Table Of ContentOptimization Theory
Applied Optimization
Volume 59
Series Editors:
Panos M. Pardalos
University of Florida, U.S.A.
Donald Hearn
University of Florida, U.S.A.
The titles published in this series are listed at the end of this volume.
Optimization Theory
Recent Developments from Matrahaza
Edited by
Franco Giannessi
Department of Mathematics,
University of Pisa,
Pisa, Italy
Panos Pardalos
Department (~f Industrial and
Systems Engineering,
University (~f Florida,
Gainesville, Florida, U.S.A.
and
Tamas Rapcsak
Computer and Automation Institute,
Hungarian Academy of Sciences,
Budapest, Hungary
KLUWER ACADEMIC PUBLISHERS
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ISBN-I3:978-1-46I3-7983-6 e-ISBN-I3:978-1-4613 -0295-7
DO 1:10.1007/978-1-4613 -0295-7
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CONTENTS
Preface Vll
Heuristics for the Process Network Synthesis Problem
z. Blazsik, Cs. HollO, Cs. Imreh and Z. Kovacs
Heuristics for Simplified Process Network Synthesis (PNS)
Problems with a Blossom-Type Algorithm for the Edge
Covering Problem
Z. Blazsik, K. Keseru and Z. Kovacs 19
Lower and Upper Bounds on the Probability of the Union of
Some Events with Applications
J. Bukszar 33
The Linear Complementarity Problem and the Class of
Generalized Positive Subdefinite Matrices
J.-P. Crouzeix and S. Komlosi 45
Computer Experiences with Successive Regression
Approximations for Solving Equations
I. Deak 65
A Bundle of Smooth Lagrangians in Mathematical
Programming
v. F. Demyanov 81
A Nontopological Two-function Minimax Theorem with
Monotone Transformations of the Functional Values
F. Forgo 99
Non-smooth Optimization with Randomization
L. Gerencser and Zs. Vago III
vi
The Shennan-Morrison Formula for the Detenninant and its
Application for Optimizing Quadratic Functions on Condition
Sets Given by Extreme Generators
G. Keri 119
Duality For D.C. Optimization Over Compact Sets
Juan-Enrique Marllinez-Legaz and Michel Volle 139
Characterization of Monotone Operators by Using a Special
Class of Preinvex Functions
S. Z. Nemeth 147
Two Approaches for Parallelizing the UEGO Algorithm
P. M. Ortigosa, I. Garcia and M. Jelasity 159
Convexification of Functions by Nonlinear Coordinate
Transfonnations
T. Rapcsak 179
Packing Equal Circles in a Square I. - Problem Setting and
BOlmds for Optimal Solutions
P. G. Szabo, T. Csendes, L. G. Casado and I. Garcia 191
Packing Equal Circles in a Square II. - New Results for up
to 100 Circles Using the TAMSASS-PECS Algorithm
L. G. Casado, I. Garcia, P. G. SzabO and T. Csendes 207
On Duality for Quasi-convex Supremization and Reverse
Convex Infimization
1. Singer 225
A Minimization Problem in Function Spaces
B. Uhrin 255
PREFACE
This volume contains refereed papers based on the lectures presented at
the XIV International Conference on Mathematical Programming held
at Matrahaza, Hungary, between 27-31 March 1999. This conference
was organized by the Laboratory of Operations Research and Deci
sion Systems at the Computer and Automation Institute, Hungarian
Academy of Sciences. The editors hope this volume will contribute to
the theory and applications of mathematical programming.
As a tradition of these events, the main purpose of the confer
ence was to review and discuss recent advances and promising research
trends concerning theory, algorithms and applications in different fields
of Optimization Theory and related areas such as Convex Analysis,
Complementarity Systems and Variational Inequalities.
The conference is traditionally held in the Matra Mountains, and
housed by the resort house of the Hungarian Academy of Sciences.
This was the 14th event of the long lasting series of conferences started
in 1973.
The organizers wish to express their thanks to the authors for their
contributions in this volume, and the anonymous referees for their valu
able comments. Special thanks are directed to our sponsors, the Hun
garian Academy of Sciences, the National Committee for Technological
Development, the Hungarian National Science Foundation, and last but
not least, the Hungarian Operational Research Society.
We would like to thank John Martindale from Kluwer Academic
Publishers for helping us produce this volume, Eva Nora Nagy for cor
rections and proof-readings, and Peter Dombi for his excellent work on
typesetting and editing the manuscript.
