High Performance Algorithms and Software for Nonlinear Optimization

418 Pages·2003·33.385 MB·English

by  M. Al-Baali (auth.), Gianni Di Pillo, Almerico Murli (eds.)

High Performance Algorithms
and Software for
Nonlinear Optimization
Applied Optimization
Volume 82
Series Editors:
Panos M. Pardalos
University ofF lorida, US.A.
Donald W. Hearn
University ofF lorida, US.A.
High Performance Algorithms
and Software for
Nonlinear Optimization
Edited by
Gianni Di Pillo
Universitd di Roma "La Sapienza", Italy
Almerico Murli
Universitd di Napoli Federico II, Italy
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Library of Congress Cataloging-in-PubUcation
Di Pillo, Gianni! Murli, Almerico
High Performance Algorithms and Software for Nonlinear Optimization
ISBN-13:978-1-4613-7956-0 e-ISBN-13:978-1-4613-0241-4
001: 10.1007/978-1-4613-0241-4
Copyright ro 2003 by Kluwer Academic Publishers
Softcover reprint of the hardcover I st edition 2003
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Contents
Preface VII
1.
Quasi-Newton Algorithms for Large-Scale Nonlinear Least-Squares
M Al-Baali
2. 23
The Extended Ritz Method in Stochastic Functional Optimizazion: An Example of
Dynamic Routing in Traffic Networks
M Baglietto, M Sanguineti, R Zoppoli
3. 57
Efficient Serial and Parallel Implementation of the Cutting Angle Method
G. Beliakov, K M Ting, M Murshed, A. Rubinov, M Bertoli
4. 75
A Globalization Strategy for Interior Point Methods for Mixed Complementarity
Problems
S. Bellavia, B. Morini
5. 95
A Comparative Study of Large-Scale Nonlinear Optimization Algorithms
H. Y. Benson, D. F. Shanno, R J. Vanderbei
6. 129
A Software System for PDE-Constrained Optimization Problems
P. T Boggs, K R. Long
7. 149
A Truncated SQP Algorithm for Solving Nonconvex Equality Constrained
Optimization Problems
L. Chauvier, A. Fuduli, J. C. Gilbert
8. 177
Newton-Type Methods for the Approximate Solution of Nonlinear Programming
Problems in Real-Time
M Diehl, H. G. Bock, J. P. Schlijder
9. 201
Fruitful Uses of Smooth Exact Merit Functions in Constrained Optimization
G. Di PilIo, G. Liuzzi, S. Lucidi, L. Palagi
10. 227
New Perspective on the Theorems of Alternative
Y. G. Evtushenko, A. 1. Golikov
II. 243
Planar-CG Methods and Matrix Tridiagonalization in Large Scale Unconstrained
Optimization
G. Fasano
12. 265
Filter-type Algorithms for Solving Systems of Algebraic Equations and Inequalities
R. Fletcher, S Leyffer
13. 285
A New Algorithm for Solving Large Scale Molecular Distance Geometry Problems
L. T Hoai An, P. D. Tao
14. 303
PENNON A generalized Augmented Lagrangian Method for Semidefinite Programming
M Kocvara, M Stingl
15. 323
Understanding Asynchronous Parallel Pattern Search
T. G. Kolda, V. J. Torczon
16. 343
Smoothing Techniques for the Solution of Finite and Semi-Infinite Min-Max-Min
Problems
E. Polak
17. 363
Exploiting Optimality Conditions in Accurate Static Circuit Tuning
C. Visweswariah, A. R. Conn, L.G. Silva
18. 383
Efficient Analysis on a Truncated Newton Method with Preconditioned Conjugate
Gradient Technique for Optimization
J. Z Zhang, N. Y. Deng, Z. Z Wang
Preface
This volume contains the edited texts of the lectures presented at the Workshop on
High Performance Algorithms and Software for Nonlinear Optimization held in Erice,
Sicily, at the "G. Stampacchia" School of Mathematics of the "E. Majorana" Centre
for Scientific Culture, June 30 - July 8, 2001.
In the first year of the new century, the aim of the Workshop was to assess the past
and to discuss the future of Nonlinear Optimization, and to highlight recent achieve
ments and promising research trends in this field. An emphasis was requested on
algorithmic and high performance software developments and on new computational
experiences, as well as on theoretical advances.
We believe that such goal was basically achieved. The Workshop was attended by
71 people from 22 countries. Although not all topics were covered, the presentations
gave indeed a wide overview of the field, from different and complementary stand
points. Besides the lectures, several formal and informal discussions took place. We
wish to express our appreciation for the active contribution of all the participants in
the meeting.
The 18 papers included in this volume represent a significant selection of the most
recent developments in nonlinear programming theory and practice. They show that
there is plenty of exciting ideas, implementation issues and new applications which
produce a very fast evolution in the field.
As a matter of fact, the new parallel computer technologies are having a significant
impact upon the development of numerical software in general, and specifically within
the optimization software, which is being doubly affected. On one hand, the most
straightforward way to parallelize is by embedding the enormous amount of existing
highly efficient parallel linear algebra software into optimization codes. On the other
hand, considerations about parallelism lead to new algorithms, especially tailored on
the new architectures, in which great care is taken in the algorithm decomposability.
An example of new parallel algorithm is given by the multidirectional line search
strategies, which are the subject of the paper by Kolda and Torczon, who are among
the founders of such class of algorithms. Their paper supplies an interesting overview
as well as a deep analysis of the main parallelization issues.
Benson, Shanno and Vanderbei give an excellent comparative analysis of the exist
ing nonlinear optimization software, made on a large set of test problems, providing a
very comprehensive overview of the efficiency of the available software for large scale
nonlinear optimization.
Strongly concerned with the new computational tools is the paper by Boggs and
Long, which deals with one of the most interesting topics, i.e. the use of metacomput
