition for quasiconvex programming in terms of Greenberg–Pierskalla subdifferential. We show some closedness properties for Greenberg–Pierskalla subdifferential. Under the Slater constraint qualification, we show a necessary and sufficient optimality condition for essentially quasiconvex programming in terms of Greenberg–Pierskalla subdifferential. Additionally, we introduce a necessary and sufficient constraint qualification of the optimality condition. As a corollary, we show a necessary and sufficient optimality condition for convex programming in terms of the subdifferential.

Nonsmooth multiobjective mathematical programming problems with equilibrium constraints (NMMPEC) are studied in this article in the Hadamard manifold setting. In the context of (NMMPEC), the generalized Guignard constraint qualification (GGCQ) is formulated within the framework of Hadamard manifolds. Moreover, Karush–Kuhn–Tucker (KKT)-type necessary optimality conditions are derived for (NMMPEC). Thereafter, we explore constraint qualifications (CQ) tailored to (NMMPEC) in the Hadamard manifold setting. Interrelations between these constraint qualifications are subsequently derived. It is further demonstrated that the proposed constraint qualifications, when satisfied, ensure that GGCQ holds. It is noteworthy that constraint qualifications and optimality conditions for (NMMPEC) have not been investigated in the Hadamard manifold setting.

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.

Nonsmooth semi-infinite programming problems with mixed constraints

In this paper we study constraint qualifications and optimality conditions for bilevel programming problems. We strive to derive checkable constraint qualifications in terms of problem data and applicable optimality conditions. For the bilevel program with convex lower level program we discuss drawbacks of reformulating a bilevel programming problem by the mathematical program with complementarity constraints and present a new sharp necessary optimality condition for the reformulation by the mathematical program with a generalized equation constraint. For the bilevel program with a nonconvex lower level program we propose a relaxed constant positive linear dependence (RCPLD) condition for the combined program.

This is a survey of high‐order necessary optimality conditions for control problems with inequality and equality terminal constraints. Both classical and Pontryagin extremals are considered. For classical extremals, the characteristics of necessary optimality conditions in terms of variations are given, and the investigations devoted to Kelley's conditions are discussed in detail. For Pontryagin extremals, two types of results are given: generalizations and extensions of matrix impulse optimality condition and high‐order maximum principle.

In the classical (“smooth”) mathematical analysis, a differentiable function is studied by means of the derivative (gradient in the multidimensional space). In the case of nondifferentiable functions, the tools of nonsmooth analysis are to be employed. In convex analysis and minimax theory, the corresponding classes of functions are investigated by means of the subdifferential (it is a convex set in the dual space), quasidifferentiable functions are treated via the notion of quasidifferential (which is a pair of sets). To study an arbitrary directionally differentiable function, the notions of upper and lower exhausters (each of them being a family of convex sets) are used. It turns out that conditions for a minimum are described by an upper exhauster, while conditions for a maximum are stated in terms of a lower exhauster. This is why an upper exhauster is called a proper one for the minimization problem (and an adjoint exhauster for the maximization problem) while a lower exhauster will be referred to as a proper one for the maximization problem (and an adjoint exhauster for the minimization problem). The directional derivatives (and hence, exhausters) provide first-order approximations of the increment of the function under study. These approximations are positively homogeneous as functions of direction. They allow one to formulate optimality conditions, to find steepest ascent and descent directions, to construct numerical methods. However, if, for example, the maximizer of the function is to be found, but one has an upper exhauster (which is not proper for the maximization problem), it is required to use a lower exhauster. Instead, one can try to express conditions for a maximum in terms of upper exhauster (which is an adjoint one for the maximization problem). The first to get such conditions was Roshchina. New optimality conditions in terms of adjoint exhausters were recently obtained by Abbasov. The exhauster mappings are, in general, discontinuous in the Hausdorff metric, therefore, computational problems arise. To overcome these difficulties, the notions of upper and lower coexhausters are used. They provide first-order approximations of the increment of the function which are not positively homogeneous any more. These approximations also allow one to formulate optimality conditions, to find ascent and descent directions (but not the steepest ones), to construct numerical methods possessing good convergence properties. Conditions for a minimum are described in terms of an upper coexhauster (which is, therefore, called a proper coexhauster for the minimization problem) while conditions for a maximum are described in terms of a lower coexhauster (which is called a proper one for the maximization problem). In the present paper, we derive optimality conditions in terms of adjoint coexhausters.

