By Michael Patriksson (auth.)
Since i began operating within the quarter of nonlinear programming and, afterward, variational inequality difficulties, i've got often been shocked to discover that many algorithms, notwithstanding scattered in different journals, monographs and books, and defined quite another way, are heavily on the topic of one another. This publication is intended to aid the reader comprehend and relate algorithms to one another in a few intuitive style, and represents, during this admire, a consolidation of the sphere. The framework of algorithms provided during this booklet is termed price Approxi mation. (The preface of the Ph.D. thesis [Pat93d] explains the heritage to the paintings that bring about the thesis, and finally to this book.) It describes, for a given formula of a variational inequality or nonlinear programming challenge, an set of rules via approximating mappings and difficulties, a precept for the replace of the generation issues, and a benefit functionality which publications and displays the convergence of the set of rules. One objective of this publication is to supply this framework as an intuitively allure ing software for describing an set of rules. one of many benefits of the framework, or any moderate framework for that topic, is that algorithms will be simply comparable and in comparison via its use. This framework is restricted in that it covers an unlimited variety of tools, whereas nonetheless being quite targeted; the extent of abstraction is in reality almost like that of the unique challenge statement.
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Additional info for Nonlinear Programming and Variational Inequality Problems: A Unified Approach
3. 2 The updating phase In general, we can of course not expect that yt solves GVIP(F, u, X). The second step of a CA algorithm utilizes the subproblem solution yt to improve upon xt. In the basic algorithm, we utilize yt in the construction of a direction of search, df := yt - x t , in which a step is taken to reduce some measure of the distance from xt to the set SOL(F, u, X). We are interested in utilizing a measure which can be evaluated at any point of dom un X, has sufficient continuity properties, and is a merit function in the sense that its set of (local) minimizers (or, stationary points) coincide with SOL(F,u,X).
We also supply formulas for the calculation of directional derivatives and gradients, when they exist. An important observation made is that the possible iteration-dependency of the cost approximating function cp, which for the most part is immaterial to results established in Chapter 3 for nonlinear programs, here does affect some results, since the merit function 'Ij; is constructed based on the current iterate. The CA algorithm contains a large class of known algorithms for variational inequalities, some of which have been used in conjunction with the gap functions that are inherent in their construction.
We show that the 28 Nonlinear Programming and Variational Inequality Problems standard form of the CA algorithm is of the form of a splitting method as well as an inexact version of a regularization method, albeit it is much more general in scope, and we derive several algorithms, among which is Korpelevich's extra-gradient method, as inexact proximal point methods. 3 considers an optimization problem with explicit constraints, and shows how the framework of cost approximation covers primal-dual methods such as sequential quadratic programming, the Arrow-Hurwicz and Uzawa methods, and dual subgradient optimization, as well as a class of augmented Lagrangean methods.