Optimization and Operations Research: Proceedings of a Workshop Held at the University of Bonn, October 2–8, 1977

264 Pages·1978·8.642 MB·English

by  Achim Bachem (auth.), Rudolf Henn, Bernhard Korte, Werner Oettli (eds.)

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continuation on page 273
Lectu re Notes
in Economics and
Mathematical Systems
Managing Editors: M. Beckmann and H. P. KUnzi
157
Optimization
and Operations Research
Proceedings of a Workshop Held at the
University of Bonn, October 2-8, 1977
Edited by
R. Henn, B. Korte, and W. Oettli
Springer-Verlag
Berlin Heidelberg New York 1978
Editorial Board
H. Albach A. V. Balakrishnan M. Beckmann (Managing Editor)
P. Dhrymes J. Green W. Hildenbrand W. Krelle
H. P. KOnzi (Managing Editor) K. Ritter R. Sato H. Schelbert
P. Schonfeld
Managing Editors
Prof. Dr. M. Beckmann Prof. Dr. H. P. Kunzi
Brown University UniversitiH Zurich
Providence, RI 02912/USA 8090 Zurich/Schweiz
Editors
Rudolf Henn Werner Oettli
Institut fOr Statistik und Fakultat fOr Mathematik
Mathematische Wirtschaftstheorie und Informatik
UniversiUit Karlsruhe Universitat Mannheim
Kollegium am SchloB SchloB
7500 Karlsruhe 1 6800 Mannheim
Bernhard Korte
Institut fOr Okonometrie
und Operations Research
Universitat Bonn
NassestraBe 2
5300 Bonn 1
AMS Subject Classifications (1970): 49-XX, 65 K 05,90 B XX, 90 C XX
ISBN-13: 978-3-540-08842-4 e-ISBN-13: 978-3-642-95322-4
DOl: 10.1007/978-3-642-95322-4
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© by Springer-Verlag Berlin Heidelberg 1978
2142/3140-543210
PREFACE
This volume constitutes the proceedings of the workshop "Optimierung
und Operations Research", held at the Elly Holterhoff Backing Stift
(Bad Honnef) of the University of Bonn, October 2-8, 1977.
This conference was devoted to recent advances in the field of mathe
matical programming, optimization techniques, and operations research.
It was attended by about 50 invited participants. Furthermore many
scholars in these areas showed a great interest in this workshop,
despite several other conferences and activities on similar topics
in the year 1977. The organizers regret that considerations of avail
able space for conference activities limited the number of participants.
This widespread interest, the high quality of the lectures presented
and the active and stimulating discussions at the conference mani
fested the breadth of the activity ongoing in the field covered by
this workshop and the necessity that this field be cultivated to a
greater extent by the scientific community.
The workshop was organized by the Institute of Operations Research
(Sonderforschungsbereich 21), University of Bonn and was generously
sponsored by the Gesellschaft der Freunde und Forderer der Rheinischen
Friedrich-Wilhelms-Universitat and by IBM Germany. Only through this
invaluable support was this workshop possible; for this the editors
wish to express their sincere thanks and appreciation.
Bonn, December 1977
R. Henn B. Korte W. Oettli
III
TABLE OF CONTENTS
A. Bachem
The theorem of Minkowski for polyhedral monoids and
aggregated linear diophantine systems • • • . • • • • • • •
S. Baum and L.E. Trotter, Jr.
Integer rounding and polyhedral decomposition for
totally unimodular systems ••..•..•..••••.. 15
D. Bierlein
25
Measure extensions according to a given function
B. Brosowski
37
On parametric linear optimization
P. Brucker
45
On the complexity of clustering problems
R.E. Burkard and U. Zimmermann
The solution of algebraic assignment and transportation
problems • • • • • • • • • • • • •• •• • • • • • • 55
L. Collatz
Application of optimization methods to the solution of
67
operator equations ••••••••• .•••• •
B. Dejon
A note on directional differentiability of flow network
equilibria with respect to a parameter ••••••• 73
U. Derigs
On solving symmetric assignment and perfect matching
79
problems with algebraic objectives • • • • • • • • • • • • •
R. Euler
Generating maximum members of independence systems 87
~·r. Gaul
95
Remarks on stochastic aspects on graphs
M. Grotschel and M.W. Padberg
On the symmetric travelling salesman problem: Theory and
computation • • • • • • • • • • • • • • • • • • •• •• 105
J. Gwinner
On the existence and approximation of solutions of
pseudomonotone variational inequalities • • • • • • • • • • 117
J. Hartung
Minimum norm solutions of convex programs • • • • • • • • • 127
D. Hausmann and B. Korte
Oracle-algorithms for fixed-point problems - an
axiomatic approach • • • •• • • • • • • • • • • • 137
