Optimization and Operations Research: Proceedings of a Conference Held at Oberwolfach, July 27–August 2, 1975

309 Pages·1976·3.429 MB·English-German

by  A. Bachem (auth.), Professor Werner Oettli, Professor Klaus Ritter (eds.)

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Lectu re Notes
in Economics and
Mathematical Systems
Managing Editors: M. Beckmann and H. P. Kunzi
Operations Research
117
Optimization and
Operations Research
Proceedings of a Conference Held at
Oberwolfach, July 27-August 2, 1975
Edited by W. Oettli and K. Ritter
Springer-Verlag
Berlin· Heidelberg· New York 1976
Editorial Board
H. Albach' A. V. Balakrishnan' M. Beckmann (Managing Editor)
P. Dhrymes . J. Green' W. Hildenbrand' W. Krelle . H. P. Kunzi
(Managing Editor) . K. Ritter' R. Sato . H. Schelbert . P. Schonfeld
Managing Editors
Prof. Dr. M. Beckmann Prof. Dr. H. P. Kunzi
Brown University UniversiUit Zurich
Providence, RI 02912/USA 8090 Zurich/Schweiz
Editors
Professor Werner Oettli
Fakultat fur Mathematik und Informatik
Universitat Mannheim
SchloB
6800 Mannheim l/BRD
Professor Klaus Ritter
Mathematisches Institut A
Universitat Stuttgart
Pfaffenwaldring 57
7000 Stuttgart 80/BRD
Library of Congress Cataloging in Publication Data
Main entry under title:
Optimization and operations research.
(Lecture notes in economics and mathematical
systems ; 117)
Bibliography: p.
Includes index.
1. Mathematical optimization--Congresses.
2. Control theory--Congresses. 3. Programming
(Mathematics)--Congress. 4. Operations research
--Congresses. I. Oettli, Werner. II. Ritter,
Klaus, 1936- III. Series.
QA402.5.0643 001.4'24 75-46562
AMS Subject Classifications (1970): 49AXX, 49BXX, 49DXX, 65K05,
90 BXX,90CXX
ISBN-13: 978-3-540-07616-2 e-ISBN-13: 978-3-642-46329-7
001: 10.1007/978-3-642-46329-7
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© by Springer·Verlag Berlin' Heidelberg 1976
Softcover reprint of the hardcover 1s t edition 1976
Offsetdruck: Julius Beltz, Hemsbach/Bergstr.
TABLE OF CONTENTS
A. Bachem
Reduction and Decomposition of Modulo Optimization Problems
H. Baier
Optimization of Elastic Structures by r·lathematical Programming 9
Techniques
P. Bod
On Closed Sets Having a Least Element 23
L.C.W. Dixon
On the Convergence of the Variable Metric Method with Numerical
Derivatives and the Effect of Noise in the Function Evaluation 35
H. Drygas
Parallel Path Strategy. Theory and Numerical Illustrations 55
J. Fi scher
On r·linimization under Linear Equality Constraints 77
W. Gaul
On Constrained Shortest-Route Problems 83
K. Glashoff
Bang-Bang Solution of a Control Problem for the Heat Equation 93
J. Gwinner
Generalized Stirling-Newton Methods 99
J. Hartung
On a Method for Computing Pseudoinverses 115
R. Hettich
Charakterisierung lokaler Pareto-Optima 127
E. Hopfinger, U. Steinhardt
On the Exact Evaluation of Finite Activity Networks with Stochastic 143
Durations of Activities
~1. Koh 1e r
Approximation of a Parabolic Boundary Control Problem by the Line 147
Method
P. Kosmol
Regularisation of Optimization Problems and Operator Equations 161
IV
V. Kovacevic
Some Extensions of Linearly Constrained Nonlinear Programming 171
w.
