Weak subdifferential/superdifferential,weak exhausters and optimality conditions

In this work, the notion of weak superdifferential is presented. Some calculation rules are given to evaluate weak subdifferential and weak superdifferential of some classes of functions represented by support functions. Moreover, some methods are obtained to calculate weak subdifferential of convex functions. In addition, the concept of weak lower and weak upper exhausters of positively homogeneous functions are introduced by using weak subdifferential and weak superdifferential, respectively. In terms of weak exhausters, some optimality conditions are given to find local or global minimizers/maximizers of some classes of functions.     

Proper exhausters and coexhausters in nonsmooth analysis

There exist many tools to analyze nonsmooth functions. For convex and max-type functions, the notion of subdifferential is used, for quasidifferentiable functions – that of quasidifferential. By means of these tools, one is able to solve, e.g. the following problems: to get an approximation of the increment of a functional, to formulate conditions for an extremum, to find steepest descent and ascent directions and to construct numerical methods. For arbitrary directionally differentiable functions, these problems are solved by employing the notions of upper and lower exhausters and coexhausters, which are generalizations of such notions of nonsmooth analysis as sub- and superdifferentials, quasidifferentials and codifferentials. Exhausters allow one to construct homogeneous approximations of the increment of a functional while coexhausters provide nonhomogeneous approximations. It became possible to formulate conditions for an extremum in terms of exhausters and coexhausters. It turns out that conditions for a minimum are expressed by an upper exhauster, and conditions for a maximum are formulated via a lower one. This is why an upper exhauster is called a proper one for the minimization problem (and adjoint for the maximization problem) while a lower exhauster is called a proper one for the maximization problem (and adjoint for the minimization problem). The conditions obtained provide a simple geometric interpretation and allow one to find steepest descent and ascent directions. In this article, optimization problems are treated by means of proper exhausters and coexhausters.     

Reducing Exhausters

The notions of upper and lower exhausters were introduced by Demyanov (Optimization 45:13–29, 1999). Upper and lower exhausters can be employed to study a very wide range of positively homogeneous functions, for example, various directional derivatives of nonsmooth functions. Exhausters are not uniquely defined; hence, the problem of minimality arises naturally. This paper describes some techniques for reducing exhausters, both in size and amount of sets. We define also a modified convertor which provides much more flexibility in converting upper exhausters to lower ones and vice versa, and allows us to obtain much smaller sets.     

Exact calculus of Fréchet subdifferentials for Hadamard directionally differentiable functions

We continue the study of the calculus of the generalized subdifferentials started in [V.F. Demyanov, V. Roshchina, Exhausters and subdifferentials in nonsmooth analysis, Optimization (2006) (in press)] and [V. Roshchina, Relationships between upper exhausters and the basic subdifferential in Variational Analysis, Journal of Mathematical Analysis and Applications 334 (2007) 261–272] and provide some basic calculus rules for the Fréchet subdifferentials via collections of compact convex sets associated with Hadamard directional derivative. The main result of this paper is the sum rule for the Fréchet subdifferential in the form of an equality, which holds for Hadamard directionally differentiable functions, and is of significant interest from the points of view of both theory and applications.     

Relationships between upper exhausters and the basic subdifferential in variational analysis

In this paper we establish a relationship between the basic subdifferential and upper exhausters of positively homogeneous and polyhedral functions. In the case of a finite exhauster this relationship is represented in a form of an equality, and in the case of a Lipschitz function an inclusion formula is obtained.     

Adjoint Coexhausters in Nonsmooth Analysis and Extremality Conditions

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.     

Optimality Conditions for Nonconvex Constrained Optimization Problems

Abstract

We present optimality conditions for a class of nonsmooth and nonconvex constrained optimization problems. To achieve this aim, various well-known constraint qualifications are extended based on the concept of tangential subdifferential and the relations between them are investigated. Moreover, local and global necessary and sufficient optimality conditions are derived in the absence of convexity of the feasible set. In addition to the theoretical results, several examples are provided to illustrate the advantage of our outcomes.     

Optimality for nonsmooth fractional multiple objective programming

In this study the necessary and sufficient optimality conditions for nonsmooth fractional multiple objective optimization problems are provided. Our idea is based on using the properties of limiting subdifferential vectors in nonsmooth analysis and a separation theorem in convex analysis.     

