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Some results on Lipschitz properties of the optimal values in semi-infinite programming

Some results on Lipschitz properties of the optimal values i...

2nd International Conference on Optimization Methods and Software/6th EUROPT Workshop on Advances in Continuous Optimization

作者： Toledo, F. J. Miguel Hernandez Univ Elche Ctr Operat Res Alicante Spain

This paper studies certain Lipschitz properties of the optimal value function of a linear semi-infinite programming problem and its dual problem in the sense of Haar. In this setting, it is already known that the optimal value function of the so-called primal problem was Lipschitz continuous around a given stable solvable problem, when perturbations of all the coefficients are allowed. Recently, a Lipschitz constant, depending only on the nominal problem data, has been computed and, the no duality gap under certain stability conditions ensures moreover that the obtained constant still holds for the dual optimal value function. Our approach here is focused on obtaining Lipschitz constants for both primal and dual optimal value functions under weaker hypothesis of stability, which do not preclude, in all the cases, the existence of duality gap. Now, the allowed perturbations are restricted to the coefficients of the objective function of the corresponding problems.

关键词： semi-infinite programming Lipschitz properties optimal value

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On a reduction line search filter method for nonlinear semi-infinite programming problems 20

On a reduction line search filter method for nonlinear semi-...

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20th International Conference/Euro Mini Conference on Continuous Optimization and Knowledge-Based Technologies (EurOPT 2008)

作者： Pereira, Ana Isabel P. N. Fernandes, Edite M. G. P. Polytech Inst Braganca Dept Math Braganca Portugal Univ Minho Dept Prod & Syst P-4710057 Braga Portugal

ISBN: (纸本)9789955282839

In this paper, a reduction-type method combined with a line search filter method to solve nonlinear semi-infinite programming problems is presented. The algorithm uses the simulated annealing method equipped with a function stretching technique to compute the maximizers of the constraint, and a penalty technique to solve the reduced finite optimization problem. The filter method is used as an alternative to merit functions to guarantee convergence from poor starting points. Preliminary numerical results with a problems test set are shown.

关键词： semi-infinite programming reduction method line search filter method

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Quadratic programming Approach for UWB Pulse Shaping Filter Design with Nonconstant Upper Bounds

Quadratic Programming Approach for UWB Pulse Shaping Filter ...

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International Conference on Advanced Communication Technology

作者： Mohd Amaluddin Yusoff Zhuquan Zang Department of Electrical and Computer Engineering Curtin University Sarawak Malaysia

ISBN: (纸本)9781467331487

In UWB pulse shaping filter design, FIR filter formulated as a semi-definite programming problem has been widely used. Recently, piece-wise continuous bounds has been incorporated into the FIR filter design to avoid power loss due to the constant spectral mask indirectly assumed in the design. In this paper, we formulate the pulse shaping filter design as a semi-infinite quadratic optimization problem using Hermite-Rodriguez basis functions. Computationally effective approach to solving the optimization problem is proposed. Through numerical design examples, we compare the spectral utilization of our approach with those FIR-based pulse shaping filter designs. We demonstrate that our proposed approach is easy to implement and achieves better spectral utilization than a recent result when a suboptimal basic pulse is used.

关键词： Pulse shaping Filter design semi-infinite programming Ultra-wideband (UWB) Hermite-Rodriguez functions

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Market price-based convex risk measures: A distribution-free optimization approach

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OPERATIONS RESEARCH LETTERS 2012年第2期40卷 128-133页

作者： Li, Jonathan Y. Kwon, Roy H. Univ Toronto Dept Mech & Ind Engn Toronto ON M5S 3G8 Canada

In Cont (2006)[1], a convex risk measure was proposed to measure the impact of uncertainty resulting from the misspecification of derivative models. Evaluation of the risk measures was illustrated on finite families of probability measures. In this paper, we consider the case of infinite families of measures that share common moments, e.g. mean and variance for European-style options. We show that the risk measure can still be efficiently evaluated based on semi-infinite programming. Examples are given that illustrate the benefits of evaluating the risk measure with infinite families of measures and shed light on the limitations of considering only finite families of measures. (C) 2011 Elsevier B.V. All rights reserved.

