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generalized convexity and the existence of finite time blow-up solutions for an evolutionary problem

NONLINEAR ANALYSIS-THEORY METHODS & APPLICATIONS 2012年第1期75卷 270-277页

作者： Niculescu, Constantin P. Roventa, Ionel Univ Craiova Dept Math RO-200585 Craiova Romania

In this paper we study a class of nonlinearities for which a nonlocal parabolic equation with Neumann-Robin boundary conditions, for p-Laplacian, has finite time blow-up solutions. (C) 2011 Elsevier Ltd. All rights re... 详细信息

关键词： Finite time blow-up solutions p-Laplacian generalized convexity Regularly varying functions

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generalized convexity - CP3 AND BOUNDARIES OF CONVEX-SETS

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PATTERN RECOGNITION 1995年第8期28卷 1191-1199页

作者： LATECKI, L ROSENFELD, A SILVERMAN, R UNIV MARYLAND CTR AUTOMAT RESCOLLEGE PKMD 20742 UNIV HAMBURG DEPT COMP SCID-22527 HAMBURGGERMANY UNIV DIST COLUMBIA DEPT COMP SCIWASHINGTONDC 20009

A set S is convex if for every pair of points P, Q is an element of S, the line segment PQ is contained in S. This definition can be generalized in various ways. One class of generalizations makes use of k-tuples, rather than pairs, of points-for example, Valentine's property P-3: For every triple of points P, Q, R of S, at least one of the line segments PQ, QR, or RP is contained in S. It can be shown that if a set has property P-3, it is a union of at most three convex sets. In this paper we study a property closely related to, but weaker than, P-3. We say that S has property CP3 (''collinear P-3'') if P-3 holds for all collinear triples of points of S. We prove that a closed curve is the boundary of a convex set, and a simple are is part of the boundary of a convex set, iff they have property CP3. This result appears to be the first simple characterization of the boundaries of convex sets;it solves a problem studied over 30 years ago by Menger and Valentine.

关键词： convexity CONVEX ARCS CONVEX CURVES generalized convexity BOUNDARIES

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generalized convexity AND CONCAVITY OF THE OPTIMAL VALUE FUNCTION IN NONLINEAR-PROGRAMMING

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MATHEMATICAL PROGRAMMING 1987年第3期39卷 285-304页

作者： KYPARISIS, J FIACCO, AV GEORGE WASHINGTON UNIV DEPT OPERAT RES WASHINGTON DC 20052 USA

In this paper we consider generalized convexity and concavity properties of the optimal value functionf * for the general parametric optimization problemP(ε) of the form min x f(x, ε) s.t.x∈R(ε). Many results on ... 详细信息

In this paper we consider generalized convexity and concavity properties of the optimal value functionf ^* for the general parametric optimization problemP(_ε) of the form min_x f(x, ε) s.t.x∈R(ε). Many results on convexity and concavity characterizations off ^* were presented by the authors in a previous paper. Such properties off ^* and the solution set mapS ^* form an important part of the theoretical basis for sensitivity, stability and parametric analysis in mathematical optimization. We give sufficient conditions for several types of generalized convexity and concavity off ^*, in terms of respective generalized convexity and concavity assumptions onf and convexity and concavity assumptions on the feasible region point-to-set mapR. Specializations of these results to the parametric inequality-equality constrained nonlinear programming problem are provided.

关键词： Nonlinear programming parametric analysis generalized convexity optimal value function point-to-set mappings

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generalized convexity in Multiple View Geometry

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JOURNAL OF MATHEMATICAL IMAGING AND VISION 2010年第1期38卷 35-51页

作者： Olsson, Carl Kahl, Fredrik Lund Univ Ctr Math Sci S-22100 Lund Sweden

Recent work on geometric vision problems has exploited convexity properties in order to obtain globally optimal solutions. In this paper we give an overview of these developments and show the tight connections between different types of convexity and optimality conditions for a large class of multiview geometry problems. We also show how the convexity properties are closely linked to different types of optimization algorithms for computing the solutions. Moreover, it is also demonstrated how convexity can be used for detection and removal of outliers. The theoretical findings are accompanied with illustrative examples and experimental results on real data.