January 2001
The Editors
vii
Heuristics for the Process Network Synthesis
Problem
Z. Blazsik, Cs. Hollo, Cs. Imreh, Z. Kovacs
Department of Informatics,
University of Szeged, Hungary
Abstract. A manufacturing system consists of operating units converting materials
of different properties into further materials. In a design problem we are to find a
suitable network of operating units which produces the desired products from the
given raw materials. By considering this network design from a structural point of
view, we obtain a combinatorial optimization problem called the Process Network
Synthesis or (PNS) problem. It is known that the PNS problem is NP-complete. In
this work two heuristic algorithms are presented for solving this problem. Moreover,
we establish some worst-case bound theorems for the algorithms under consideration,
and we present an empirical analysis on randomly generated PNS problems.
Key words: Heuristics, Process Design
1. Introduction
In a manufacturing system materials of dUferent properties are con
sumed through various mechanical, physical and chemical transforma
tions to yield desired products. Devices in which these transformations
are carried out are called operating units, e.g., a lathe or a chemical
reactor. Thus, a manufacturing system can be considered as a network
of operating units, which is called process network. The process de
sign problem in general, and fiowsheeting in particular, is to construct
a manufacturing system. A design problem is defined from a struc
tural point of view by the raw materials, the desired products, and the
available operating units. This triplet determines the structure of the
problem as a process graph containing the corresponding interconnec
tions among the operating units. Consequently, the appropriate process
networks can be described by some subgraphs of the process graph be
longing to the design problem under consideration. Naturally, the cost
minimization of a process network is indeed essential.
The importance of process network synthesis (PNS) arises from the
fact that networks like this are ubiquitous in the chemical and related
industries. The foundations of PNS and the background of the combina
torial model studied here can be found in [5], [6], [7], and [8]. Therefore,
1 This work has been supported by the Ministry of Culture and Education of
Hungary, Grant FKFP 0008/1999, and by the Hungarian National Foundation for
Scientific Research, Grant T030074.
F Giannessi et al. (eds.), Optimization Theory, 1-17.
© 2001 Kluwer Academic Publishers.
2 Z. Bhizsik, Cs. Hollo, Cs. Imreh, Z. Kovacs
we recall only the definitions in Section 2 where the necessary notions
and notation are introduced.
It has recently been proven (see [2], [4], [11]) that the PNS problem is
NP-complete. For NP-hard problems, the construction and analysis of
heuristic algorithms is a fastly developing area. By heuristic algorithms
we mean fast, (polynomial time) algorithms which do not guarantee an
optimal solution in general, but always result in a feasible solution
which is close to the optimal solution in some sense. Heuristic algo
rithms are important for several reasons. The feasible solutions found
by these algorithms can be used in procedures based on branch and
bound techniques. Moreover, in practical problems often there is not
enough time to find an optimal solution by an exponential algorithm,
or the size of the problem is too large to use an exponential algorithm.
In these cases, heuristic algorithms can be useful again. It can also oc
cur that we do not need an optimal solution, it is sufficient to find a
feasible solution the cost of which is not far from the optimal cost. One
can find further details on heuristic algorithms in [9], [13]. For the PNS
problem, heuristic algorithms have not been studied yet. In Section 3
we introduce two heuristic algorithms for this problem. In Section 4
tight worst-case bounds are presented for these heuristics for particu
lar classes of PNS problems. Moreover, in Section 5 the results of an
empirical analysis for the effectiveness of these heuristics are given.
2. Preliminaries
In the combinatorial approach the structure of a process can be de
scribed by the process graph (see [6]) defined as follows.
Let M be a finite nonempty set, the set of the materials. Moreover,
let 0 i= 0 ~ p'(M) x p'(M) with M n 0 = 0, where p'(M) denotes
the set of all nonempty subsets of M. The elements of 0 are called
operating units. For an operating unit (a, (3) EO, a and (3 are called
the input-set and output-set of the operating unit, respectively. Pair
(M,O) is defined to be a process graph or P-graph in short. The set
of vertices of this directed graph is M U 0, and the set of arcs is A =
Al U A2 where Al = {(X, Y) : Y = (a,{3) E 0 and X E a} and
A2 = {(Y,X) : Y = (a,{3) EO and X E {3}. In the rest of the paper,
by path and cycle we always mean directed path and cycle.
Now, let 0 ~ 0 be arbitrary. Let us define the following functions
on the set
0:
U U
matin(o) = a, matOut(o) = {3,
(o.,{3)Eo (o.,{3)Eo
