Combinatorial and Global Optimization

Series on Applied Mathematics Volume 14

COMBINATORIAL AND GLOBAL OPTIMIZATION Editors

Panos M. Pardalos Athanasios Migdalas Rainer E. Burkard

World Scie

This page is intentionally left blank

COMBINATORIAL AND GLOBAL OPTMZATION

SERIES ON APPLIED MATHEMATICS Editor-in-Chief: Frank Hwang Associate Editors-in-Chief: Zhong-ci Shi and U Rothblum

Vol. 1

International Conference on Scientific Computation eds. T. Chan and Z.-C. Shi

Vol. 2

Network Optimization Problems — Algorithms, Applications and Complexity eds. D.-Z. Du and P. M. Pandalos

Vol. 3

Combinatorial Group Testing and Its Applications by D.-Z. Du and F. K. Hwang

Vol. 4

Computation of Differential Equations and Dynamical Systems eds. K. Feng and Z.-C. Shi

Vol. 5

Numerical Mathematics eds. Z.-C. Shi and T. Ushijima

Vol. 6

Machine Proofs in Geometry by S.-C. Chou, X.-S. Gao and J.-Z. Zhang

Vol. 7

The Splitting Extrapolation Method by C. B. Liem, T. Lu and T. M. Shih

Vol. 8

Quaternary Codes by Z.-X. Wan

Vol. 9

Finite Element Methods for Integrodifferential Equations by C. M. Chen and T. M. Shih

Vol. 10

Statistical Quality Control — A Loss Minimization Approach by D. Trietsch

Vol. 11

The Mathematical Theory of Nonblocking Switching Networks by F. K. Hwang

Vol. 12

Combinatorial Group Testing and Its Applications (2nd Edition) by D.-Z. Du and F. K. Hwang

Vol. 13

Inverse Problems for Electrical Networks by E. B. Curtis and J. A. Morrow

Vol. 14

Combinatorial and Global Optimization eds. P. M. Pardalos, A. Migdalas and R. E. Burkard

Series on Applied Mathematics Volume 14

COMBINATORIAL AND GLOBAL OPTIMIZATION Editors

Panos M. Pardalos Department of Industrial Systems Engineering University of Florida, USA

Athanasios Migdalas Department

of Production Engineering and Management Technical University of Crete, Greece

Rainer E. Burkard Institute of Mathematics Technical University of Graz, Austria

V f e World Scientific m

New Jersey 'London • Singapore • Hong Kong

Published by World Scientific Publishing Co. Pte. Ltd. P O Box 128, Farrer Road, Singapore 912805 USA office: Suite IB, 1060 Main Street, River Edge, NJ 07661 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE

Library of Congress Cataloging-in-Publication Data Combinatorial and global optimization / editors Panos M. Pardalos, Athanasios Migdalas, Rainer E. Burkard. p. cm. — (Series on applied mathematics ; v. 14) Includes bibliographical references. ISBN 9810248024 (alk. paper) 1. Combinatorial optimization — Congresses. 2. Mathematical optimization—Congresses. 3. Nonlinear programming—Congresses. I. Pardalos, P. M. (Panos M.), 1954- II. Migdalas, Athanasios. III. Burkard, Rainer E. IV. Series. QA402.5 .C5435 2001 511'.6-dc21

2001046899

British Library Cataloguing-in-Pubiication Data A catalogue record for this book is available from the British Library.

Copyright © 2002 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher.

For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher.