关键词： generalized convexity Multiple view geometry Computer vision

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generalized convexity and efficiency for non-regular multiobjective programming problems with inequality-type constraints

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NONLINEAR ANALYSIS-THEORY METHODS & APPLICATIONS 2010年第8期73卷 2463-2475页

作者： Hernandez-Jimenez, Beatriz Osuna-Gomez, Rafaela Arana-Jimenez, Manuel Ruiz Garzon, Gabriel Univ Pablo Olavide Dept Econ Metodos Cuantitat & Ha Econ Area Estadist & Invest Operat Seville 41013 Spain Univ Seville Dept Estadist & Invest Operat Fac Matemat E-41012 Seville Spain Univ Cadiz Dept Estadist & Invest Operat Escuela Super Ingn Cadiz 11002 Spain

For multiobjective problems with inequality-type constraints the necessary conditions for efficient solutions are presented. These conditions are applied when the constraints do not necessarily satisfy any regularity assumptions, and they are based on the concept of 2-regularity introduced by Izmailov. In general, the necessary optimality conditions are not sufficient and the efficient solution set is not the same as the Karush-Kuhn-Tucker points set. So it is necessary to introduce generalized convexity notions. In the multiobjective non-regular case we give the notion of 2-KKT-pseudoinvex-II problems. This new concept of generalized convexity is both necessary and sufficient to guarantee the characterization of all efficient solutions based on the optimality conditions. (C) 2010 Elsevier Ltd. All rights reserved.

关键词： generalized convexity Non-regularity Constraint qualifications Optimality conditions

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generalized convexity-based inexact projection method for multiple kernel learning

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JOURNAL OF INTELLIGENT & FUZZY SYSTEMS 2014年第4期27卷 1825-1835页

作者： Luo, Lin Su, Hongye Zheng, Baofen Zhang, Junfeng Zhejiang Univ Inst Cyber Syst & Control State Key Lab Ind Control Technol Hangzhou 310003 Zhejiang Peoples R China

In many applications of kernel method, such as support vector machine (SVM), the performance greatly depends on the choice of kernel. It is not always clear what is the most suitable kernel for one specified task. Multiple kernel learning (MKL) allows to optimize over linear combinations of kernels. A two-step approach, which alternately optimizes the standard SVM and kernel weights, is proposed to solve the non-convex MKL problem. The generalized convexity of MKL is studied which guarantees the strong duality of the corresponding optimization problem. A further contribution is the utilization of a computationally efficient inexact-projection-based method to optimize the standard SVM. In addition, a nonmonotone gradient method is proposed to optimize the kernel weights and the Hessian matrix is approximated with a variation of Limited-memory Broyden-Fletcher-Goldfarb-Shanno (L-BFGS) approach. The global convergence of the proposed two-step approach is analyzed. The utility of the proposed scheme is demonstrated by empirical study with several datasets, and its performance is compared with state-of-the-art methods in terms of accuracy and scalability. The results show that the proposed method performs as well as existing methods in accuracy, but takes much less time to converge to the stationary point.

关键词： Multiple kernel learning generalized convexity inexact projection method

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generalized convexity and inequalities involving special functions

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JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS 2007年第2期336卷 768-776页

作者： Chu Yuming Wang Gendi Zhang Xiaohui Qiu Songliang Hunan City Univ Dept Math Yiyang 413000 Peoples R China Huzhou Teachers Coll Dept Math Huzhou 313000 Peoples R China Zhejiang Sci Tech Univ Coll Sci Hangzhou 310018 Peoples R China

In this paper, the authors show a relation between the generalized convexity and super- (sub-)multiplicative property, and discuss some generalized convexity and inequalities involving the Gaussian hypergeornetric function, the generalized eta-distortion function eta(a)(K)(x) and the generalized Grotzsch function mu(a)(r). (c) 2007 Elsevier Inc. All rights reserved.

