Proper and adjoint exhausters in nonsmooth analysis: optimality conditions

Demyanov

2012

J Optim Theory Appl

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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. J Optim Theory ApplThe 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.

Section: Discussionmentioning

confidence: 99%

“…In the present paper, which is a generalization and extension of results of [11], we derive optimality conditions in terms of adjoint coexhausters.…”

Section: Introductionmentioning

confidence: 90%

See 2 more Smart Citations

Adjoint Coexhausters in Nonsmooth Analysis and Extremality Conditions

Demyanov

2012

J Optim Theory Appl

Self Cite

“…Exhausters are families of convex compact sets by means of which the main part of the augment can be represented in the form of min-max or max-min of linear functions. Optimality conditions were described in terms of exhausters [9,10]. This promoted the construction of optimization algorithms.…”

Section: Introductionmentioning

confidence: 99%

Second-Order Minimization Method for Nonsmooth Functions Allowing Convex Quadratic Approximations of the Augment

2015

J Optim Theory Appl

Self Cite

Second-order methods play an important role in the theory of optimization. Due to the usage of more information about considered function, they give an opportunity to find the stationary point faster than first-order methods. Well-known and sufficiently studied Newton's method is widely used to optimize smooth functions. The aim of this work is to obtain a second-order method for unconstrained minimization of nonsmooth functions allowing convex quadratic approximation of the augment. This method is based on the notion of coexhausters-new objects in nonsmooth analysis, introduced by V. F. Demyanov. First, we describe and prove the second-order necessary condition for a minimum. Then, we build an algorithm based on that condition and prove its convergence. At the end of the paper, a numerical example illustrating implementation of the algorithm is given.

Geometric conditions of reduction of exhausters

2018

J Glob Optim

Exhausters are families of convex compact sets. They allow one to represent the principal part of the increment of a studied function in the form of minimax or maximin of linear functions. The calculus of exhausters was developed in the last decade. It gives formulas for building these families for a wide class of functions. There have been developed a number of optimality conditions that are described in terms of exhausters. This led to emergence of new optimizations algorithms. So exhausters became an effective tool in the study of nonsmooth functions. Since exhausters are not uniquely defined an important problems of their minimality and reduction arise. These problems were studied by researchers for decades.In this paper we propose new conditions for the verification of exhauster minimality and develop procedures for their reduction. The main advantage of our approach is its transparent geometric meaning.