关键词： Convex risk measures Model uncertainty Derivative pricing semi-infinite programming semidefinite programming

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Radiofrequency ablation planning: An application of semi-infinite modelling techniques

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EUROPEAN JOURNAL OF OPERATIONAL RESEARCH 2012年第3期218卷 856-864页

作者： Haase, Sabrina Suess, Philipp Schwientek, Jan Teichert, Katrin Preusser, Tobias Fraunhofer Inst Bildgestutzte Med MEVIS D-28359 Bremen Germany Fraunhofer Inst Techno & Wirtschaftsmath ITWM D-67663 Kaiserslautern Germany Jacobs Univ Bremen D-28759 Bremen Germany

In radiofrequency (RF) ablation a needle-shaped probe is inserted into the patient's body in order to heat and subsequently destroy the malignant tissue around the needle tip. The determination of the optimal probe position poses an intricate problem, as it requires the modelling of the tumour destruction depending on the attained temperature as well as the incorporation of constraints that prohibit puncturing bones or other risk structures. In this work we present a new optimization procedure that reflects both aspects adequately. We assess tumour destruction by solving the underlying system of partial differential equations using a finite element method. Next we show how the probe's position and orientation can be optimized by gradient-based methods. Ensuring that risk structures are not harmed by the probe is easily modelled using semi-infinite constraints in the optimization problem. Techniques to reduce the semi-infinite problem to a standard nonlinear constrained optimization problem are introduced and demonstrated as a proof-of-concept on real clinical data. The results indicate the high potential of this combination of PDE-based simulation and numerical optimization for RF ablation planning. (C) 2011 Elsevier B.V. All rights reserved.

关键词： semi-infinite programming Radiofrequency ablation Heat equation Simulation and optimization

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A REGULARIZED EXPLICIT EXCHANGE METHOD FOR semi-infinite PROGRAMS WITH AN infinite NUMBER OF CONIC CONSTRAINTS

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SIAM JOURNAL ON OPTIMIZATION 2012年第3期22卷 1009-1028页

作者： Okuno, Takayuki Hayashi, Shunsuke Fukushima, Masao Kyoto Univ Dept Appl Math & Phys Grad Sch Informat Kyoto 6068501 Japan

The semi-infinite program (SIP) is normally represented with infinitely many inequality constraints and has been studied extensively so far. However, there have been very few studies on the SIP involving conic constraints, even though it has important applications such as Chebyshev-like approximation, filter design, and so on. In this paper, we focus on the SIP with infinitely many conic constraints, called an SICP for short. We show that under the Robinson constraint qualification a local optimum of the SICP satisfies the KKT conditions that can be represented only with a finite subset of the conic constraints. We also introduce two exchange type algorithms for solving the convex SICP. We first provide an explicit exchange method and show that it has global convergence under the strict convexity assumption on the objective function. We then propose an algorithm combining a regularization method with the explicit exchange method and establish global convergence of the hybrid algorithm without the strict convexity assumption. We report some numerical results to examine the effectiveness of the proposed algorithms.

关键词： semi-infinite programming conic constraints exchange method

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Symmetry of convex sets and its applications to the extremal ellipsoids of convex bodies

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OPTIMIZATION METHODS & SOFTWARE 2012年第4-5期27卷 735-759页

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

A convex body K in R-n has around it a unique circumscribed ellipsoid CE(K) with minimum volume and within it a unique inscribed ellipsoid IE(K) with maximum volume. The modern theory of these ellipsoids is pioneered by Fritz John in his 1948 seminal paper. This paper has two related goals. First, we investigate the symmetry properties of a convex body by studying its (affine) automorphism group Aut(K) and relate this group to the automorphism groups of its ellipsoids. We show that if Aut(K) is large enough, then the complexity of determining the ellipsoids CE(K) and IE(K) is greatly reduced, and in some cases, the ellipsoids can be determined explicitly. We then use this technique to compute the extremal ellipsoids associated with some classes of convex bodies that have important applications in convex optimization, namely when the convex body K is the part of a given ellipsoid between two parallel hyperplanes and when K is a truncated second-order cone or an ellipsoidal cylinder.

