Table Of ContentState of the Art in Global Optimization
No nconvex Optimization and Its Applications
Volume 7
Managing Editors:
Panos Pardalos
University ofF lorida, U.S.A.
Reiner Horst
University of Trier, Germany
Advisory Board:
Ding-ZhuDu
University ofM innesota, U.S.A.
C. A. Floudas
Princeton University, U.S.A.
G. Infanger
Stanford University, U.S.A.
J. Mockus
Lithuanian Academy of Sciences, Lithuania
H. D. Sherali
Virginia Polytechnic Institute and State University, U.S.A.
The titles published in this series are listed at the end oft his volume.
State of the Art
in Global Optimization
Computational Methods and Applications
Edited by
C. A. Floudas
Princeton University
and
P. M. Pardalos
University ofF lorida
KLUWER ACADEMIC PUBLISHERS
DORDRECHT I BOSTON I LONDON
A C.I.P. Catalogue record for this book is available from the Library of Congress.
ISBN-13: 978-1-4613-3439-2 e-ISBN-13: 978-1-4613-3437-8
001: 10.1007/978-1-4613-3437-8
Published by Kluwer Academic Publishers,
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Contents
Preface ...................................................... ix
Lagrange Duality in Partly Convex Programming
S. Zlobec .................................................... 1
Global Optimization using Hyperbolic Cross Points
E. Novak and K. Ritter. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 19
Global Minimization of Separable Concave Functions under Linear Constraints
with Totally Unimodular Matrices
R. Horst and N. van Thoai ....................................... 35
On Existence of Robust Minimizers
S. Shi, Q. Zheng and D. Zhuang ................................... 47
A Branch and Bound Algorithm for the Quadratic Assignment Problem using a
Lower Bound Based on Linear Programming
K.G. Ramakrishnan, M.G.C. Resende and P.M. Pardalos .................. 57
Dynamic Matrix Factorization Methods for using Formulations Derived from
Higher Order Lifting Techniques in the Solution of the Quadratic Assignment
Problem
B. Ramachandran and I.F. Pekny .................................. 75
Conical Coercivity Conditions and Global Minimization on Cones.
An Existence Result
G. Isac ..................................................... 93
The use of Ordinary Differential Equations in Quadratic Maximization with
Integer Constraints
P. Maponi, M. C. Recchioni and F. Zirilli ............................. 109
Adaptive Control via Non-Convex Optimization
G.H. Staus, L.T. Biegler and B.E. Ydstie .............................. 119
A Decomposition-Based Global Optimization Approach for Solving Bilevel Linear
and Quadratic Problems
V. Visweswaran, C.A. Floudas, M.G. Ierapetritou and E.N. Pistikopoulos ....... 139
vi CONTENTS
Generalized TRUST Algorithms for Global Optimization
J. Barhen and V. Protopopescu . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163
Test Results for an Interval Branch and Bound Algorithm for Equality-Constrained
Optimization
R.B. Kearfott ................................................. 181
Equivalent Methods for Global Optimization
D. MacLagan, T. Sturge and W. Baritompa ........................... 201
A C++ Class Library for Interval Arithmetic in Global Optimization
K. Holmqvist and A. Migdalas ..................................... 213
On the Convergence of Localisation Search
D. W. Bulger and G.R. Wood ...................................... 227
Stochastic Approximation with Smoothing for Optimization of an Adaptive
Recursive Filter
W. Edmonson, K. Srinivasan, G. Wang and J. Principe ................... 235
The Grouping Genetic Algorithm
E. Falkenauer ................................................ 249
Accelerating Convergence of Branch-and-Bound Algorithms for Quadratically
Constrained Optimization Problems
T. Van Voorhis and F. Al-Khayyal .................................. 267
Distributed Decomposition-Based Approaches in Global Optimization
I.P. Androulakis, V. Visweswaran and G.A. Floudas ..................... 285
A Finite Algorithm for Global Minimization of Separable Concave Programs
J.P. Shectman and N. V. Sahinidis .................................. 303
A Pseudo E-Approximate Algorithm for Feedback Vertex Set
T. Qian, Y. Ye and P.M. Pardalos .................................. 341
Iterative Topographical Global Optimization
A. Tom and S. Viitanen ......................................... 353
Global Optimization for the Chemical and Phase Equilibrium Problem using
Interval Analysis
K.I.M. McKinnon, C. Millar and M. Mongeau .......................... 365
Nonconvex Global Optimization of the Separable Resource Allocation Problem
with Continuous Variables
E. Haddad .................................................. 383
CONTENTS vii
A. D.C. Approach to the Largest Empty Sphere Problem in Higher Dimension
I. Shi and Y. Yoshitsugu ......................................... 395
A General D.C. Approach to Location Problems
H. Tuy ..................................................... 413
Global Optimization by Parallel Constrained Biased Random Search
I. Garcia and G.T. Herman . ...................................... 433
Global Optimization Problems in Computer Vision
P. Sussner, P.M. Pardalos and G.X. Ritter ............................ 457
An Application of Optimization to the Problem of Climate Change
1.A. Filar, P.S. Gaertner and M.A. lanssen ............................ 475
Dynamic Visualization in Modelling and Optimization of TIl Defined Problems
W.F. Eddy and A. Mockus ....................................... 499
A New Global Optimization Algorithm for Batch Process Scheduling
L. Mockus and G. V. Reklaitis ..................................... 521
Nonconvexity and Descent in Nonlinear Programming
A. Lucia and I. Xu . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 539
Global Optimization of Chemical Processes using Stochastic Algorithms
I.R. Banga and W.D. Seider ............................... . . . . . . . 563
Logic-Based Outer-Approximation and Benders Decomposition Algorithms for
the Synthesis of Process Networks
M. Tiirkay and I.E. Grossmann .................................... 585
Combinatorially Accelerated Branch-and-Bound Method for Solving the MIP
Model of Process Network Synthesis
F. Friedler, 1.B. Varga, E. Feher and L.T. Fan ......................... 609
Discrete Optimization using String Encodings for the Synthesis of Complete
Chemical Processes
E.S. Fraga .................................................. 627
Preface
Optimization problems abound in most fields of science, engineering, and tech
nology. In many of these problems it is necessary to compute the global optimum
(or a good approximation) of a multivariable function. The variables that define the
function to be optimized can be continuous and/or discrete and, in addition, many
times satisfy certain constraints.
