Recent Advances in Optimization (Lecture Notes in Economics and Mathematical Systems)

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: [email protected]

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

5 4 3 2 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: 1. 2. 3. 4. 5. 6. 7. 8. 9.

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

3

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 3 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 [email protected]

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 [email protected]

Stephan Dempe Tech. University Bergakademie Freiberg/Dep. of Mathematics and Computer Sciences Akademiestr. 6 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

XII

List of Contributors

Edite M.G.P. Fernandes Universidade do Minho Campus de G u a l t a r / D e p a r t a m e n t o de Producao e Sistemas, Escola de Engenharia 4710-057 Braga, Portugal emgpfQdps.uminho.pt Jose Fernandez University of Murcia/Dep. Statistics and Operations Research Murcia, Spain M a s s i m i l i a n o Ferrara University of Messina/Faculty of Economics Via dei Verdi 75 98122 Messina, Italy mferraraQunime.it Eugenio C. Ferreira Minho University/Centre of Biological Engineering Braga, Portugal ecferreiraQdeb.uminho.pt Steven Gabriel University of Maryland, College P a r k / D e p a r t m e n t of Civil and Environmental Engineering 20742 Maryland, U.S.A. sgabrielOumd.edu Joseph Gergaud E N S E E I H T - I R I T / U M R CNRS 5505 2, rue Camichel 31071 Toulouse, France gergaudSenseeiht.fr Ivan Ginchev Technical University of Varna/Dep. of Mathematics Studentska Str. 1 9010 Varna, Bulgaria ginchevQmsS.tu-varna.acad.bg

Angelo Guerraggio University of Insubria/Department of Economics Via Ravasi 2 21100 Varese, Italy aguerraggioQeco.uninsubria.it Martin Gugat Universitat ErlangenNiirnberg/Lehrstuhl 2 fiir Angewandte Mathematik Martensstr. 3 91058 Erlangen, Germany gugatOam.uni-erlangen.de Elvira H e r n a n d e z U N E D / Depto. de Matematica Aplicada, E.T.S.I. I n d u s t r i a l s c / J u a n del Rosal 12 28040 Madrid, Spain ehernandezQind.uned.es Alexey F. Izmailov Moscow State University/ Dep. of Operations Research Leninskiye Gori, GSP-2 119992 Moscow, Russia izmafQccas.ru Bienvenido Jimenez Universidad de Salamanca/Depto. de Economia e Historia Economica, Facultad de Economia y Empresa Campus Miguel de Unamuno, s/n, 37007 Salamanca, Spain bj i m e n l Q e n c i n a . p n t i c . m e c . e s Alexander Kaplan University of Trier/Department of Mathematics 54286 Trier, Germany Al.KaplanOtiscali.de Igor V . K o n n o v Kazan University/Department of Applied Mathematics Kazan, Russia ikonnovQksu.ru

List of Contributors Valeriya Lykina Brandenburg Univ. of Technology Cottbus, Germany lykinaQmath.tu-cottbus.de Sebastian Lohse Tech. University Bergakademie Freiberg/Dep. of Mathematics and Computer Sciences Akademiestr. 6 09596 Freiberg, Germany Alexander Manz ASPECTA Lebensversicherung AG Germany AMaenzQaspecta.com

Juan Enrique Martinez-Legaz Universitat Autonoma de Barcelona/Departament d'Economia i d'Historia Economica 08193 Bellaterra, Spain juanenrique.martinezQuab.es Pierre Martinon ENSEEIHT-IRIT/UMR CNRS 5505 2, rue Camichel 31071 Toulouse, France martinonQenseeiht.fr Christian Meyer Technische Universitat Berlin/Institut fur Mathematik Str. des 17. Juni 136 D-10623 Berlin, Germany cmeyerOmath.tu-berlin.de M . Teresa T. Monteiro University of Minho Portugal tmOdps.uminho.pt