关键词： circumscribed ellipsoid inscribed ellipsoid John ellipsoid Lowner ellipsoid minimum-volume ellipsoid maximum-volume ellipsoid optimality conditions semi-infinite programming contact points automorphism group symmetric convex bodies Haar measure

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Sample average approximation of stochastic dominance constrained programs

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MATHEMATICAL programming 2012年第1-2期133卷 171-201页

作者： Hu, Jian Homem-de-Mello, Tito Mehrotra, Sanjay Northwestern Univ Dept Ind Engn & Management Sci Evanston IL 60208 USA Univ Illinois Dept Mech & Ind Engn Chicago IL 60607 USA

In this paper we study optimization problems with second-order stochastic dominance constraints. This class of problems allows for the modeling of optimization problems where a risk-averse decision maker wants to ensure that the solution produced by the model dominates certain benchmarks. Here we deal with the case of multi-variate stochastic dominance under general distributions and nonlinear functions. We introduce the concept of -dominance, which generalizes some notions of multi-variate dominance found in the literature. We apply the Sample Average Approximation (SAA) method to this problem, which results in a semi-infinite program, and study asymptotic convergence of optimal values and optimal solutions, as well as the rate of convergence of the feasibility set of the resulting semi-infinite program as the sample size goes to infinity. We develop a finitely convergent method to find an -optimal solution of the SAA problem. An important aspect of our contribution is the construction of practical statistical lower and upper bounds for the true optimal objective value. We also show that the bounds are asymptotically tight as the sample size goes to infinity.

关键词： Stochastic programming Stochastic dominance Sample average approximation semi-infinite programming Convex programming Cutting plane algorithms

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SIP: critical value functions have finite modulus of non-convexity

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MATHEMATICAL programming 2012年第1期136卷 133-154页

作者： Dorsch, D. Guerra-Vazquez, F. Guenzel, H. Jongen, H. Th. Rueckmann, J. -J. Univ Birmingham Sch Math Birmingham B15 2TT W Midlands England Rhein Westfal TH Aachen Dept Math C D-52056 Aachen Germany Univ Americas Puebla Dept Math & Phys Cholula 72820 Mexico

We consider semi-infinite programming problems depending on a finite dimensional parameter . Provided that is a strongly stable stationary point of , there exists a locally unique and continuous stationary point mapping . This defines the local critical value function , where denotes the objective function of for a given parameter vector . We show that is the sum of a convex function and a smooth function. In particular, this excludes the appearance of negative kinks in the graph of .

关键词： SIP semi-infinite programming Parametric optimization Strong stability Modulus of non-convexity Critical value function

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Tangential extremal principles for finite and infinite systems of sets, I: basic theory

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MATHEMATICAL programming 2012年第1期136卷 3-30页

作者： Mordukhovich, Boris S. Phan, Hung M. Wayne State Univ Dept Math Detroit MI 48202 USA

In this paper we develop new extremal principles in variational analysis that deal with finite and infinite systems of convex and nonconvex sets. The results obtained, unified under the name of tangential extremal principles, combine primal and dual approaches to the study of variational systems being in fact first extremal principles applied to infinite systems of sets. The first part of the paper concerns the basic theory of tangential extremal principles while the second part presents applications to problems of semi-infinite programming and multiobjective optimization.

关键词： Variational analysis Extremal systems Extremal principles Tangent and normal cones semi-infinite programming Multiobjective optimization

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