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作者：Nguyen, T. T. V.; Strodiot, J. J.; Nguyen, V. H.
作者单位：University of Namur; Vietnam National University Ho Chi Minh City (VNUHCM) System
摘要：We present a bundle method for solving nonsmooth convex equilibrium problems based on the auxiliary problem principle. First, we consider a general algorithm that we prove to be convergent. Then we explain how to make this algorithm implementable. The strategy is to approximate the nonsmooth convex functions by piecewise linear convex functions in such a way that the subproblems are easy to solve and the convergence is preserved. In particular, we introduce a stopping criterion which is satisf...
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作者：Sorin, Sylvain
作者单位：heSam Universite; Conservatoire National Arts & Metiers (CNAM); Institut Polytechnique de Paris; Ecole Polytechnique; Sorbonne Universite
摘要：The exponential weight algorithm has been introduced in the framework of discrete time on-line problems. Given an observed process {X-m}(m)=1,2,... the input at stage m + 1 is an exponential function of the sum S-m = Sigma(m)(l=1) X-l. We define the analog algorithm for a continuous time process X-t and prove similar properties in terms of external or internal consistency. We then deduce results for discrete time from their counterpart in continuous time. Finally we compare this approach to an...
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作者：van Ngai, Huynh; Thera, Michel
作者单位：Universite de Limoges; Centre National de la Recherche Scientifique (CNRS)
摘要：In this paper, using the Frechet subdifferential, we derive several sufficient conditions ensuring an error bound for inequality systems in Asplund spaces. As an application we obtain in the context of Banach spaces a global error bound for quadratic nonconvex inequalities and we derive necessary optimality conditions for optimization problems.
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作者：Lasserre, Jean B.
作者单位：Centre National de la Recherche Scientifique (CNRS)
摘要：Given a compact basic semi-algebraic set K subset of R-n, a rational fraction f : R-n -> R, and a sequence of scalars y = (y(alpha)), we investigate when y(alpha) = integral K x(alpha) f d mu holds for all alpha is an element of N-n, i.e., when y is the moment sequence of some measure f d mu, supported on K. This yields a set of (convex) linear matrix inequalities(LMI). We also use semidefinite programming to develop a converging approximation scheme to evaluate the integral integral f d mu wh...
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作者：Borwein, Jonathan M.; Hamilton, Chris H.
作者单位：Dalhousie University
摘要：Of key importance in convex analysis and optimization is the notion of duality, and in particular that of Fenchel duality. This work explores improvements to existing algorithms for the symbolic calculation of subdifferentials and Fenchel conjugates of convex functions defined on the real line. More importantly, these algorithms are extended to enable the symbolic calculation of Fenchel conjugates on a class of real-valued functions defined on the real line. More importantly, these algorithms ...
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作者：Hare, Warren; Sagastizabal, Claudia
作者单位：Instituto Nacional de Matematica Pura e Aplicada (IMPA); McMaster University
摘要：The proximal point mapping is the basis of many optimization techniques for convex functions. By means of variational analysis, the concept of proximal mapping was recently extended to nonconvex functions that are prox-regular and prox-bounded. In such a setting, the proximal point mapping is locally Lipschitz continuous and its set of fixed points coincide with the critical points of the original function. This suggests that the many uses of proximal points, and their corresponding proximal e...
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作者：Karas, Elizabeth; Ribeiro, Ademir; Sagastizabal, Claudia; Solodov, Mikhail
作者单位：Universidade Federal do Parana
摘要：For solving nonsmooth convex constrained optimization problems, we propose an algorithm which combines the ideas of the proximal bundle methods with the filter strategy for evaluating candidate points. The resulting algorithm inherits some attractive features from both approaches. On the one hand, it allows effective control of the size of quadratic programming subproblems via the compression and aggregation techniques of proximal bundle methods. On the other hand, the filter criterion for acc...
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作者：Penot, J. -P.; Zalinescu, C.
作者单位：Universite de Pau et des Pays de l'Adour; Centre National de la Recherche Scientifique (CNRS); Alexandru Ioan Cuza University
摘要：We use representations of maximal monotone operators for studying recession (or asymptotic) operators associated to maximal monotone operators. Such a concept is useful for dealing with unboundedness.
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作者：Todd, M. J.
作者单位：Cornell University
摘要：Suppose (x) over bar and (s) over bar lie in the interiors of a cone K and its dual K*, respectively. We seek dual ellipsoidal norms such that the product of the radii of the largest inscribed balls centered at (x) over bar and (s) over bar and inscribed in K and K *, respectively, is maximized. Here the balls are defined using the two dual norms. When the cones are symmetric, that is self-dual and homogeneous, the solution arises directly from the Nesterov-Todd primal-dual scaling. This shows...
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作者：Jofre, Alejandro; Wets, Roger J. -B.
作者单位：University of California System; University of California Davis; Universidad de Chile
摘要：We explore convergence notions for bivariate functions that yield convergence and stability results for their maxinf (or minsup) points. This lays the foundations for the study of the stability of solutions to variational inequalities, the solutions of inclusions, of Nash equilibrium points of non-cooperative games and Walras economic equilibrium points, of fixed points, of solutions to inclusions, the primal and dual solutions of convex optimization problems and of zero-sum games. These appli...