Stefano Moretti University of Genoa/Department of Mathematics Genoa, Italy morettiQdima.unige.it

XIII

Henk Norde Tilburg University/CentER and Department of Econometrics and Operations Research Tilburg, The Netherlands [email protected] Vicente Novo UNED/ Depto. de Matematica Aplicada, E.T.S.I. Industriales c/ Juan del Rosal 12 28040 Madrid, Spain vnovoOind.uned.es Zsolt Pales University of Debrecen/Institute of Mathematics 4029 Debrecen, Pf. 12, Hungary palesQmath.kite.hu Bias Pelegrin University of Murcia/Dep. Statistics and Operations Research Murcia, Spain Sabine Pickenhain Brandenburg Univ. of Technology Cottbus, Germany sabineQmath.tu-cottbus.de Frank Plastria Vrije Universiteit Brussel/MOSIDep. of Mathematics, Operational Research and Information Systems for Management Brussel, Belgium Maria Albina Puente Polytechnic University of Catalonia/ Dep. of Applied Mathematics III and Polytechnic School Manresa, Spain m.albina.puenteQupc.edu

XIV

List of Contributors

Gerhard Reinelt IWR Heidelberg Heidelberg, Germany Matteo Rocca University of Insubria/Department of Economics Via Ravasi 2 21100 Varese, Italy mroccaQeco.uninsubria.it Helena Sofia Rodrigues University of Minho Portugal helena.rodriguesOipb.pt

Sebastian Sager IWR Heidelberg Heidelberg, Germany Sebastian.sager Oiwr.uni-heidelberg.de Joliannes P. Schloder IWR Heidelberg Heidelberg, Germany Yves Smeers Universite catholique de Louvain/ Dep. of Mathematical Engineering and Center for Operations Research and Econometrics Louvain-la-Neuve, Belgium smeersOcore.ucl.ac.be Mikhail V. Solodov Instituto de Matematica Pura e Aplicada Estrada Dona Castorina 110, Jardim Botanico, RJ 22460-320, Rio de Janeiro, Brazil. solodovOimpa.br

lonel T^vy University Politehnica of Bucharest/ Department of Mathematics Splaiul Independen^ei 313 060042 Bucharest, Romania

Rainer Tichatschke University of Trier/Department of Mathematics 54286 Trier, Germany tichatOuni-trier.de Stef Tijs Tilburg University/CentER and Department of Econometrics and Operations Research Tilburg, The Netherlands [email protected] Boglarka Toth University of Murcia/Dep. Statistics and Operations Research Murcia, Spain Fredi Troltzsch Technische Universitat Berlin/Institut fiir Mathematik Str. des 17. Juni 136 D-10623 Berlin, Germany troeltzQmath.tu-berlin.de Constantin Udri§te University Politehnica of Bucharest/ Department of Mathematics Splaiul Independen^ei 313 060042 Bucharest, Romania udristeQmathem.pub.ro A. Ismael F. Vaz Universidade do Minho Campus de Gualtar/Departamento de Producao e Sistemas, Escola de Engenharia, 4710-057 Braga, Portugal a i vazOdps. ximinho. pt Silvia Vogel Technische Universitaet Ilmenau Ilmenau, Germany Silvia.VogelQtu-ilmenau.de

List of Contributors Marcus Wagner Cottbus University of Technology/Dep. of Mathematics Karl-Marx-Str. 17, P.O. Box 101344 D-03013 Cottbus, Germany wagnerQmath.tu-cottbus.de

XV

Alexander J. Zaslavski Technion, Dep. of Mathematics Haifa, Israel aj zaslQtechunix.technion.ac.il