Necessary Conditions in Nonsmooth Minimization via Lower and Upper Subgradients

The paper concerns first-order necessary optimality conditions for problems of minimizing nonsmooth functions under various constraints in infinite-dimensional spaces. Based on advanced tools of variational analysis and generalized differential calculus, we derive general results of two independent types called lower subdifferential and upper subdifferential optimality conditions. The former ones involve basic/limiting subgradients of cost functions, while the latter conditions are expressed via Fréchet/regular upper subgradients in fairly general settings. All the upper subdifferential and major lower subdifferential optimality conditions obtained in the paper are new even in finite dimensions. We give applications of general optimality conditions to mathematical programs with equilibrium constraints deriving new results for this important class of intrinsically nonsmooth optimization problems.     

Comparison Between Quasidifferentials and Exhausters

Directional derivatives play one of the major roles in optimization. Optimality conditions can be described in terms of these objects. These conditions, however, are not constructive. To overcome this problem, one has to represent the directional derivative in special forms. Two such forms are quasidifferentials and exhausters proposed by V.F. Demyanov. Quasidifferentials were introduced in 1980s. Optimality conditions in terms of these objects were developed by L.N. Polyakova and V.F. Demyanov. It was described how to find directions of steepest descent and ascent when these conditions are not satisfied. This paved a way for constructing new optimization algorithms. Quasidifferentials allow one to treat a wide class of functions. V.F. Demyanov introduced the notion of exhausters in 2000s to expand the class of functions that can be treated. It should be noted that a great contribution to the emergence of this notion was made by B.N. Pshenichny and A.M. Rubinov. In this work it is shown that exhausters not only allow one to treat a wider class of functions than quasidifferentials (since every quasidifferentiable function has exhausters) but is also preferable even for quasidifferentiable functions when solving nonsmooth optimization problems.     

Optimality conditions for nonsmooth mathematical programs with equilibrium constraints,using convexificators

In this paper, we deal with constraint qualifications, stationary concepts and optimality conditions for a nonsmooth mathematical program with equilibrium constraints (MPEC). The main tool in our study is the notion of convexificator. Using this notion, standard and MPEC Abadie and several other constraint qualifications are proposed and a comparison between them is presented. We also define nonsmooth stationary conditions based on the convexificators. In particular, we show that GS-stationary is the first-order optimality condition under generalized standard Abadie constraint qualification. Finally, sufficient conditions for global or local optimality are derived under some MPEC generalized convexity assumptions.     

Some relationships among quasidifferential,weak subdifferential and exhausters

In this study, some relationships between quasidifferentiability and weakly subdifferentiability of a function are investigated, and some results on calculating the quasidifferential in terms of weak exhausters are presented. Moreover, under some assumptions weak subdifferentiability of a quasidifferentiable function is proved and the weak subdifferential is obtained by using the quasidifferential. Under some assumptions, it is proved that a positively homogeneous and lower semicontinuous function is quasidifferentiable. It is also shown that directional differentiability and weak subdifferentiability imply quasidifferentiability. Furthermore, a method to evaluate the quasidifferential is given by using reduced weak lower exhausters.     

Extremum conditions for a nonsmooth function in terms of exhausters and coexhausters

The notions of upper and lower exhausters and coexhausters are discussed and necessary conditions for an unconstrained extremum of a nonsmooth function are derived. The necessary conditions for a minimum are formulated in terms of an upper exhauster (coexhauster) and the necessary conditions for a maximum are formulated in terms of a lower exhauster (coexhauster). This involves the problem of transforming an upper exhauster (coexhauster) into a lower exhauster (coexhauster) and vice versa. The transformation is carried out by means of a conversion operation (converter). Second-order approximations obtained with the help of second-order (upper and lower) coexhausters are considered. It is shown how a secondorder upper coexhauster can be converted into a lower coexhauster and vice versa. This problem is reduced to using a first-order conversion operator but in a space of a higher dimension. The obtained result allows one to construct second-order methods for the optimization of nonsmooth functions (Newton-type methods).     

Abstract convex approximations of nonsmooth functions

In the article we use abstract convexity theory in order to unify and generalize many different concepts of nonsmooth analysis. We introduce the concepts of abstract codifferentiability, abstract quasidifferentiability and abstract convex (concave) approximations of a nonsmooth function mapping a topological vector space to an order complete topological vector lattice. We study basic properties of these notions, construct elaborate calculus of abstract codifferentiable functions and discuss continuity of abstract codifferential. We demonstrate that many classical concepts of nonsmooth analysis, such as subdifferentiability and quasidifferentiability, are particular cases of the concepts of abstract codifferentiability and abstract quasidifferentiability. We also show that abstract convex and abstract concave approximations are a very convenient tool for the study of nonsmooth extremum problems. We use these approximations in order to obtain various necessary optimality conditions for nonsmooth nonconvex optimization problems with the abstract codifferentiable or abstract quasidifferentiable objective function and constraints. Then, we demonstrate how these conditions can be transformed into simpler and more constructive conditions in some particular cases.     

