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Saddle Points Theory of Two Classes of Augmented Lagrangians and Its Applications to Generalized semi-infinite programming

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APPLIED MATHEMATICS AND OPTIMIZATION 2009年第3期59卷 413-434页

作者： Wang, Changyu Zhou, Jinchuan Xu, Xiuhua Qufu Normal Univ Inst Operat Res Qufu 273165 Peoples R China Beijing Jiaotong Univ Dept Math Sch Sci Beijing 100044 Peoples R China Zibo Expt High Sch Zibo 255090 Shandong Peoples R China

In this paper, we develop the sufficient conditions for the existence of local and global saddle points of two classes of augmented Lagrangian functions for nonconvex optimization problem with both equality and inequality constraints, which improve the corresponding results in available papers. The main feature of our sufficient condition for the existence of global saddle points is that we do not need the uniqueness of the optimal solution. Furthermore, we show that the existence of global saddle points is a necessary and sufficient condition for the exact penalty representation in the framework of augmented Lagrangians. Based on these, we convert a class of generalized semi-infinite programming problems into standard semi-infinite programming problems via augmented Lagrangians. Some new first-order optimality conditions are also discussed.

关键词： Saddle point Augmented Lagrangian function semi-infinite programming First-order optimality conditions

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DUALITY OF ELLIPSOIDAL APPROXIMATIONS VIA semi-infinite programming

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SIAM JOURNAL ON OPTIMIZATION 2009年第3期20卷 1421-1438页

作者： Guertuna, Filiz Univ Maryland Baltimore Cty Dept Math & Stat Baltimore MD 21250 USA

Inner and outer approximations of complicated sets by simpler sets of a particular form is a fundamental approach in applied mathematics. In this context, ellipsoids are very useful objects with such remarkable properties as simple representation, an associated convex quadratic function, and invariance, that is, an affine transformation of an ellipsoid is an ellipsoid. Therefore, ellipsoidal approximations are easier to work with than polyhedral approximations, and their invariance property can be a gain over spherical approximations in some applications such as clustering.

关键词： inscribed ellipsoid circumscribed ellipsoid minimum volume maximum volume duality semi-infinite programming optimal experimental design John ellipsoid

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Augmented Lagrangians in semi-infinite programming

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MATHEMATICAL programming 2009年第1-2期116卷 499-512页

作者： Ruckmann, Jan-J. Shapiro, Alexander Georgia Inst Technol Sch Ind & Syst Engn Atlanta GA 30332 USA Univ Birmingham Sch Math Birmingham B15 2TT W Midlands England

We consider the class of semi-infinite programming problems which became in recent years a powerful tool for the mathematical modeling of many real-life problems. In this paper, we study an augmented Lagrangian approach to semi-infinite problems and present necessary and sufficient conditions for the existence of corresponding augmented Lagrange multipliers. Furthermore, we discuss two particular cases for the augmenting function: the proximal Lagrangian and the sharp Lagrangian.

关键词： semi-infinite programming augmented Lagrangian duality gap augmented Lagrange multiplier proximal Lagrangian sharp Lagrangian

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Optimality conditions and duality for semi-infinite programming involving B-arcwise connected functions

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JOURNAL OF GLOBAL OPTIMIZATION 2009年第4期45卷 615-629页

作者： Zhang, Qingxiang Yanan Univ Inst Math Yanan 716000 Shaanxi Peoples R China Yanan Univ Coll Math & Comp Sci Yanan 716000 Shaanxi Peoples R China

In this paper, a class of functions called B-arcwise connected (BCN) and strictly B-arcwise connected (STBCN) functions are introduced by relaxing definitions of arcwise connected function (CN) and B-vex function. The differential properties of B-arcwise connected function (BCN) are studied. Their two extreme properties are proved. The necessary and sufficient optimality conditions are obtained for the nondifferentiable nonlinear semi-infinite programming involving B-arcwise connected (BCN) and strictly B-arcwise connected (STBCN) functions. Mond-Weir type duality results have also been established.

关键词： B-arcwise connected function (BCN) semi-infinite programming Optimality condition Duality

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A reduction method for semi-infinite programming by means of a global stochastic approach

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OPTIMIZATION 2009年第6期58卷 713-726页

作者： Pereira, Ana I. P. N. Fernandes, Edite M. G. P. Polytech Inst Braganca Braganca Portugal Univ Minho Braga Portugal

We describe a reduction algorithm for solving semi-infinite programming problems. The proposed algorithm uses the simulated annealing method equipped with a function stretching as a multi-local procedure, and a penalty technique for the finite optimization process. An exponential penalty merit function is reduced along each search direction to ensure convergence from any starting point. Our preliminary numerical results seem to show that the algorithm is very promising in practice.

关键词： semi-infinite programming reduction method simulated annealing penalty method exponential function

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Nonsmooth semi-infinite programming problems with mixed constraints

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JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS 2009年第1期351卷 170-181页

作者： Kanzi, N. Nobakhtian, S. Univ Isfahan Dept Math Esfahan Iran

We consider a nonsmooth semi-infinite programming problem with a feasible set defined by inequality and equality constraints and a set constraint. First, we Study some alternative theorems which involve linear and sublinear functions and a convex set and we propose several generalizations of them. Then, alternative theorems are applied to obtain, under different constraint qualifications, several necessary optimality conditions in the type of Fritz-John and Karush-Kuhn-Tucker. (c) 2008 Elsevier Inc. All rights reserved.