Printed in Singapore by Mainland Press

Tr\ yX&aaa \iou SSaav eXXrjVixr) TO OTUTI (pico)(ix6 axle, d^ijaouSiic; xou 'Ofdrpou. Mova)(rj cyvoia f] yXSaaa [iou GXIC djj^iouSiec; xou 'Ojarpou. 'Exei oTtdpoi xou 7iepxe 0 for

°^ all all

ie R j eC (i,j)eE

(P)

It is well known that (P) is feasible if and only if an assignment exists. The dual problem of (P) is : max s.t

E?=i ui + E"=i Vj itiij = Cij — Ui — Vj > 0

for

all

(i,j) 6 E

(D)

where each node i € Ror € C is associated with a dual variable M, or Vj respectively. A subgraph F of G, which contains all nodes and no cycle, is called spanning forest or simply forest. Given a spanning forest F we can associate dual variables (u(F),v{F)) = {Ul{F),u2(F),• • ,un{F)MF),MF),• • ,vn(F)) and primal variables x(F) = (iij{F)) not satisfying all equations of (P) such that w^j{F) = d.j - Ui(F) - Vj(F) = 0 for all (i,j) G F and xitj(F) = 0 for all (i,j) 0 F, respectively. Clearly, (u(F),v(F)) and x(F) being not unique (up to a constant) satisfy the complementarity slackness condition. A forest F is called dual feasible if and only if there is a (u(F),v(F)) which is feasible for (D). F is called primal feasible if and only if there is an x(F) which is feasible for (P). Clearly, if F is both primal and dual feasible then x(F) is optimal for (P) and (u(F),v(F)) is optimal for (D). A forest that is both primal and dual feasible is simply called an optimal forest. A forest, from which it can be concluded that (P) is not feasible, is called an infeasible forest. The basic idea of the algorithm to be described in the next section is as follows: (i). Start with a forest i*o (ii). By a special pivoting rule construct a sequence of forests Fo, • • •, Ft such that Ft is either feasible or infeasible.

3

Description of the algorithm

The algorithm does not update primal variables. A dual vector (u(Fm),v(Fm)) is associated with every forest Fm = 0,1,2, ...,t. Consequently, if Ft is optimal,

4 (u(Ft),v(Ft)) is an optimal solution for (D). If it is desirable, an optimal solution for (P) or an optimal assignment can easily be computed from Ft. The forest F m has special structure. As in [1], we call them super forests. F is a super forest if and only if : (i) every connected component of F has a unique root, (ii) every column node in F is at most 2-valent, (iii) every column node, which is not 2-valent is a root and every 2-valent column node is not a root. Given a super forest F , we can partition it into two subforests : Fs (the surplus forest), where F s is the union of those components of F whose root is at least 2-valent, which contains all the components of F rooted at a row node, and FD = F/Fs (the deficit forest), which contains all the components of F rooted at a column node. With such a partition at hand we can define the sets of edges : A(F) = {(i,j)

eE:ie

Fs,j

eFD,ie D

K,j e C} s

B(F) = {(i,j) eE-.ie F ,j eF ,ie

and

», j e C}.

The initial forest FQ contains no edge. Hence, F 0 S = R, FP = C, A(F0) = E and B(F0) = 0. Since the initial forest contains no edge, (u(F0), v(F0)) can be arbitrary. However, it is cheaper to set (u(Fo),v(Fo)) = (0,0) and we do this in our algorithm. With this terminology we can now describe formally an iteration of the algorithm. Step 1 If F ^ = 0, STOP (F m is optimal) Step 2 If A(Fm) = 0, STOP (F m is infeasible) Step 3 Let Sm = Wgh{Fm) = min{wij(Fm)

: (i,j) 6

A(Fm)}

a) Update F m , its roots and F%, F® according to the following cases : (1) h is not isolated root and (a) h is a root (hence 1-valent) (b) h is not a root (hence 2-valent) (2) h is an isolated root (hence 0-valent) Case (la): Let F ^ be the component of F ^ which contains h. Set F ^ + 1 = F^/F^ and F ^ + 1 = F ^ = F ^ U F ^ U (g, h). The column node h is no longer a root.

5 Case (lb): Let (k, h) be the edge of Fm where k is the father of h with respect to its root. Let F^ be the component of F^/(k, ft) which contains ft. Set F°+l=F°/{F*mU{k,h))

and

F^+l = F^U F^U

(g,h).

The roots do not change. Case (2): Let F^ be the component of F^ which contains g. Set ^m+i = F% U K U (