Part I

Optimization Theory and Algorithms

On t h e Asymptotic Behavior of a System of Steepest Descent Equations Coupled by a Vanishing M u t u a l Repulsion* F. Alvarez^** and A. Cabot^ ^ Departamento de Ingenieria Matematica and Centre de Modelamiento Matematico, Universidad de Chile, Casilla 170/3, Correo 3, Santiago, Chile. falvarezOdim.uchile.cl ^ Laboratoire LACO, Universite de Limoges, Limoges, Prance. alexcindre. cabotQunilim. f r S u m m a r y . We investigate the behavior at infinity of a special dissipative system, which consists of two steepest descent equations coupled by a non-autonomous conservative repulsion. The repulsion term is parametrized in time by an asymptotically vanishing factor. We show that under a simple slow parametrization assumption the limit points, if any, must satisfy an optimality condition involving the repulsion potential. Under some additional restrictive conditions, requiring in particular the equilibrium set to be one-dimensional, we obtain an asymptotic convergence result. Finally, some open problems are listed.

1 Introduction Throughout this paper, H is a, real Hilbert space with scalar product and norm denoted by (•, •) and || • ||, respectively. Let (j): H -^Rhe a, C^ function and suppose that the set of critical points of (j) is nonempty, that is, S:={xeH\

V0(x) = 0} 7^ 0.

A standard first-order method for finding a point in S consists in following the "Steepest Descent" trajectories: (SD)

X -f V0(2;) = 0 ,

t > 0.

The evolution equation SD defines a dissipative dynamical system in the sense that every solution x{t) satisfies •^(j){x{t)) = — ||V 00, and moreover, it is apparent t h a t the adequate condition is

/

£{t) dt = 00.

(3)

Such a "slow parametrization" condition has already been pointed out by many authors in various contexts (cf. [3, 4, 9, 11]). Since £{t) vanishes when t - ^ 00, it is quite easy to prove the convergence of the gradients V(j){x) and V(j)(y) toward 0. T h e examples above show t h a t under unboundedness of S we may observe divergence to infinity. Divergence can be prevented under coercivity of 0 and the natural question t h a t arises is the convergence of the trajectory {x(t)^y(t)) as t ^ 00. This is a difficult problem due to the non convexity of the repulsive potential V (see [10] for positive results in a convex framework). In this direction, a one-dimensional convergence result has been obtained in [11] for a second-order in time version of SDVR. T h e paper is organized as follows. In section 2, we state some general convergence properties for the SDVR system and we show t h a t the slow parametrization assumption (3) forces the limit points to satisfy an optimality condition involving W and the normal cone of S. This normal condition^ is new and allows to reformulate some results of [11] in a more elegant way. In section 3 we derive a sharp convergence result when the equilibrium set S is one-dimensional. In the last section, we precise our results when (j) is the square of a distance function. Due to the first-order (in time) structure of SDVR, our asymptotic selection results are sharper t h a n in [11]. ^ This optimality condition has been found independently by M.-O. Czarnecki (University Montpellier II).

6

F. Alvarez and A. Cabot

Notations, We use the standard notations of convex analysis. In particular, given a convex set C C i/, we denote by dc{x) (resp. Pc{x)) the distance of the point x E H to the set C (resp. the best approximation to x from C). For every x G C, the set Nc{x) stands for the normal cone of C at x. Given any set D C H, the closed convex hull of D is denoted by c6{D). Given a,b E: H^ we define [a, 6] = {a-\-X{b-a) | AG [0,1]} and ]a,6[= {a + A(fe-a) | A G ] 0 , 1 [ } .

2 General Asymptotic Results From now on, suppose that the functions 0 : i J - ^ R , F : i 7 — ) > M and e : IR+ -^ M+, which are assumed to be of class C^, satisfy the following set of hypotheses (H): (n-/ \ / ^ ~ ^ ^^^ ^ ^^^ bounded from below on H, with inf V = 0. ^ ^^ ]^ii — V0 and W are Lipschitz continuous on bounded sets of H.

{

i — The map £ is non-increasing, i.e. €{t) < 0 Vt G M-f.. ii — The map e is Lipschitz continuous on R+. in — lim e(t) = 0.