Global optimization problems belong to the complexity class of NP-hard prob
lems. Such problems are very difficult to solve. Traditional descent optimization
algorithms based on local information are not adequate for solving these problems.
In most cases of practical interest the number of local optima increases, on the aver
age, exponentially with the size of the problem (number of variables). Furthermore,
most of the traditional approaches fail to escape from a local optimum in order to
continue the search for the global solution.
Global optimization has received a lot of attention in the past ten years, due
to the success of new algorithms for solving large classes of problems from diverse
areas such as engineering design and control, computational chemistry and biology,
structural optimization, computer science, operations research, and economics.
This book contains refereed invited papers presented at the conference on "State
of the Art in Global Optimization: Computational Methods and Applications" held
at Princeton University, April 28-30, 1995. The conference presented current re
search on global optimization and related applications in science and engineering.
The papers included in this book cover a wide spectrum of approaches for solving
global optimization problems and applications.
We feel that the book will definitely be a valuable source of information to faculty,
students and researchers in optimization, engineering, mathematics, computer sci
ences, and computational chemistry. We would like to take the opportunity to thank
the authors of the papers, the referees, the school of engineering and applied science
of Princeton University for supporting this effort, and the publisher for assisting us
in producing this book.
Christodoulos A. Floudas and Panos M. Pardalos
Princeton University and University of Florida
June 1995
ix
Lagrange Duality in Partly Convex Programming
SANJO ZLOBEC zlobec @math.mcgill.ca
Department oj Mathematic. and Stati,tic" McGill Uni1Jer,ity, Montreal, Quebec, Canada H9A
M<6
Abstract. A Lagrangian duality theory is formulated for a large class of nonconvex optimization
prob1ems. The theory is based on the recent characterizations of global and local optima for partly
convex: programs. It uses basic notions from convex and parametric programming, such as the
minimal index set of active constraints and continuity of the feasible set point-to-set mapping.
The results are applied to the classic navigation problem of Zermelo. The primal problem CQnsists
of finding a steering angle that minimizes the sailing time to a target. Its dual solution is a
function that associates, with every steering angle, the sensitivity of the sailing time relative to
small perturbations of the target. The unconstrained minima of the dual solution yield the most
''robust" steering angles.
Keywords: Dual problem, partly convex programming, characterizations of optimality, global
optimum, local optimum, Zermelo's navigation problem.
AMS Classification: 90C25, 90C30, 90C90.
1. Introduction
Consider the mathematical program of the form
Min I(z)
(P) s.t.
li(Z)::;O, iEP={l, ... ,m}.
where I, Ii : RN -+ R, i E P are continuous functions. IT there exists a splitting
z = (z,9), z ERn, 9 E RP, n + p = N, such that l(e,9), li(e,9) : Rn -+ R, i E
P are convex functions for every 9 E RP, then (P) is termed a "partly convex
program" (abbreviation: PCP). (In [8] such programs were called "convexifiable
by a splitting".) These programs are essentially different from convex and usual
generalized convex programs. The feasible set
is generally nonconvex (often disjoint) and the local and global minima generally
do not coincide.
Many theoretical and real-life problems can be formulated as PCPs. They range
from the classic problem of isoperimetry of triangles (see, e.g., [10]) and the classic
navigation problems on the water and under the water of Zermelo ([5,9,11]) to
the problems of optimal pooling and blending in oil refineries and minimal energy
problems in chemical physics (see, e.g., [11] and the references given therein).
1
CA. Floudas and P.M. Pardalos (eds.), State oft he Art in Global Optimiwtion 1-17.
© 1996 Kluwer Academic Publishers.
Description:Optimization problems abound in most fields of science, engineering, and tech nology. In many of these problems it is necessary to compute the global optimum (or a good approximation) of a multivariable function. The variables that define the function to be optimized can be continuous and/or discr