关键词： Optimality conditions semi-infinite programming Mixed constraints Nonsmooth analysis

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Local reduction based SQP-type method for semi-infinite programs with an infinite number of second-order cone constraints

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JOURNAL OF GLOBAL OPTIMIZATION 2014年第1期60卷 25-48页

作者： Okuno, Takayuki Fukushima, Masao Tokyo Univ Sci Div 1 Dept Management Sci Fac Engn Tokyo 1628601 Japan Nanzan Univ Fac Informat Syst & Engn Dept Informat Syst & Math Sci Nagoya Aichi 4890863 Japan

The second-order cone program (SOCP) is an optimization problem with second-order cone (SOC) constraints and has achieved notable developments in the last decade. The classical semi-infinite program (SIP) is represented with infinitely many inequality constraints, and has been studied extensively so far. In this paper, we consider the SIP with infinitely many SOC constraints, called the SISOCP for short. Compared with the standard SIP and SOCP, the studies on the SISOCP are scarce, even though it has important applications such as Chebychev approximation for vector-valued functions. For solving the SISOCP, we develop an algorithm that combines a local reduction method with an SQP-type method. In this method, we reduce the SISOCP to an SOCP with finitely many SOC constraints by means of implicit functions and apply an SQP-type method to the latter problem. We study the global and local convergence properties of the proposed algorithm. Finally, we observe the effectiveness of the algorithm through some numerical experiments.

关键词： semi-infinite programming Second-order cone constraints SQP-type method Local reduction method

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Calmness of the Feasible Set Mapping for Linear Inequality Systems

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SET-VALUED AND VARIATIONAL ANALYSIS 2014年第2期22卷 375-389页

作者： Canovas, M. J. Lopez, M. A. Parra, J. Toledo, F. J. Miguel Hernandez Univ Elche Ctr Operat Res Alicante 03202 Spain Univ Alicante Dept Stat & Operat Res Alicante 03071 Spain Federat Univ Australia Ballarat Vic Australia

In this paper we deal with parameterized linear inequality systems in the n-dimensional Euclidean space, whose coefficients depend continuosly on an index ranging in a compact Hausdorff space. The paper is developed in two different parametric settings: the one of only right-hand-side perturbations of the linear system, and that in which both sides of the system can be perturbed. Appealing to the backgrounds on the calmness property, and exploiting the specifics of the current linear structure, we derive different characterizations of the calmness of the feasible set mapping, and provide an operative expresion for the calmness modulus when confined to finite systems. In the paper, the role played by the Abadie constraint qualification in relation to calmness is clarified, and illustrated by different examples. We point out that this approach has the virtue of tackling the calmness property exclusively in terms of the system's data.

关键词： Calmness Local error bounds Variational analysis semi-infinite programming Linear programming Feasible set mapping

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A filter trust region method for solving semi-infinite programming problems

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JOURNAL OF APPLIED MATHEMATICS AND COMPUTING 2009年第1-2期29卷 311-324页

作者： Yi-Gui, Ou Hainan Univ Coll Informat Sci & Technol Dept Math Haikou 570228 Peoples R China

This paper is concerned with numerical methods for solving a semi-infinite programming problem. We first reformulate the KKT system derived from the problem into a system of semismooth equations by using the F-B NCP function. Under some conditions, a solution of the system of semismooth equations is a solution of the problem. Then we develop a filter-trust-region method for solving this system of semismooth equations. An advantage of the proposed method is that at each iteration, only a system of linear equations is solved. We prove that under standard assumptions, the iterate sequence generated by this proposed method converges globally and superlinearly. Numerical tests are also reported.

关键词： semi-infinite programming Filter technique semismooth equations Trust region method

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Calmness of the Argmin Mapping in Linear semi-infinite Optimization

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JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS 2014年第1期160卷 111-126页

作者： Canovas, M. J. Hantoute, A. Parra, J. Toledo, F. J. Miguel Hernandez Univ Elche Ctr Operat Res Elche 03202 Alicante Spain Univ Chile CMM Dept Ingn Matemat Santiago Chile

This paper characterizes the calmness property of the argmin mapping in the framework of linear semi-infinite optimization problems under canonical perturbations;i.e., continuous perturbations of the right-hand side of the constraints (inequalities) together with perturbations of the objective function coefficient vector. This characterization is new for semi-infinite problems without requiring uniqueness of minimizers. For ordinary (finitely constrained) linear programs, the calmness of the argmin mapping always holds, since its graph is piecewise polyhedral (as a consequence of a classical result by Robinson). Moreover, the so-called isolated calmness (corresponding to the case of unique optimal solution for the nominal problem) has been previously characterized. As a key tool in this paper, we appeal to a certain supremum function associated with our nominal problem, not involving problems in a neighborhood, which is related to (sub)level sets. The main result establishes that, under Slater constraint qualification, perturbations of the objective function are negligible when characterizing the calmness of the argmin mapping. This result also states that the calmness of the argmin mapping is equivalent to the calmness of the level set mapping.

关键词： Calmness Local error bounds Variational analysis semi-infinite programming Linear programming

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