Let us begin our study of SDVR by noticing that it can be rewritten as a single vectorial equation in H^ = H x H. Indeed, let us set X = {x,y) e H^, ^(X) = (j){x) + (l){y) and U{X) = V{x - y). With such notations, SDVR is equivalent to X + V^(X) + £{t)VU{X) = 0, (4) where ^ and U are differentiable functions on H^ satisfying the analogue to (7^1), that is (njvec\ { i — ^ and U are bounded from below on i/^, with miU = 0. ^ ^ \^ii — V ^ and Vt/ are Lipschitz continuous on bounded sets of H^. Set E{t) = ^{X{t)) + s{t)U{X{t)) = ct>{x{t)) + cl>{y{t)) + e{t)V{x{t) - y{t)). By differentiating E with respect to time, we obtain E = -\\Xf

+ iU{X)

= -\\xf

- ||y||2 +iV{x-y)
iJ^ of (4), which is of class C^ and satisfies X{0) = XQ. Moreover, X G L^([0, CXD); iif^). {ii) Assuming additionally that {X{t)}t>o is bounded in H^ (which is the case for example if ^ is coercive, i.e. lim ^{X) = oo^, then lim X{t) = 0 ||X||—i>oo

t—>oo

and lim V ^ ( X ( t ) ) = 0. t—>'00

{iii)If ^ is convex and {X{t)}t>o

is hounded then lim ^{X{t))

= inf^.

t—^'OO

T h e natural question t h a t arises is the convergence of the trajectory X{t) as t —> oo. When £ = 0, (4) reduces to the steepest descent dynamical system associated with ^ . In t h a t case, there are different conditions ensuring the asymptotic convergence towards an equilibrium. For instance, it is well-known t h a t under convexity of ^ , the trajectories weakly converge to a minimum of ^ (cf. Bruck [8]). This last result can be generalized when e tends to zero fast enough; indeed, we have P r o p o s i t i o n 2. In addition to (Til^^) and (H2), assume that ^ is convex with argmin ^ 7^ 0. / / JQ e{t) dt < 00 then every solution X(t) of (4) weakly converges to a minimum of ^ as t ^^ 00. We omit the proof of this result because it is similar to t h a t given in [1] for second-order in time systems, which has been revisited with slight variants in [4, 5, 9, 11]. Notice t h a t under the conditions of Proposition 2, any minimizer of # is asymptotically attainable. As the following result shows, t h a t is not the case when the parametrization £{t) satisfies (3). L e m m a 1. Assume that {Hi^^), {H2) o,nd (3) hold. Let X{t) (4) and suppose that X{t) —^ XQQ strongly as t ^^ 00. Then, (i) (Convex case)^ If ^ is convex, then X^o G argmin ^ and -VU{Xcx>)

be a solution

G Nargmin ^{X00).

to

(5)

(ii) (General case) We have X^o G C := {X G i7^ | V ^ ( X ) = 0} and -VU{Xoo)e

n

CO(M+Voo V ^ ( X ( t ) ) = 0 and hence XQ© G argmin ^ . Let w G argmin ^ so t h a t V^{w) = 0. By convexity, V ^ is monotone and we have V^G^^ (V^{v),v-w)>0. (7) ^ This result has been obtained simultaneously by M.-O. Czarnecki (University Montpellier II).

8

F. Alvarez and A. Cabot

Taking the scalar product of (4) by X(-) — w and integrating on [0,^], we obtain '^\\X{t)-wf-^\\X{0)-wf+f\v^{Xis))+sis)VU{X{s)),Xis)-w)ds

= 0.