ing environments: heterogeneous computational and software resources are effectively
combined in a user friendly solver for the design of complex physical systems. The
efficient use of networking computing resources is going to make available to a vast
V1ll
community the optimization technology, as well as an unbelievable quantity of com
puting power. It is easy to forecast, in the next few years, a large amount of work
in that direction which is probably going to produce a revolution in the numerical
software and more generally in the problem solving, only comparable to the one that
in the sixties did follow the use of computers.
Of course the progress of numerical optimization relies either on the development
of new algorithms or in the evolution of well known strategies, which improve in terms
of accuracy and efficiency the existing ones. Al Baali considers a modification of the
L-BFGS algorithm for large scale nonlinear least-squares, in which he considers an
updating strategy for the Hessian which takes advantage of the structure of the ob
jective function; the numerical results show the effectiveness of the strategy on a large
set of test problems. Baglietto, Sanguineti and Zoppoli present an extension of the
Ritz method in stochastic functional optimization, and they show how their method
can be fruitfully applied in some large scale optimization problems. A very simple,
but apparently very effective, truncated SQP for equality constrained problems is
presented by Chauvier, Fuduli and Gilbert; especially interesting are the computa
tional performances of the method on some real life problems. The paper by Fasano
about tridiagonalization of a symmetric matrix through a Planar CG approach, and
its use in nonlinear programming, can be seen in the general framework of the strong
relationship existing between numerical linear algebra and optimization. Fletcher and
Leyffer show how the NLP filter approach can be extended to the solution of systems
of algebraic equations and inequalities.
Several papers in the volume show how solving real-world problems is a main
source of inspiration for large scale optimization algorithms; the development of effi
cient special purpose software has been strongly stimulated in the last years by the
need of optimization algorithms ad hoc devised or tuned for specific applications. A
clear example of that is the static optimization problem which arises in the circuit
tuning problem described in the paper by Visweswariah, Conn and Silva; optimality
conditions are derived, which are strongly related to the inherent structure of the
static problem, and which allow a solution of the problem apparently much more
accurate than in the past. Engineering design requirements motivate also the theo
retical and computational analysis of the min-max-min problems considered in the
paper by Polak: smoothed and consistent approximations are introduced for their
solution. Of interest for possible applications in real-life problems is the Newton type
approach for the real time optimization of nonlinear control systems presented by
Diehl, Bock and Schloder.
Combinatorial, and more generally global optimization is the area which more
benefited by parallel computing; in particular, deterministic algorithms tend to be
excellent candidates for parallel computing, as shown by Beliakov, Ting, Murshed,
Rubinov and Bertoli with their parallel version of the cutting angle method. The
molecular distance geometry, a very challenging global optimization problem which
has attracted a large amount of work in the last few years, is considered by Hoai
ix
An and Tao, who propose a new problem formulation, based on a DC scheme, which
appears very competitive with respect to more classical approaches for such problem.
Finally, several papers in this volume show that also in classical strategies there
is still some room for improvement, and how theoretical analysis can help in better
understanding the numerical behavior of algorithms, forecast scenarios and suggest
appropriate choices in developing efficient software. Bellavia and Morini endow in
terior point methods for mixed complementarity problems with a new globalization
strategy, with successful computational results. Di Pillo, Liuzzi, Lucidi and Palagi
describe algorithms for constrained problems based on an augmented Lagrangian
function which enjoyes improved exactness properties. Kocvara and Stingl extend a
penaltyj barrier multiplier approach to convex semidefinite programs in their PEN
NON algorithm, with interesting computational results on a very large set of test
problems. Zhang, Deng and Wang investigate the efficiency of the Truncated Newton
method with preconditioning from a theoretical point of view, by making a com
parison with the Newton's method with Cholesky factorization, thus explaining the
computational comparative results obtained in the past. Also an apparently more the
oretical paper, such as the one by Evtushenko and Golikov, has relevance from the
computational point of view: new theorems of the alternative give us an opportunity
to construct new numerical methods for linear and nonlinear programming.
We are indebted to many anonymous referees for the care taken in reviewing the
papers submitted for publication.
Our gratitude is due to the "E. Majorana" Centre in Erice, which offered its
facilities and rewarding environment: its staff was certainly instrumental for the
success of the Workshop.
Our gratitude is due to Giovanni Fasano and Massimo Roma for the effort and
time devoted as members of the Organising Committee.
Our gratitude is also due to the University of Rome "La Sapienza" , and to the I tal
ian National Research Council through its "Research Center on Parallel Computing
and Supercomputers" , for their financial support.
Finally, we wish to thank Kluwer Academic Publishers for publishing this volume.
Gianni Di Pillo
Almerico Murli