In this talk, some variants of reduction-type method combined with a line search filter
method to solve nonlinear semi-infinite programming problems are presented. We use the
stretched simulated annealing method and the branch and bound technique to compute
the maximizers of the constraint. The filter method is used as an alternative to merit
functions to promote convergence from poor starting points

Fernandes, Edite M.G.P.

Pereira, Ana I.

Biblioteca Digital do IPB

I COTA8 ~8JljHml~A'HJtftjJ1tlkw.JH*~ T he 8th Intemational Conference on Optimization: Techniques and Application Decetnber 10-13, 2010 Shanghai Fudan University Book of Abstracts s Convergence Analysis of an Explicit E xchange M ethod for Convex Semi-infinite Programming Problems with Second-order Cone Constraints Shunsuke Hayashi*, Soon-Yi Wu, Liping Zhang . ... . .. .. .. .. . ... ... . ... . . . . ..... 88 A N ew Exchange Met hod for Convex Semi-infinite Programming Liping Zhang*, 80on-Yi Wu, Marco A. López . . ... .. . ... .. ... .. . . . . .. . .... .. . .. 90 On Saddle Points in Nonconvex Semi-infinite Progra mming J an-J. Rückmann*, Francisco Guerra-Vázquez, Ralf Werner . . ... . .. . .... . ... . .. 91 A Regularized Explicit Ex change Method for Semi-infinite Programs with an Infinite Number of Second-order Cone Constraints Takayuki Okuno*, Shunsuke Hayashi , Masao Fukushima . . . . .. ......... .. ....... 92 An Algorithm for Semi-infinite Transportation Problems 800n-Yi Wu*, Shen-Yu Chen . ...... ... . .. .. . ........... . ......... . ..... . ..... .. 94 Relaxed Cutting Plane M ethod with Convexification to Solve Nonlinear Semi-infinite Programming Problems Ting-Jang 8hiu*, Soon-Yi Wu ......... . ... ..... .. ....... . ......... . ... .... .. ... 95 Experiences with Reduction Method to Solve Semi-infinite Programming Problems Ana L Pereira* , Edite M.G.P. Fernandes .. . ... . .. .. ....... . ... . ...... , .. .. . ... . . 96 Capacity-achieving Probability Measure in Communication Channels Satoshi !to*, Shiro Ikeda . . .. .. ...... .. .. . . . . .. ~ . . . . . . .. .......... . ... . ... . . .... 98 A Smoothing Penalized Sample Average Approximation Method for Stochastic Programs with Second Order Stochastic Dominance Constraints Hailin Sun*, Huifu Xu, Yong Wang .. .. . ... ........ . . ... . ............ . . . . ....... 99 Inertial Relaxed CQ Algorithm for Solving Split Feasibility Problem Yazheng Dang*, Yan Gao, Jin Dong . ...... .. . . . .... . .. .. .. . .. .. ... . . . . .... .. . . 101 Nonlinear Extension of Multi-objective Multiclass Support Vector Machine Maximizing Margins Based on One-against-all M ethod K. Tatsumi*, M. Tai , T . Tanino ... . .... . ........ . ... . .. . .... . . . ........ . . .... . 103 Global Quadratic and Polynomial Optimizat ion (SSOI ) Quadratic Optimization under Uncertainty with Applications V. Jeyakumar .......... . ....... . .. .. . .. ....... .. . .. . . . . .. ... .. . .. ....... . .. . .. 105 Semi-algebraic Geometry and Global Optimization Problems with Polynomial Constraints Guoyin Li ...... . .. .......... .. ... .. . ....... . . .. .. .. .. ... . . ... . ... . ... . . ...... . 106 On Stability Results for Quadratic Optimization Problems Gue Myung Lee*, Nguyen Nang Tam, Nguyen Dong Yen . .. .. ... ...... . . ...... 107 Solving Large-sized Sensor Network Localization Problems using Sparsity and Continuation Sunyoung Kim*, Masakazu Kojima .. .. . .. . .... . ...... .. ..... . . . . . .. .. ... . ... . 108 Combinatorial Optimizat ion-Theory and Applications (SSM04-A ~ SSM04-B) The 'Backup 2-Median Problem on B10ck Graphs Yukun Cheng*, Liying Kang, Yan Hong ..... .. ... . .. ..... ... . ... . .. . .... ... . .. 1l0 A Lower Bound of Some Classica l Ramsey N umbers R (3, t) Decha Samana*, Vites Longani .... . . .. ........ . .......... . ... . ... .. ....... . ... 1l2 The Hamiltonian Cycle Problem and Some Challenging Non-Convex Programs Jerzy A. Filar . . . .... .. . . . .. . . ... .. ... .. ... .... .. .. . ............... . .. ... .. .. .. 113 96 T11e 8t11 International Conference on Optimization: Tec11niques and Applications /, Experiences with reduction method to solve semi-infinite programming problems Ana I. Pereira Department of Mathematics - ESTiG; Polytechnic Institute of Braganca, Campus de Sta Apolónia, 5301-854 Braganca, Portugal apereira@ipb.pt Edite M.G.P. Fernandes Department of Production and Systems, University of Minha, Campus de Gualtar, 4710-057 Braga, Portugal emgpf@dps.uminho.pt Abstract In this talk, some variants of reduction-type method combined with a line search filter method to solve nonlinear semi-infinite programming problems are presented. We use the stretched simulated annealing method and the branch and bound technique to compute the maximizers of the constraint. The filter method is used as an alternative to merit functions to promote convergence from poor starting points. Keywords: Semi-infinite programming. Reduction method, Line search filter method. 