Jo

Using (7), we get s{s)(VU{X{s)),X{s)

-w)ds
to, X{t) G W, and consequently yt>to,

{V^{X{t)),v) {X{to) - X{t),v),

{VU(Xoo),v) = lim {VU{X{t)),v)

Wt > to. By

> 0.

t—^oo

This proves that, for every v G H^ and w G W^ if {V^{w),v) (V[/(Xoo),f) > 0, which amounts to V^; G {R+V^{W)y,

{VU{Xoo),v) > 0,

< 0 then (9)

where {R^V^{W)y stands for the polar cone of the conic hull of V^(VF). By (9), the vector -VU{Xoo) belongs to (]R+V^(T^))^^, the polar cone of (IR+V^(VF))^. Pinally the bipolar theorem (cf., for example, [6]) ensures that -VU{Xoo) e CO (R+V^(H^)), which completes the proof. D Remark 1. Condition (5) for the convex case expresses a necessary condition for XQO to be a local minimum of the function U on the set argmin #. In the general case, the set arising in (6) is closely related to the normal cone to C at XQO' However, Lemma l(i) cannot be viewed as a special case of Lemma i(ii).

Steepest Descent Equations Coupled by a Vanishing Mutual Repulsion

9

3 Convergence for a One-Dimensional Equilibrium Set When (j) has non-isolated critical points, the general results of the previous section for the vectorial form (4) of SDVR do not ensure the asymptotic convergence of the solution [x{t)^y{t)) under the slow parametrization condition (3). If 0 and V are both convex then it is possible to prove the asymptotic convergence to a pair (a:oo)2/oo) that minimizes [x^y] — i ^ V{x — y) on argmin (j) x argmin 0 (see [10]). Although the repulsion condition (1) is not compatible with the convexity of V ^ the asymptotic selection principle given by Lemma 1 establishes that the "candidates" to be limit points must satisfy an analogous extremality condition depending on U{x,y) = V{x — y). In a one-dimensional scalar setting, a convergence theorem for a second-order in time system involving a repulsion term has been proved in [11]. Next, we show that this type of result is valid for SDVR. To our best knowledge, convergence in the general higher dimensional case is an open problem. From now on, we assume the following hypotheses on the function (f>: for every bounded sequence (xn) C H, lim ||V0(xn)|| = 0 =^ lim ds{xn) = 0, n^oo

(10)

n—>oo

the map 4> is coercive and S = [a,b] for some a,b E H. If a 7^ 6 then we suppose that for every x £ H^ if PA{X) ^ S then V0(a:) is orthogonal to Z\,

(11) (12)

where A is the straight line A\=^ {a-V X{h — a) | A G M}. Remark 2. Condition (10) holds automatically when diuiH < oo, but (11) and (12) are stringent. Take 0 := fod[a,b] where / G C-^(IR+;M) and d[a,b] refers to the distance function to the segment [a, 6]. If the function / is such that f'ip) = 0 and f{x) > 0 for every a; > 0, then the function (j) satisfies (10), (11) and (12). Note that the function (f) is a, C^ function due to the assumption /'(O) = 0. On the repulsion potential V, we assume that there exists a scalar function 7 : iJ ^ ' I^++ such that Vx G H, VV{x) = -j{x)x.

(13)

Theorem 1. Under hypotheses {H), let {x{t),y{t)) (10)-(13) hold, then:

be a solution to SDVR. If

(i) There exists (xoo.yoo) ^ [o^.W' ^^^^ ^^^^ \\mt^oo{x(t),y{t)) = (xoo^yoo)(ii) Suppose that a ^ b and let us denote by Fa (resp. Fb) the connected component ofcl{A\S) such that a G Fa (resp. b G Fb). Assume thatx^Q = y^o = ^ and P^(x(0)) ^ P/^(y(0)). Then £ equals a or b and • i = a implies (PA{x{t)),PA{yit))) G F^ for every t > 0. • i = b implies (^P^(a:(t)),P^(y(t))^ G F^ for every t > 0.