Multiplicity of nontrivial solutions for elliptic equations with nonsmooth potential and resonance at higher eigenvalues

We consider a semilinear elliptic equation with a nonsmooth, locally Lipschitz potential function (hemivariational inequality). Our hypotheses permit double resonance at infinity and at zero (double-double resonance situation). Our approach is based on the nonsmooth critical point theory for locally Lipschitz functionals and uses an abstract multiplicity result under local linking and an extension of the Castro-Lazer-Thews reduction method to a nonsmooth setting, which we develop here using tools from nonsmooth analysis.     

Exhausters af a positively homogeneous function *

Notions of upper exhauster and lower exhauster of a positively homogeneous (of the first degree) function h: ? ⁿ →? are introduced. They are linked to exhaustive families of upper convex and lower concave approximations of the function h. The pair of an upper exhauster and a lower exhauster is called a biexhauster of h. A calculus for biexhausters is described (in particular, a composition theorem is formulated). The problem of minimality of exhausters is stated. Necessary and sufficient conditions for a constrained minimum and a constrained maximum of a directionally differentiable function f: ? ⁿ →? are formulated in terms of exhausters of the directional derivative of f. In general, they are described by means of exhausters of the Hadamard upper and lower directional derivatives of the function f. To formulate conditions for a minimum, an upper exhauster is employed while conditions for a maximum are formulated via a lower exhauster of the respective directional derivative (the Hadamard lower derivative for a minimum and the Hadamard upper derivative for a maximum).

If a point x _o is not stationary then directions of steepest ascent and descent can also be calculated by means of exhausters.     

Optimality and duality in nonsmooth multiobjective fractional programming problem with constraints

In this paper, we establish some quotient calculus rules in terms of contingent derivatives for the two extended-real-valued functions defined on a Banach space and study a nonsmooth multiobjective fractional programming problem with set, generalized inequality and equality constraints. We define a new parametric problem associated with these problem and introduce some concepts for the (local) weak minimizers to such problems. Some primal and dual necessary optimality conditions in terms of contingent derivatives for the local weak minimizers are provided. Under suitable assumptions, sufficient optimality conditions for the local weak minimizers which are very close to necessary optimality conditions are obtained. An application of the result for establishing three parametric, Mond–Weir and Wolfe dual problems and several various duality theorems for the same is presented. Some examples are also given for our findings.

     

非光滑最优化问题的充分条件

§l引言 考虑如下最优化问题: fNP、) /n““,(。) 、 Iz E R一{z E E‘/口(z)≤0,^(z)一0},其中,:驴一E,g一(9∥“,‰)’:E。一13",h一(^∥一,^,)”:驴一E’并且f,g。(1≤。≤,,,‘),hjL I≤j≤p)均是E。上的局部Lipsehitz函数. 最近,唐焕文等在[1]中提出了广义伪凸函数并在[2]中利用这类函数讨论了非光滑最优化问题解的充分条件.在这篇文章里,我们提出几类,“义凸性函数,在这些凸性条件下我们证明了非光滑最优化问题(NP)的解的允分条件,它包括Kuhn—Tucker允分条件和FritzJohn允分条件.§2概念 设D是驴的一个开子集,一个实值函数…     

THE CHECK OF OPTIMALITY CONDITIONS IN NONSMOOTH PROGRAMMING

In this paper,the geometrical meaning of two basic optimality conditions for a class of common cases in nonsnaooth programming is exposed. Thus it makes the check of optimalityconditions for this class of nonsmooth programming problem much easier.     

Comments on: Farkas’ Lemma: three decades of generalizations for mathematical optimization

In these comments on the excellent survey by Dinh and Jeyakumar, we briefly discuss some recently developed topics and results on applications of extended Farkas’ lemma(s) and related qualification conditions to problems of variational analysis and optimization, which are not fully reflected in the survey. They mainly concern: Lipschitzian stability of feasible solution maps for parameterized semi-infinite and infinite programs with linear and convex inequality constraints indexed by arbitrary sets; optimality conditions for nonsmooth problems involving such constraints; evaluating various subdifferentials of optimal value functions in DC and bilevel infinite programs with applications to Lipschitz continuity of value functions and optimality conditions; calculating and estimating normal cones to feasible solution sets for nonlinear smooth as well as nonsmooth semi-infinite, infinite, and conic programs with deriving necessary optimality conditions for them; calculating coderivatives of normal cone mappings for convex polyhedra in finite and infinite dimensions with applications to robust stability of parameterized variational inequalities. We also give some historical comments on the original Farkas’ papers.