Lecture Notes in Economics and Mathematical Systems Founding Editors: M Beckmann H P Kunzi Managing Editors: Prof Dr G Fandel FachbereichWirtschaftswissenschaften Femuniversitat Hagen Feithstr 140/AVZII, 58084 Hagen, Germany Prof Dr W Trockel Institut fur Mathematische Wirtschaftsforschung (IMW) Universitat Bielefeld Universitatsstr 25, 33615 Bielefeld, Germany Editorial Board: A Basile, A Drexl, H Dawid, K Inderfurth, W Kursten, U Schittko 563 Alberto Seeger (Ed.) Recent Advances in Optimization Springer Editor Prof Alberto Seeger University of Avignon Department of Mathematics 33, rue Louis Pasteur 84000 Avignon, France E-mail: alberto.seeger@univavignon.fr ISSN 0075-8442 ISBN-10 3-540-28257-2 Springer Berlin Heidelberg New York ISBN-13 978-3-540-28257-0 Springer Berlin Heidelberg New York This work is subject to copyright AH rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in any other way, and storage in data banks Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9,1965, in its current version, and permission for use must always be obtained from Springer-Verlag Violations are liable for prosecution under the German Copyright Law Springer is a part of Springer Science+Business Media springeronline com © Springer-Verlag Berlin Heidelberg 2006 Printed in Germany The use of general descriptive names, registered names, trademarks, etc in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use Typesetting: Camera ready by author Cover design: Erich Kirchner, Heidelberg Printed on acid-free paper 42/3130Jo 10 Preface This volume contains the Proceedings of the Twelfth French-German-Spanish Conference on Optimization held at the University of Avignon in 2004 We refer to this conference by using the acronym FGS-2004 During the period September 20-24, 2004, about 180 scientists from around the world met at Avignon (France) to discuss recent developments in optimization and related fields The main topics discussed during this meeting were the following: smooth and nonsmooth continuous optimization problems, numerical methods for mathematical programming, optimal control and calculus of variations, differential inclusions and set-valued analysis, stochastic optimization, multicriteria optimization, game theory and equilibrium concepts, optimization models in finance and mathematical economics, optimization techniques for industrial applications The Scientific Committee of the conference consisted of F Bonnans (Rocquencourt, France), J.-B Hiriart-Urruty (Toulouse, France), F Jarre (Diisseldorf, Germany), M.A Lopez (Alicante, Spain), J.E Martinez-Legaz (Barcelona, Spain), H Maurer (Miinster, Germany), S Pickenhain (Cottbus, Germany), A Seeger (Avignon, France), and M Thera (Limoges, France) The conference FGS-2004 is the 12th of the series of French-German meetings which started in Oberwolfach in 1980 and was continued in Confolant (1981), Luminy (1984), Irsee (1986), Varetz (1988), Lambrecht (1991), Dijon (1994), Trier (1996), Namur (1998), Montpellier (2000), and Cottbus (2002) Since 1998, this series of meetings has been organized under the participation of a third European country In 2004, the guest country was Spain The conference promoted, in particular, the contacts between researchers of the three VI Preface involved countries and provide a forum for sharing recent results in theory and applications of optimization The conference FGS-2004 was organized by the "Group of Nonlinear Analysis and Optimization" of the University of Avignon As chairman of the Organizing Committee, I would like to acknowledge the following institutions for their financial or material support: • • • • • Region Provence-Alpes-Cote d'Azur Universite d'Avignon et des Peiys de Vaucluse Agroparc: Technopole Regional d'Avignon Mairie d'Avignon Institut National de Recherche en Informatique et en Automatique For the sake of convenience, the contributions appearing in this volume are splitted in four different groups: Part Part Part Part I Optimization Theory and Algorithms, II Optimal Control and Calculus of Variations, III