关键词： generalized convexity super (sub-)multiplicative Gaussian hypergeometric function generalized eta-distortion function generalized grotzsch function

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generalized convexity in non-regular programming problems with inequality-type constraints

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JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS 2009年第2期352卷 604-613页

作者： Hernandez-Jimenez, B. Rojas-Medar, M. A. Osuna-Gomez, R. Beato-Moreno, A. Univ Pablo Olavide Area Estadist & Invest Operat Dept Econ Metodos Cuantitativos & H Econ Seville Spain Univ Seville Dept Estadist & Invest Operat E-41080 Seville Spain Univ Bio Bio Fac Ciencias Dept Ciencias Basicas Casilla 447 Chillan Chile

convexity plays a very important role in optimization for establishing optimality conditions. Different works have shown that the convexity property can be replaced by a weaker notion, the invexity. In particular, for problems with inequality-type constraints, Martin defined a weaker notion of invexity, the Karush-Kuhn-Tucker-invexity (hereafter KKT-invexity), that is both necessary and sufficient to obtain Karush-Kuhn-Tucker-type optimality conditions. It is well known that for this result to hold the problem has to verify a constraint qualification, i.e., it must be regular or non-degenerate. In non-regular problems, the classical optimality conditions are totally inapplicable. Meaningful results were obtained for problems with inequality-type constraints by lzmailov. They are based on the 2-regularity condition of the constraints at a feasible point. In this work. we generalize Martin's result to non-regular problems by defining an analogous concept, the 2-KKT-invexity, and using the characterization of the tangent cone in the 2-regular case and the necessary optimality condition given by lzMailov. (c) 2008 Elsevier Inc. All rights reserved.

关键词： Non-regular problem generalized convexity KKT-invexity Optimality conditions

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generalized convexity for non-regular optimization problems with conic constraints

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JOURNAL OF GLOBAL OPTIMIZATION 2013年第3期57卷 649-662页

作者： Hernandez-Jimenez, B. Osuna-Gomez, R. Rojas-Medar, M. A. Batista dos Santos, L. Univ Pablo de Olavide Area Estadist & Invest Operat Dept Econ Metodos Cuantitat & Ha Econ Seville Spain Univ Seville Dept Estadist & Invest Operat E-41080 Seville Spain Univ Bio Bio Fac Ciencias Dept Ciencias Basicas Chillan Chile Univ Fed Parana Dept Matemat BR-81531990 Curitiba Parana Brazil

In non-regular problems the classical optimality conditions are totally inapplicable. Meaningful results were obtained for problems with conic constraints by Izmailov and Solodov (SIAM J Control Optim 40(4):1280-1295, 2001). They are based on the so-called 2-regularity condition of the constraints at a feasible point. It is well known that generalized convexity notions play a very important role in optimization for establishing optimality conditions. In this paper we give the concept of Karush-Kuhn-Tucker point to rewrite the necessary optimality condition given in Izmailov and Solodov (SIAM J Control Optim 40(4):1280-1295, 2001) and the appropriate generalized convexity notions to show that the optimality condition is both necessary and sufficient to characterize optimal solutions set for non-regular problems with conic constraints. The results that exist in the literature up to now, even for the regular case, are particular instances of the ones presented here.

关键词： generalized convexity Regularity Constraints qualifications Optimality conditions

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generalized convexity AND PASSAGE FROM LOCAL TO GLOBAL VIA DIFFERENTIAL INEQUALITIES

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REAL ANALYSIS EXCHANGE 2022年第1期47卷 103-120页

作者： Kocak, Sahin Limoncu, Murat Anadolu Univ Dept Math TR-26470 Eskisehir Turkey Eskisehir Tech Univ Dept Math TR-26470 Eskisehir Turkey

We give an elementary framework of generalized convexity in terms of comparison of a given function with reference functions from a chosen set. As important examples we consider reference sets used in Alexandrov geome... 详细信息

关键词： generalized convexity differential inequality Alexandrov geometry

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