10

F. Alvarez and A. Cabot

(iii) Suppose that the slow parametrization condition (3) holds. If P^{x{0)) 7^ P^(y(0)) then (a^ocl/oo) ^ {^)^}^- When in addition a ^ h, if Xoo = 2/oo = a (resp. 0:00 = 2/00 = b), then we have (P^(a;(t)),P/i(2/(t))) € F^ (resp. {PA{x{t)),PA{y{t))) e r^) for every t > 0. Proof, (i) Prom the coercivity of ^, we deduce the boundedness of the map 11—> {x{t),y{t)) and hence in view of Proposition 1 (ii), we have Umt_oo V0(x(t)) = lim^-^oo V(/)(y(t)) = 0. From assumption (10), it ensues that lim ds(x(t)) = lim ds(y{t)) = 0. t—>oo

(14)

t—>oo

If a = 6 the set 5 is reduced to the singleton {a} and the convergence of x{t) and y(t) toward a is immediate. Now assume that the segment line S is not trivial. Since S C Z\, we have for every x G H, ||a: — Pzi(^)|| = d^ix) < ds(x). Hence, in view of (14), we obtain lim \\x{t) - PA{xm\ T—>00

= lim \\y{t) - PAivim

= 0-

t—>00

As a consequence, the convergence of x{t) (resp. y{t)) as t —> ) oo is equivalent to the convergence of P^(a:(t)) (resp. PA{y{t))i which amounts to the convergence of {x(t),b — a) (resp. {y{t),b — a)) as t -^ oo. For every t > 0, set a{t) := {x{t),b- a) and P{t) := {y{t),b- a). From SDVR, we obtain a{t) + (V0(x(t)), 6 - a) - e{t) j{x{t) - y{t)) {a{t) - m) m

= 0.

(15)

+ (V0(2/(t)), 6 - a) + e{t) j{x{t) - y{t)) {a{t) - I3{t)) = 0.

(16)

We have that {{x,b — a) \ x e S} = [A,yu] for some X < fi. It is immediate to check that, for every x ^ H^ {x,b — a) G [A,^] is equivalent to PA{X) G 5*, so that we can reformulate assumption (12) as (x, b-a)

e [A, fj] =^ (V0(a:), b-a)=0.

(17)

In particular, for every ^ > 0, we have that a(t) G [A,//] (resp. /3(t) G [A,)u]) implies (V0(a:(t)),6 — a) = 0 (resp. (V(y(t)),6 — a) = 0). Since the cj-limit sets of {x{t)}t>o and {2/(0}t>o are included in 5, it is clear that: lim inf a{t), lim sup a(t) C [A,/x] t--*oo

t—*oo

and

lim inf/3{t), lim sup /3{t) c[A,/i] t—»oo

t—>oo

We are now going to prove the convergence of a{t) and ^(t) as t —> oo by distinguishing three cases: Case 1: For all t > 0, we have min{a(t),/5(^)} > fi or max{a(^),^(^)} < A. Without loss of generality, we can assume that for every t > 0, a{t) > fi and P{t) > fi. We deduce that liminft^oo jj,. Since limsup^_^QQ a(t) < /x and lim sup^_^^ f3{t) < /x, we conclude that hmt-^oo oi{t) = limt_>oo P(t) = /x.

Steepest Descent Equations Coupled by a Vanishing Mutual Repulsion

11

Case 2: There exist c G]A,yu[ and to > 0 such that either a(to) < c < (3{to) or /3(to) < c < a(to). Suppose a(to) < c < /3(to). Let us first prove that Vt>to, a{t) A. Let us now use the differential equation (15) satisfied by a. Since a(t) G [A,c] for every t G [ti,too], we deduce from (17) that (V0(x(t)),6 — a) = 0. On the other hand, the sign of a — /? is negative on [ti, too], so that equation (15) yields Vt G [ti, too], Q;(t) < 0. As a consequence, we have c = a (too) ^ Q;(ti), which contradicts (19). Therefore, we conclude that too = +00, which ends the proof of (18). Case 2.a: First assume that a{t) > A for every t > to. From (17) and the fact that a(t) G [A,c], we deduce that (V