Game Theory, IV Modeling and Numerical Testing Each contribution has been examined by one or two referees The evaluation process has been more complete and thorough for the contributions appearing in Parts I, II, and III The papers in Part IV are less demanding from a purely mathematical point-of-view (no theorems, propositions, etc) Their principal concern is either the modeling or the computer resolution of specific optimization problems arising in industry and applied sciences I would like to thank all the contributors for their effort and the anonymous referees for their comments and suggestions The help provided by Mrs Monique Lefebvre (Secretarial Office of FGS-2004) and the staff of SpringerVerlag is also greatly appreciated Avignon, September 2005 Alberto Seeger Contents Part I Optimization Theory and Algorithms On the Asymptotic Behavior of a System of Steepest Descent Equations Coupled by a Vanishing Mutual Repulsion F Alvarez, A Cabot Inverse Linear Programming S Dempe, S Lohse 19 Second-Order Conditions in C^'^ Vector Optimization with Inequality and Equality Constraints Ivan Ginchev, Angela Guerraggio, Matteo Rocca 29 Benson Proper Efficiency in Set-Valued Optimization on Real Linear Spaces E Hernandez, B Jimenez and V Novo 45 Some Results About Proximal-Like Methods A Kaplan, R Tichatschke 61 Application of the Proximal Point Method to a System of Extended Primal-Dual Equilibrium Problems Igor V Konnov 87 On Stability of Multistage Stochastic Decision Problems Alexander Mdnz, Silvia Vogel 103 Nonholonomic Optimization C Udri§te, O Dogaru, M Ferrara, I T^vy 119 A Note on Error Estimates for some Interior Penalty Methods A F Izmailov, M V Solodov 133 VIII Contents Part II Optimal Control and Calculus of Variations L^—Optimal Boundary Control of a String to Rest in Finite Time Martin Gugat 149 An Application of PL Continuation Methods to Singular Arcs Problems Pierre Martinon and Joseph Gergaud 163 On an Elliptic Optimal Control Problem with Pointwise Mixed Control-State Constraints Christian Meyer, Fredi Troltzsch 187 On Abstract Control Problems with Non-Smooth Data Zsolt Pales 205 Sufficiency Conditions for Infinite Horizon Optimal Control Problems Sabine Pickenhain, Valeriya Lykina 217 On Nonconvex Relaxation Properties of Multidimensional Control Problems Marcus Wagner 233 Existence and Structure of Solutions of Autonomous Discrete Time Optimal Control Problems Alexander J Zaslavski 251 Numerical Methods for Optimal Control with Binary Control Functions Applied to a Lot ka-Volt err a Type Fishing Problem Sebastian Sager, Hans Georg Bock, Moritz Diehl, Gerhard Reinelt, Johannes P Schloder 269 Part III Game Theory Some Characterizations of Convex Games Juan Enrique Martmez-Legaz 293 The Bird Core for Minimum Cost Spanning Tree Problems Revisited: Monotonicity and Additivity Aspects Stef Tijs, Stefano Moretti, Rodica Branzei, Henk Norde 305 A Parametric Family of Mixed Coalitional Values Francesc Carreras, Maria Albina Puente 323 Contents IX Part I V Industrial Applications and Numerical Testing C o m p l e m e n t a r i t y P r o b l e m s in R e s t r u c t u r e d N a t u r a l G a s Markets Steven Gabriel, Yves Smeers 343 R e c o n c i l i n g Franchisor a n d l^Vanchisee: A P l a n a r B i o b j e c t i v e Competitive Location and Design Model Jose Fernandez, Bogldrka Toth, Frank Plastria, Bias Pelegrin 375 T o o l s for R o b o t i c Trajector^'^ P l a n n i n g U s i n g C u b i c S p l i n e s and Semi-Infinite Programmiing A Ismael F Vaz, Edite M,G.P Fernandes 399 Solving Mathematical Programs w i t h ComplementarityConstraints w i t h Nonlinear Solvers Helena Sofia Rodrigues, M Teresa T Monteiro 415 A F i l t e r A l g o r i t h m a n d O t h e r N L P Solvers: P e r f o r m a n c e Comparative Analysis Antonio Sanches Antunes, M Teresa T Monteiro 425 H o w Wastewater Processes can be Optimized Using LOQO LA C P Espirito-Santo, Edite M G P Fernandes, M M Araujo, E C Ferreira 435 List of Contributors Felipe Alvarez Universidad de Chile/Departamento de Ingenieria