The bilevel programming problem (BLPP) is a sequence of two optimization problems where the constraint region of the upper level problem is determined implicitly by the solution set to the lower level problem. To obtain optimality conditions, we reformulate BLPP as a single level mathematical programming problem (SLPP) which involves the value function of the lower level problem. For this mathematical programming problem, it is shown that in general the usual constraint qualifications do not hold and the right constraint qualification is the calmness condition. It is also shown that the linear bilevel programming problem and the minmax problem satisfy the calmness condition automatically. A sufficient condition for the calmness for the bilevel programming problem with quadratic lower level problem and nondegenerate linear complementar¬ity lower level problem are given. First order necessary optimality condition are given using nonsmooth analysis. Second order sufficient optimality conditions are also give...

Higher-order KKT optimality conditions through contingent derivatives for constrained nonsmooth vector equilibrium problems

Constraint qualification in a general class of nondifferentiable mathematical programming problems

Mathematical programs with equilibrium constraints (MPEC) are nonlinear programs which do not satisfy any of the common constraint qualifications (CQ). In order to obtain first-order optimality conditions, constraint qualifications tailored to the MPECs have been developed and researched in the past. In this paper, we introduce a new Abadie-type constraint qualification for MPECs. We investigate sufficient conditions for this new CQ, discuss its relationship to several existing MPEC constraint qualifications, and introduce a new Slater-type constraint qualifications. Finally, we prove a new stationarity concept to be a necessary optimality condition under our new Abadie-type CQ.

In this paper, we study optimality conditions for vector optimization problems of a difference of convex mappings $$ (VP)\left\{\matrix{{\mathbb R}_{+}^{p} &-\hbox{Minimize} f(x)-g(x),\cr &\hbox{subject to the constraints} \cr & x\in C,l(x)\in -Q,Ax=b\cr &\hbox{and } h(x)-k(x)\in -{\mathbb R}_{+}^{m},}\right. $$ where \(f:=(f_{1},\ldots,f_{p}),g:=(g_{1},\ldots,g_{p}) h:=(h_{1},\ldots,h_{m}), k:=(k_{1},\ldots,k_{m}), Q\) is a closed convex cone in a Banach space Z, l is a mapping Q-convex from a Banach space X into Z, A is a continuous linear operator from X into a Banach space \(W, {\mathbb R}_{+}^{p}\) and \({\mathbb R}_{+}^{m}\) are respectively the nonnegative orthants of \({\mathbb R}^{p}\) and \({\mathbb R}^{m}\), C is a nonempty closed convex subset of X, b∈W, and the functions f i ,g i ,h j and k j are convex for i=1,...,p and j=1,ldots,m. Necessary optimality conditions for (VP) are established in terms of Lagrange-Fritz-John multipliers. When the set of constraints for (VP) is convex and under the generalized Slater constraint qualification introduced in Jeyakumar and Wolkowicz [11] , we derive necessary optimality conditions in terms of Lagrange-Karush-Kuhn-Tucker multipliers which are also sufficient whenever the functions g i ,i=1,...,p are polyhedrals. Our approach consists in using a special scalarization function. A necessary optimality condition for convex vector maximization problem is derived. Also an application to vector fractional mathematical programming is given. Our contribution extends the results obtained in scalar optimization by Hiriart-Urruty [9] and improve substantially the few results known in vector case (see for instance: [11], [12] and [14]).

A class of nonsmooth vector optimization problems are considered, where the feasible set defined by cone constraint and the objective and constraint functions are locally Lipschitz. The concept of the Clarke's generalized directional derivative for a locally Lipschitz function is introduced. By using this concept, necessary optimality condition for the unconstrained optimization problem is established. Furthermore, a Slater-type constraint qualification is given in such a way that they generalize the classical one, when the constraint functions are differentiable. Then, Kuhn–Tucker necessary optimality condition in terms of the Clarke subdifferentials is obtained.

For nonconvex optimization problems in Rn , a counterexample was recently given by Anitescu [SIAM J. Optim., 10 (2000), pp. 1116--1135] that showed that the Mangasarian--Fromovitz constraint qualification (MFCQ) is not sufficient for the classical necessary second-order optimality conditions to hold. We prove these optimality conditions with the assumptions that the set of Lagrange multipliers is a bounded line segment and a relaxed strict complementary slackness (SCS) condition holds. A new constraint qualification is presented in this paper: We assume that the MFCQ holds and the active constraint rank deficiency is 1. This assumption relaxes the linear independence constraint qualification and enforces the MFCQ. In addition, the set of Lagrange multipliers is a bounded line segment.

We consider a multi-objective semi-infinite programming problem with a feasible set defined by inequality constraints. First, we present a Fritz–John type necessary optimality condition. Then, we introduce two constraint qualifications and derive the weak and strong Karush–Kuhn–Tucker types necessary conditions for (weakly) efficient solution of the considered problem. Finally, an extension of a Caristi–Ferrara–Stefanescu result for the (\(\Phi\), \(\rho\))-invexity is proved, and some sufficient conditions are presented under this weak assumption. All results are given in term of Clarke subdifferential.