Krabs
Boundary Control of the Higher-Dimensional Wave Equation 183
C. Lemarechal
Nondifferentiable Optimisation. Subgradient and E - Subgradient 191
t4ethods
K. t,iarti
Approximations to Stochastic Optimization Problems 201
B. Meister
Models in Resource Allocation - Trees of Queues 215
C.P. Ortlieb
Dua 1 l'lethods in Convex Control Problems 225
S.I.,. Robinson
A Subgradient Algorithm for Solving K-convex Inequalities 237
E. Sachs
Computation of Bang-Bang-Controls and Lower Bounds for a
Parabolic Boundary-Value Control Problem 247
K. Schittkowski
Numerical Solution of Systems of Nonlinear Inequalities 259
K. Schumacher
Lower Bounds and Inclusion Balls for the Solution of Locally
Uniformly Convex Optimization Problems 273
K. Spremann
Ober Vektormaximierung und Analyse der Gewichtung von Subzielen 283
W. Stadler
Preference Optimality 297
J. Stahl
Decomposition Procedures for Convex Programs 307
Reduction and Dekomposition
of Modulo Optimization Problems
A. Bachem
Institut fUr Operations Research
Universitat Bonn, Germany
1. Introduction
Let G1, G2 be subgroups of the additive group (Zm,+), let S C Zm,
~, f be maps from S into G2, ~ resp. and let ~: G1 + G2 be a homo
morphism. We consider the problem
min f(x)
(1. 1) s.t. ~(x) e: kernel ~
x e: S
where kernel ~ denotes the annulator of ~. Since "~(x) e: kernel ~"
is usually a congruence relation, we look at (1.1) as a generalized
"modulo problem" and call it a modulo optimization problem.
Semigroup optimization (1), problems in coding theory (3) and other
problems (4) may be reformulated as modulo optimization problems,
though there will be no advantage from an algorithmic point of view.
Here we are going to investigate a special homomorphism
~(x): = x - A [A+ xJ , where A+ denotes the Moore-Penrose inverse of
an (m,n) integer matrix A and" [x]" denotes the integer part of x.
The homomorphism ~ generalizes in some sense a congruence rela
tion as: "~(x): = x(=mod A)".
If we denote by G(A) the group ~(Zm) and by G(~/A) =
I
{~(x) x =w (A),Ae: Zn} the subgroup generated by W we are
interesting in the following three questions:
1) Characterize isomorphic groups G(A).
2) Find a reduced representation of G(w/A) in case G(w/A) is not
isomorphic to G(A)
3) Decompose (1.1) in subproblems G(wi/A), and link the optima of
the subproblems to an optimum of problem (1.1)
2
2. Reduction of Modulo Optimization Problems
If A is an (m,n) integer matrix and rank of A equals m, we denote by
H(A) the uniquely determined matrix B, such that for an unimodular
matrix K, AK = (B,O) is in Hermite normal form.
Definition (2.1)
Let A be an (m,n) integer matrix, rank A E {m,n}
{
if rank A = n
x - H(A) H(A)"x otherwise
where A+ denotes the Moore-Penrose inverse of A; and "LX]" the
integer part of x.
Proposition (2.2)
Let G be an additiv subgroup of zm. The map ~A: G + ~A(G) is a
homomorphism onto (~A(G).Eb). where XEby . - ~A(x+y),
= =
kerneZ ~A {XEG/X AA. AEZn}
Proof:
Let's first take the case "rank A = n". Since A+A=(A'A)+A'A =
In and A'A regular, ~A is clearly a homomorphism. Let x be an element
of kernel ~A' that is x = A~+xJ. Since A'A is regular,
A+x = (A'A)-lA,x =1J.,.+x] = Aand we have x = AA.
Let now x be an element of the set {xEG/x = AA, AEZn}.
The rank of A equals n, so we have A+ = (A'A)-lA, which means that
the system A+AA= Ais consistent for all integers A and x = AA is a
J
solution of A+ z = A. Since A is integer , it follows A+ x = [A+ or
[!.
A(A + x - +x J) = 0 and because of AA + x = AA =x we get the result
~A(x) = x - A[!,.\J= 0, which completes the proof. The case "rank A=m"
is almost the same as the first case and we don't want to treat it
here.
Definition (2.3)
Let A be an (m,n) integer matrix such that the row or column rank
is maximal. Because of proposition (2.2), (~A(Zm),Eb) is an additive
group and we will denote this group by G(A). We call the product of
the invariant factors of B the
k
determinant of B (det B). If d = II + p.E. is a representation
j=l - J ]
of d = det B as a product of prime factors ,and p a function from ID 2
into ID defined recursively as
3
p(n,m) 1 < n < m
= {
p(n,m):
p(n,m-l) + p(n-m,m) n > m > 1
= =
p (0 ,m): 1, P (n ,0) : 0 n, me: IN
we define
k
=
sup TI p(e:. ,m)
J
me:JN j =1
Theorem (2.4)
The number of nonisomorphic groups G(A). where A is an (m.n) integer
matrix with maximal. row rank. equal.s K(d). (K(d) is finite. d=det A).