Matematica and Centre de Modelamiento Matematico Casilla 170/3, Correo Santiago, Chile Francesc Carreras Polytechnic University of Catalonia/ Dep of Applied Mathematics II and Industrial Engineering School Terrassa, Spain f reuacesc carrerasQupc edu falvarezQdim.uchile.cl Antonio Sanches Antunes University of Minho Portugal asanches@ipg.pt M Madalena Araujo Minho University/Systems and Production Department Braga, Portugal mmarauj oQdps.uminho.pt Rodica Branzei Alexandru loan Cuza University/Faculty of Computer Science lasi, Romania branzeirQinfo.uaic.ro Hans Georg Bock IWR Heidelberg Heidelberg, Germany Alexandre Cabot Universite de Limoges/Laboratoire LACO Limoges, France alexandre.cabot@unilim.fr Stephan Dempe Tech University Bergakademie Freiberg/Dep of Mathematics and Computer Sciences Akademiestr 09596 Freiberg, Germany dempeSmath.tu-freiberg.de Moritz Diehl IWR Heidelberg Heidelberg, Germany Oltin Dogaru University Politehnica of Bucharest/ Department of Mathematics Splaiul Independen^ei 313 060042 Bucharest, Romania Isabel A.C.P Espiritu-Santo Minho University/Systems and Production Department Braga, Portugal iapinhoQdps.uminho.pt How Wastewater Processes can be Optimized Using LOQO 441 Soluble COD S = Si-^Ss; (14) Total COD COD = X + S; (15) VSS=^] tcv (16) TSS = VSS^ISS; (17) Volatile suspended solids Total suspended solids Biochemical oxygen demand BOD = fBOD {Ss + X5 + XBH + XBA) ; (18) Total nitrogen of Kjeldahl TKN = SNH-^SND-^XND+ixs i^BH + ^BA)+^Xp (Xp + Xi); (19) Total nitrogen N = TKN + SNO- (20) 2.3 Quality Constraints Quality constraints are usually derived from environmental law restrictions The most used are related with limits in the chemical oxygen demand (COD), total nitrogen (AT), and total solids (TSS) at the effluent In mathematical terms, these constraints are defined as: CODef < CODiau, (21) Nef < Nia^ (22) TSSef < TSSiaw' (23) 2.4 Constraints of the Secondary Settler Traditionally the secondary settler is underestimated when compared with the aeration tank However, it plays a crucial role in the activated sludge system When the wastewater leaves the aeration tank, where the biological treatment took place, the treated water should be separated from the biological sludge, otherwise, the COD would be higher than it is at the entry of the system The most common way of achieving this purpose is by sedimentation in tanks A good settler tank has to accomplish three different functions As a thickener, it aims to produce a continuous underflow of thickened sludge to return to the aeration tank; as a clarifier, it produces a good quality final effluent I A C, P Espirito-Santo et al 442 and as a storage t a n k it allows the conservation of the sludge in peak flow events None of these functions could fail If t h a t happens the effluent will be of poor quality and the overall behavior of the system can be compromised T h e behavior of a settling tank depends on its design and operation, namely the hydraulic features, as the flow rate, the physical features, as inlet and sludge collection arrangements, site conditions, as temperature and wind, and sludge characteristics The factors that most influence the size of the t a n k are the wastewater flow and the characteristics of the sludge As the former is known, the optimization of the sedimentation area and depth must rely on the sludge characteristics, which in t u r n are related with the performance of the aeration tank So, the operation of the biological reactor influences directly the performance of the settling tank and for t h a t reason, one should never be considered without the other T h e ATV design procedure contemplates the peak wet weather flow ( P W W F ) events, in which the sludge mass transferred from the biological reactor is AXVa, where AX is the change in the sludge concentration within the aeration tank A reduction of 30% on the sludge concentration for a P W W F event is considered A higher reduction of the sludge concentration into the biological reactor may compromise the entire process Water Iw el