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EFFICIENT EVALUATION OF SCALED PROXIMAL OPERATORS

ELECTRONIC TRANSACTIONS ON NUMERICAL ANALYSIS 2017年 46卷 1-22页

作者： Friedlander, Michael P. Goh, Gabriel Univ British Columbia Dept Comp Sci Vancouver BC V6T 1Z4 Canada Univ British Columbia Dept Math Vancouver BC V6T 1Z4 Canada Univ Calif Davis Dept Math Davis CA 95616 USA

Quadratic-support functions, cf. Aravkin, Burke, and Pillonetto [J. Mach. Learn. Res., 14 (2013)], constitute a parametric family of convex functions that includes a range of useful regularization terms found in applications of convex optimization. We show how an interior method can be used to efficiently compute the proximal operator of a quadratic-support function under different metrics. When the metric and the function have the right structure, the proximal map can be computed with costs nearly linear in the input size. We describe how to use this approach to implement quasi-Newton methods for a rich class of nonsmooth problems that arise, for example, in sparse optimization, image denoising, and sparse logistic regression.

关键词： support functions proximal-gradient quasi-Newton interior method

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A generalization of the Minkowski embedding theorem and applications

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FUZZY SETS AND SYSTEMS 1999年第2期102卷 263-269页

作者： Rojas-Medar, M Bassanezi, RC Román-Flores, H UNICAMP IMECC BR-13081 Campinas SP Brazil Univ Tarapaca Dept Matemat Arica Chile

Purl and Ralescu (1985) gave, recently, an extension of the Minkowski Embedding Theorem for the class E-L(n) of fuzzy sets u on R-n with the level application alpha --> L(alpha)u Lipschitzian on the C([0, 1] x Sn-1) space. In this work we extend the above result to the class E-C(n) of level-continuous applications. Moreover, we prove that E-C(n) is a complete metric space with E-L(n) not subset of or equal to E-C(n) and <(E-L(n))over bar> = E-C(n). To prove the last result, we use the multivalued Bernstein polynomials and the Vitali's approximation theorem for multifunction. Also, we deduce some properties in the setting of fuzzy random variable (multivalued). (C) 1999 Elsevier Science B.V. All rights reserved.

关键词： fuzzy sets set-convergences Hausdorff metric integration of multifunctions multivalued Bernstein polynomial Minkowski Embedding Theorem support functions

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GENERALIZED ARROW-BARANKIN-BLACKWELL THEOREMS IN LOCALLY CONVEX-SPACES

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JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS 1995年第1期84卷 93-101页

作者： CHEN, GY Professor Institute of Systems Science Academia Sinica Beijing China

This paper deals with generalizations of the Arrow-Barankin-Blackwell theorem in locally convex spaces, partially ordered by cones whose duals have nonempty quasi-interiors.

关键词： LOCALLY CONVEX SPACES QUASI-INTERIORS MINIMAL ELEMENTS support functions DENSITY

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A new numerical approach for solving shape optimization fourth-order spectral problems among convex domains

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COMPUTERS & MATHEMATICS WITH APPLICATIONS 2023年 149卷 171-189页

作者： Chakib, Abdelkrim Khalil, Ibrahim Sultan Moulay Slimane Univ Fac Sci & Tech Appl Math Team AMT Beni Mellal Morocco

In this work, we deal with a new numerical approach for solving some shape optimization eigenvalue problems governed by the bi-harmonic operator, under volume and convexity constraints. This is based on the new shape derivative formula, recently established in [10,11], which allows to express the shape derivative of optimal shape design problems of minimizing volume cost functionals, in term of support functions. This avoids some computational disadvantages required for the classical shape derivative method involving the vector fields [16, 18], when one use the finite elements discretization for approximating the auxiliary boundary value problems in shape optimization processes. So, we first show the existence of the shape derivative of the eigenvalues for these fourth-order problems with respect to a family of convex domains and express its formula by means of support functions. Thereby we propose a new numerical shape optimization process based on the gradient method performed with the finite element method for approximating the auxiliary eigenvalue boundary value problems. The so obtained numerical results show the efficiency and the ability of the proposed approach in producing good quality solutions for the first ten optimal eigenvalues and their associated optimal shapes.

关键词： Shape optimization Convex domains Bi-Laplacian Eigenvalue problem support functions Gradient method

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ALGORITHM FOR FINDING A SOLUTION TO THE INCLUSION 0-EPSILON-F(CHI)

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JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS 1990年第3期67卷 509-531页

作者： DIEN, PH PHUNG, HT UNIV HUE HUEVIETNAM

Using the steepest-descent method combined with the Armijo stepsize rule, we give an algorithm for finding a solution to the inclusion O-epsilon F(x), where F is a set-valued map with smooth support function. As an example, we consider the special case F(x) = g(x)+ K, with K being a convex cone and g a single-valued function. The relation between the present algorithm and that given by Burke and Han is also discussed.

关键词： SET-VALUED MAPS support functions INCLUSIONS GENERALIZED INEQUALITIES

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RELAXATION AND EXISTENCE OF OPTIMAL CONTROLS FOR SYSTEMS GOVERNED BY EVOLUTION INCLUSIONS IN SEPARABLE BANACH-SPACES

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JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS 1990年第3期64卷 573-594页

作者： PAPAGEORGIOU, NS 1. Department of Mathematics University of California Davis California

In this paper, we examine relaxed control systems governed by evolution inclusions in a separable Banach space. First, we establish the existence of admissible trajectories, correcting an earlier result of Ahmed. Then, we obtain a compactness result for the set of admissible trajectories. Using this compactness result, we prove the existence of optimal solutions for optimal control problems; furthermore, we show that the values of the original and relaxed problems are equal. Finally, we show that the original trajectories are dense in the set of relaxed trajectories. An example is worked out.