For a proof of the theorem we refer to (2). Additionally in (2) a
list of some of the numbers K(d) up to determinants 100 000 is given.
=
We have restricted ourselves to affine maps W, so let W(x) Nx - b,
where N is an (m,r) integer matrix and be:Zm• We denote the under
lying group of problem (1.1) by
G(N/B) : = {~B(x) I x = NA, Ae:Zn}
To get a canonical representation of G(N/B) we need
Definition (2.5)
Let pe:JN~ and rank B = n. We call oe:JNm the reduction of B under p
if G(diag(p)-lB) is isomorphic to G(diag (0».
=
In this case we denote 0: Red (B/p)
Proposition (2.6)
For every (m.n)-integer matrix B with maximal. col.umn rank and every
p e:m m there exists a 0 e:m m. such that
o = Red (Blp)
For a proof we refer to (2).
Theorem (2.7) (Reduction Procedure)
Let N be a (m.r) integer matrix and WPNQ = (diag (Pl •••.• P • 0 )
" m1 n-m1
the Smith mormal. form of N up to zero rows. Let B be an (m. n) integer
matrix. rank Be:{m.n}. If 0 is the reduction of H(WPB) under p we have
G(NIB) : G(diag(o))
Proof:
Let us denote by {N}: ={xe:Zm/x=NA, Ae:Zn}.
Since ~B is a homomorphism (proposition 2.2), G(N/B) is isomorphic
4
to the factor group{N}/K ~.
ern 't'B
The map WP : {N} + Kern cjld' () is a surjective homomorphism, which
lag p
yields an isomorphism between the factor groups {N}/Kern cjlB
and Kern cjldia ( )/ Let A : = H(WPB) and
g P Kern cjlWPB
F:=diag (p)-l A be matrices, then we have shown above the
isomorphism
m
G(N/B) - Z 1
/Kern F,
and defining 0'- Red (H(WPB)/P)
we get the desired result applying proposition (2.6).
CoroHary (2.8)
Let us denote by gl •... ~gk the group elements of G(N/B) whiah are
generated by the matrix N. The aorresponding group elements in the
isomorphia representation G(diag(o)) have the form
=
cjldiag(o) (Agi ) i 1 ••.. ,k
=
where A: U diag (p)-l WP and U denotes the left hand side unimodular
matrix, whoiah transforms E in Smith normal form.
Example (2.9)
Consider the problem
(~ (~)
) mod
s.t.
Theorem (2.7) yields
o (~), U diag (p)-l P
and we get the equivalent problem
5
min 5x1 + 6x2 + xa
s.t. 4x1 + x2 + aXa - 6 mod 7
xl' x2 • xa -> 0 integer
It is well known that the condition "(B.N) contains an (m.m) uni
modular submatrix" is sufficient but not necessary for G(N/B) and
G(B) to be isomorphic.
a
0 5 0
For instance the matrices N = (0 6) and B = (010) yields G(N/B)=G(B).
but (B.N) contains no unimodular submatrix.
If G(N/B) is isomorphic to G(B). we obtain that the ith invariant
factor of B and the matrix elements N . •.•. • N. are relatively prime
11 1n
for all i=l ••.•• m as a necessary condition.
a. Partitioning of Modulo Optimization Problems
Because of theorem (2.7) we now need only consider problems where
the matrix B is reduced in such a way that G(N/B) is isomorphic to
G(B). Let us denote the set of feasible solutions of problem (1.1) by
LG(N.b/B) : = {XES I Nx - b E Kern ~B}
Proposition (3.1)
Let io' jo E{l •..•• n} - . L such that , N:1. 6;"E"
denotes the j~h coZumn of the matrix N.
x is an optimaZ soZution to
min f (x)
(pl)
if and onZy if L. (x) is an optimaZ soZution to
:10
min I:
iEL
(P2)
where
AElN + and N. = 4> B (AN. ), and
Jo 10
i:
= and
Ii
otherwise