Test Examples for Nonlinear Programming Codes

187 Pages·1981·4.618 MB·English

by  Willi Hock, Klaus Schittkowski (auth.)

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continuation on page 185
Lectu re Notes
in Economics and
Mathematical Systems
Managing Editors: M. Beckmann and H. P. Kunzi
187
Willi Hock
Klaus Schittkowski
Test Exam pies for
Nonlinear Programming Codes
Springer-Verlag
Berlin Heidelberg New York 1981
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Preface
The performance of a nonlinear programming algorithm can only
be ascertained by numerical experiments requiring the collection
and implementation of test examples in dependence upon the desired
performance criterium. This book should be considered as an assis
tance for a test designer since it presents an extensive collec
tion of nonlinear programming problems which have been used in
the past to test or compare optimization programs. He will be in
formed about the optimal solution, about the structure of the
problem in the neighbourhood of the solution, and, in addition,
about the usage of the corresp,onding FORTRAN subroutines if he is
interested in obtaining them -ofi a magnetic tape.
Chapter I shows how the test examples are documented. In par
ticular, the evaluation of computable information about the solu
tion of a problem is outlined. It is explained how the optimal
solution, the optimal Lagrange-multipliers, and the condition
number of the projected Hessian of the Lagrangian are obtained.
Furthermore, a classification number is defined allowing a formal
description of a test problem, and the documentation scheme is
described which is used in Chapter IV to present the problems.
Chapter II contains detailed information about the usage of the
corresponding FORTRAN subroutines for all programmers who want to
receive them on a magnetic tape from the authors. A comprehensive
list of the test problems and some solution properties are given
in Chapter III. Furthermore, there is a list of the identification
numbers of some famous, frequently used test examples to simplify
their search. Together with their solution properties, all test
problems are presented then in Chapter IV using a common documen
tation scheme. Appendix A contains constant data for more exten
sive problems and the numerical test results computed by some
well-known optimization programs are listed in Appendix B. Some
further results are encluded in Appendix C where the optimal re
striction function values and the corresponding Lagrange-multi
pliers are listed.
IV
The coding of the test problem functions together with their
gradients requires about 10,000 FORTRAN statements making it
impossible to publish the corresponding listings. But the authors
are willing to provide an interested user with all subroutines
on a magnetic tape. An order form is included at the end of the
book.
The authors would like to express their gratitude to the
Rechenzentrum of the University of Wtirzburg for the generous
support and to M. Bahr, R. Ruppert, and H. Megerle for their
assistance when implementing and testing the programs.
....................................................................
Contents
Chapter I Description of the documentation
1. Introduction
2. Computable information about a solution 3
3. The classification number 8
4. The documentation scheme 9
Chapter II Usage of the FORTRAN subroutines 12
Chapter III Condensed information about the test 17
problems
Chapter IV The test problems 23
Appendix A Constant data 128
Appendix B Some numerical test results 141
Appendix C Restriction function values and Lagrange 157
multipliers
References 171
..................................................................
Chapter I
DESCRIPTION OF THE DOCUMENTATION
This chapter shows how the information about the solution
properties of the test problems are obtained, and the documen
tation scheme is described.
1. Introduction
The development of algorithms for solving the nonlinear pro
gramming problem
min f(x)
g j (x) ~ 0, j =1 , ••• , m1
(NLP)
gj(x) 0 j=m1+1, .•. ,m
xl S x S ~
with continuously differentiable functions f and g1, .•• ,gm requires
the preparation of test examples. Our intention is to present an
extensive collection of o·ptimization problems which can be used
by a test designer to choose from. All problems are found in the
literature and have been used in the past to develop, test, or
compare optimization software. There is no attempt to classify the
problems in the sense of 'good' or 'bad' test examples, since this
question depends on the performance profile of the test designer.
It should be noted, however, that all problems have been executed
to compare 26 optimization programs numerically, cf. Hock [31J.
The same set of nonlinear programming codes was investigated by
Schittkowski [55J in the scope of a comparative study based on
randomly generated test problems with predetermined solutions.
2