1 Introduction The semi-infinite programming (SIP) problem is considered to be of the form minf(x) subject to g(x,t)::; o, for every t E T (1) where T ç IRm is a nonempty set defined by T = {t E IRm : a ::; t ::; b}. Here, we assume that the set T does not depend on x. The nonlinear functions f : IRn -t IR and 9 : IRn x T -t IR are twice continuously differentiable with respect to x and 9 is continuously differentiable functions with respect to t. There are many problems in the engineering area that can be formulated as SIP prob-lems. For a review of some applications, the reader is referred to [9, 5, lOJ. The numerical methods that are mostly used to solve SIP problems generate a sequence of finite problems. Some examples can be found in [9, 2, 3, 4, 8, 10J. This work aims to describe a reduction method for SIP. Conceptually, the method is based on the local reduction theory. The focus of our proposal is to compare the perfomance of some methods in global programming theory: the simulated annealing method and the branch and bound method. This work comes in the sequence of a previous reduction-type method presented in [6J. 2 Reduction method A reduction approach, based on the local reduction theory, describes the feasible set of the SIP locally by finitely many inequality constraints. Thus, the SIP problem can be locally reduced to a finite one (at least conceptually, see [4]). Given a feasible point Xk E IRn , consider the so-called lower levei problem O(Xk): maxg(Xk, t) , tET (2) where n = {tI, ... ,tL.} is the set of its local solutions satisfying the following condition (3) T1>e 8t1> InternationaJ Conference on Optimization: Techniques and Applications 97 where Lk represents the cardinality of n, (jML is a positive constant and g' is the global solution value of (2). Under some conditions, it is possible to replace the infinite set of constraints by a finite set that is locally suflicient to define the feasible region. Thus the problem (1) is locally equivalent to the so-called reduced optimization problem min f(x) subject to 9l(x) == g(x, t/(x)) :S O, 1= 1, ... , Lk' xEUk (4) where Uk is a open neighborhood of Xk. A reduction method then emerges when any method for finite programming is applied to solve the locally reduced problem (4). Thus, at each iteration k, the main procedures of the algorithm are: 1. based on an approximation to the SIP problem, Xk, compute the set Tk, solving problem (2), with condition (3); 2. based on the set Tk , implement at most i max iterations to get an approximation Xk ,;, by solving the reduced problem (4); 3. use a globalization technique to compute a new approximation xk+l that improves significantIy over Xk; 4. use termination criteria to decide if the iterative process should terminate. The remaining part of tbis work presents our proposals for the four steps of a global reduction method for SIP. An algorithm to compute the set Tk is known in the literature as a multi-local procedure. Our proposals are the stretched simulated annealing method· anti branch and bound method. To solve the reduced problem (4) an interior point method is proposed. Finally, convergence of the overall reduction method to a SIP solution is encouraged by implementing a filter !ine search technique. References [1) R. Hettich and KO. Kortanek, Semi-infinite progmmming: Theory, methods and applications, SIAM Review, 35, (1993), pp. 380-429. [2) D-H. Li, L. Qi, J. Tam, and S-Y. Wu, A smoothing Newton method for semi-infinite progmmming, Journal of Global Optimization, 30 (2004) , pp. 169- 194. [3) C. Ling, Q. Ni, L. Qi, and S-Y. Wu, A new smoothing Newton-type algorithm for semi-infinite progmmming, Journal of Global Optimization, 47 (2010), pp. 133-159. [4) G-x. Liu, A homotopy interior point method for semi-infinite progmmming problems, Journal of Global Optimization, 37 (2007) , pp. 631-646. [5) M. López, G. Still, Semi-infinite programming, European Journal of Operations Re-search, 180, (2007), pp. 491-518. [6) A.I.P.N. Pereira and E.M.G.P. Fernandes, A reduction method for semi-infinite pro-gramming by means of a global stochastic approach, Optimization, 58, (2009), pp. 713-726. [8) L. Qi, C. Ling, X. Tong, and G. Zhou, A smoothing projected Newton-type algo-rithm for semi-infinite programming, Computational Optimization and Applications, 42 (2009) , 1- 30. [9) R. Reemtsen and J.-J. Rückmann, Semi-infinite progmmming, Nonconvex Optimiza-tion and Its Applications, Vol 25, 1998, Kluwer Academic Pub!ishers. ['10) O. Yi-gui , A filter trust region method for solving semi-infinite progmmming problems, Journal of App!ied Mathematics and Computing, 29 (2009), pp. 311-324. 

Experiences with reduction method to solve semi-infinite programming problems

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Experiences with reduction method to solve semi-infinite programming problems

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