关键词： Transition probabilities Polish spaces integrably bounded multifunctions u.s.c. multifunctions l.s.c. multifunctions Hausdorff continuity support functions

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Computationally Tractable Counterparts of Distributionally Robust Constraints on Risk Measures

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SIAM REVIEW 2016年第4期58卷 603-650页

作者： Postek, Krzysztof den Hertog, Dick Melenberg, Bertrand Tilburg Univ CentER NL-5000 LE Tilburg Netherlands Tilburg Univ Dept Econometr & Operat Res NL-5000 LE Tilburg Netherlands

In optimization problems appearing in fields such as economics, finance, or engineering, it is often important that a risk measure of a decision-dependent random variable stays below a prescribed level. At the same time, the underlying probability distribution determining the risk measure's value is typically known only up to a certain degree and the constraint should hold for a reasonably wide class of probability distributions. In addition, the constraint should be computationally tractable. In this paper we review and generalize results on the derivation of tractable counterparts of such constraints for discrete probability distributions. Using established techniques in robust optimization, we show that the derivation of a tractable robust counterpart can be split into two parts, one corresponding to the risk measure and the other to the uncertainty set. This holds for a wide range of risk measures and uncertainty sets for probability distributions defined using statistical goodness-of-fit tests or probability metrics. In this way, we provide a unified framework for reformulating this class of constraints, extending the number of solvable risk measure-uncertainty set combinations considerably, also including risk measures that are nonlinear in the probabilities. To provide a clear overview for the user, we provide the computational tractability status for each of the uncertainty set-risk measure pairs, some of which have been solved in the literature. Examples, including portfolio optimization and antenna array design, illustrate the proposed approach in a theoretical and numerical setting.

关键词： risk measure robust counterpart nonlinear inequality robust optimization support functions

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A new family of metrics for compact, convex (fuzzy) sets based on a generalized concept of mid and spread

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INFORMATION SCIENCES 2009年第23期179卷 3964-3972页

作者： Trutschnig, Wolfgang Gonzalez-Rodriguez, Gil Colubi, Ana Angeles Gil, Maria European Ctr Soft Comp Res Unit Intelligent Data Anal & Graph Models Mieres 33600 Spain Vienna Univ Technol Inst Stat & Probabil Theory A-1040 Vienna Austria Univ Oviedo Dpto Estadist & IO & DM Oviedo 33007 Spain

One of the most important aspects of the (statistical) analysis of imprecise data is the usage of a suitable distance on the family of ail compact, convex fuzzy sets, which is not too hard to calculate and which reflects the intuitive meaning of fuzzy sets. On the basis of expressing the metric of Bertoluzza et al. [C. Bertoluzza, N. Corral, A. Salas, On a new class of distances between fuzzy numbers, Mathware Soft Comput. 2 (1995) 71-84] in terms of the mid points and spreads of the corresponding intervals we construct new families of metrics on the family of all d-dimensional compact convex sets as well as on the family of all d-dimensional compact convex fuzzy sets. It is shown that these metrics not only fulfill many good properties, but also that they are easy to calculate and easy to manage for statistical purposes, and therefore, useful from the practical point of view. (C) 2009 Elsevier Inc. All rights reserved.

关键词： Distance Generalized mid and spread Convex compact sets Imprecise data analysis Fuzzy sets support functions

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Parallel reachability analysis of hybrid systems in XSpeed

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INTERNATIONAL JOURNAL ON SOFTWARE TOOLS FOR TECHNOLOGY TRANSFER 2019年第4期21卷 401-423页

作者： Gurung, Amit Ray, Rajarshi Bartocci, Ezio Bogomolov, Sergiy Grosu, Radu Natl Inst Technol Meghalaya Shillong Meghalaya India Vienna Univ Technol Vienna Austria Australian Natl Univ Canberra ACT Australia

Reachability analysis techniques are at the core of the current state-of-the-art technology for verifying safety properties of cyber-physical systems (CPS). The current limitation of such techniques is their inability to scale their analysis by exploiting the powerful parallel multi-core architectures now available in modern CPUs. Here, we address this limitation by presenting for the first time a suite of parallel state-space exploration algorithms that, leveraging multi-core CPUs, enable to scale the reachability analysis for linear continuous and hybrid automaton models of CPS. To demonstrate the achieved performance speedup on multi-core processors, we provide an empirical evaluation of the proposed parallel algorithms on several benchmarks comparing their key performance indicators. This enables also to identify which is the ideal algorithm and the parameters to choose that would maximize the performances for a given benchmark.

关键词： Hybrid systems Reachability analysis support functions Parallel algorithms Multi-core processors Breadth-first-search

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OPTIMAL-CONTROL OF NONLINEAR EVOLUTION INCLUSIONS

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JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS 1990年第2期67卷 321-354页

作者： PAPAGEORGIOU, NS 1. Department of Mathematical Sciences National Technical University of Athens 15773 Athens Greece

In this paper, we study the optimal control of nonlinear evolution inclusions. First, we prove the existence of admissible trajectories and then we show that the set that they form is relatively sequentially compact and in certain cases sequentially compact in an appropriate function space. Then, with the help of a convexity hypothesis and using Cesari's approach, we solve a general Lagrange optimal control problem. After that, we drop the convexity hypothesis and pass to the relaxed system, for which we prove the existence of optimal controls, we show that it has a value equal to that of the original one, and also we prove that the original trajectories are dense in an appropriate topology to the relaxed ones. Finally, we present an example of a nonlinear parabolic optimal control that illustrates the applicability of our results.

关键词： Evolution inclusions dense embedding transition probability support functions relaxed systems Caratheodory integrands Arzela-Ascoli theorem monotone operators parabolic systems

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