Another intention is to provide a user with information about
the structure of a test problem at the optimal solution to get
more insight in the numerical behaviour of optimization codes.
The fundamental tool is the Lagrange-function
m
L(x,u):= f(x) - B uJogo(x) (1)
j=1 J
defined for all x E ~ and u = (u1, ••• ,um)T E i m • The vari
ables u1, ••• ,um are called 'Lagrange-multipliers'. To simplify
the notation in this and the following section, let us assume
that there are no bounds on the variables or, in other words,
that they are formulated as general restrictions in the form
gm +i(x):= Xi - Xli
1
(2)
gm +n+i(x):= -xi + ~
1 i
T T
for i=1, ••• ,n, where x = (x1, ••• ,xn) , Xl = (xl , ••• ,xl ) ,
1 n
x = (xu , ••• ,x )T. The Lagrange-function (1) allows to charac-
u 1 -~n
terize the optimality conditions which can be stated in the follow-
ing form, cf. McCormick [40J.
Theorem (sufficient second order optimality condition): Let f
and g1, ••• ,gm be twice continuously differentiable functions.
A point x* E Rn is an isolated local minimizer of (NLP) , if
the re eX10 S t s a vec t or u * = (u*1 ""'UID*) T suc h tha t the
following conditions are valid:
a) Kuhn-Tucker-conditions:
,
gj(x*) ;:, 0 j=1 , ... ,m1
gj (x*) 0 j=m1+ 1 , ••• ,m
,
uit ;:, 0 j="1 , ••• ,m1
J
Ujgj(x*) = 0 j=1, ... ,m1
VxL(x*,u*) = 0 (3)
condition: For every nonzero vector y with
o for all j with Uj > 0 , j=1, ••• ,m1, and
o , j=m1, ••• ,m, it follows that
3
Under an additional assumption (constrained qualification),
the reversal of the theorem is also valid. Since the Kuhn-Tucker
condition (3) indicates that the gradient of the objective func
tion is a linear combination of the gradients of the restriction
functions, we can consider the Lagrange-multipliers as a measure
for the scaling or ascent properties of the restrictions at the
optimal solution. In addition, a vanishing Lagrange-multiplier
uj with gj(x*) = 0 shows that the restriction gj is redundant.
In this situation, we say that a test problem is degenerate and
the quotient of the absolutely greatest and absolutely smallest
non-zero Lagrange-multipliers is called 'degree of degeneracy'.
The second order condition (4) is interpreted in the sense
that the Lagrange-function projected on the subspace of the
binding constraints is locally convex. A measure for the curva
ture of the Lagrangian in this subspace is given by the condition
number of the Hesse-matrix, i.e. the quotient of the greatest
and smallest eigenvalue. They are positive if and only if con
dition (4) is valid.
2. Computable information about a solution
This describes the procedure which was used to compute
sectio~
an optimal solution and to check the optimality conditions (3)
and (4). For many test examples, the exact solution x* of (NLP)
is known a priori within machine precision. However, in all other
cases, we have to approximate x* numerically, that means by an
optimization code. But since none of the implemented optimization
programs could solve all problems successfully, since different
codes give different numerical solutions on return, and since, in
particular, no optimization program can guarantee having found
the global solution of (NLP), we executed the test problems by
the routines of Table 1 which encludes the name of the author
and the underlying mathematical method.
4
Code Author I1ethod
)
VF02AD Powell
Quadratic approximation
OPRQP Bartholomew-Biggs ~
GRGA Abadie Generalized reduced gradient
VF01A Fletcher }
Multiplier
FUNMIN Kraft
FI1IN Kraft , Lootsma Penalty
Table 1: Optimization programs for evaluating the optimal
solutions.
More detailed information about these programs is contained in
Schittkowski [55). The purpose of performing these test runs was
not to compare the efficiency of the programs (has been done
separately, cf. Hock (31)), but only to compute one solution as
precise as possible. Therefore, the programs were executed with
very low stopping tolerances and a decision was made which of the
obtained results could be accepted and published as a solution
of the test problem. This choice has been based on the objective
function value f(x*) and the sum of constrained violations r(x*),
i. e. of
m
1 m
r(x*):= - D min(O,gJo(x*)) + Digj (x*) I ,
j=1 j=m1+1
where x* denotes the current computed approximation of the exact,
but unknown solution. A numerical solution was considered to be
the best one which had the lowest objective function value among
all results with a constraint violation below a tolerance of 10-7•
The procedure described so far yielded for each test problem
of this collection the data x*, f(x*), and r(x*), where x* is
considered now to be the optimal solution of the optimization
problem. But these results give too less information about the
structure of the problem in the neighbourhood of the solution x*.
In particular, one has to determine the active constraints allowing
a computation of the optimal Lagrange-multipliers u*, i.e. of the
index set