v
K. Hinderer and D. Kadelka
The general solution of a classical stochastic
inventory problem and its generalization 157
R. Kannan and C.L. Monma
On the computational complexity of integer
programming problems • • • • • • • • • • • • • • • • • •• 161
P. Kosmol
On stability of convex operators ••••••••- . • • •• 173
F. Lempio and H. Maurer
Differentiable perturbations of infinite
optimization problems • • • • • • • • • • • • • •• 181
L. Matthews
Matroids on the edge sets of directed graphs • • • • • •• 193
Ph. Michel
A disc~ete time maximum-principle in a stochastic
case • 201
• • • • • e. • • • • • • •
E.J. Muth and K. Spremann
A class of stationary EOQ problems and learning effects •• 209
E. Sachs
Optimal control for a class of integral equations •••• 223
M. Sch1il
On the M/G/l-queue with controlled service rate 233
C.P. Schnorr
Multiterminal network flow and connectivity in
......
unsymmetrical networks " ••••. • • • •
241
M. Vlach
Augmented penalty function technique for optimal
control problems • • • • • • • • •• •• • • • • • • • 255
J. Zowe and H. Maurer
Optimality conditions for the programming problem
in infinite dimensions • • • • • • • • • • • • • • • • •• 261
VI
THE THEOREM OF MINKOWSKI FOR POLYHEDRAL MONO IDS
AND AGGREGATED LINEAR DIOPHANTINE SYSTEMS
Achim Bachem
Institut fur Okonometrie
und Operations Research
Universitat Bonn
Nassestr. 2
D-5300 Bonn
Abstract
We study polyhedral mOnO ids of the form M = {XEZn / Ax ~ a} for
(m,n) integer matrices with rank m and prove in an elementary and
constructive way that M has a finite basis, i.e. every XEM is the
nonnegative integer linear combination of a finite set of vectors. We show
that this theorem holds also for monoids M(N,B)={XEZ! / Nx + By=o, YEZn}.
We consider the aggregated system GNxtGBy=o where G is an (r,m)
aggregation matrix and show how the cardinality of a span of M(GN,GB) and M(N,B)
relate to each other. Moreover we show how the group order of the Gomory
group derived from M(N,B) changes if we aggregate Nx+By=o to
GNx+GBy=o.
1. A Basistheorem For Polyhedral Monoids
Definition oEMcZn is a monoid iff x,YEM in~lies x+YEM. The monoid
is called polyhedral if there is an (m,n) integer matrix A such that
M={XEZn / Ax ~ o}. By (x)={nx/nEZ+} (XEZn and Z+=Nu{o}) we denote an
integer ray. (Note that two integer rays (x) and (y) are equal iff x=y).
An integer ray (x) c: M is an integer extreme ray of the monoid M iff for
all (y), (z)cM (y)*(z) implies (x)*(y+z). If there are integer rays
(xl), •.• ,(xk)cM such that M=(xl)+ ••• +(xk), {x\ .•• ,xk} is called a
span for M. If in addition xi(i=l, ... ,k) are pairwise distinct and
integer extreme rays, {xl, •.. ,xk} is called a basis of M.
The well known theorems of Minkowski (1896) and Weyl (1935) state that
Kc~n is a polyhedral cone iff K equals the sum of its finite extreme
rays. Do these theorems have an integer analogon, i. e. is M c Zn a
polyhedral monoid iff M equals the sum of its finite integer extreme
rays? David Hilbert (1890) was probably the first who gave a partial answer
(namely the "only if" part) that is he proved that every polyhedral monoid
has a finite basis. More direct and elegant proofs can be found in
Jeroslow (1975) and implicitly in Graver (1975). A drawback of these proofs
is that they are either not constructive (Hilbert, Graver) or in using the
theorems of Minkowski, Weyl and Caratheodory they show no way to construct
- 2 -
a finite basis in an efficient way. Here we give (for the case (rank(A}=m})
an elementary and constructive proof without using Minkowski's or
Caratheodory's theorems and show how to construct (in a very easy way) an
integer span of a polyhedral monoid M. The proof is based on a result of
Fiorot (1972). We show in theorem 1.2 how Fiorot's approach can be
generalized in a way to minimize the cardinality of a span of a polyhedral
monoid. Moreover we give a bound on the cardinality of a basis.
Concerning the 'if part' i.e. the integer version of the Weyl theorem it
is very easy (as Jeroslow (1975) remarked) to give a counterexample.
Consider for instance the monoid
with Basis (021) which is obviously not polyhedral.
201
Definition If x,tEZn and T=diag(t}, x(=modT} denotes a vector whose
-1
components are taken modulo t (i.e. x(=modT}=x-Tl1 ~, where y
denotes the largest integer vector not greater than y. If A is an (m,n)
integer matrix
denotes the Smith normal form of A. Here UA and KA are unimodular
matrices which transform A into Smith normalform. -S-+-(A-)- = diag(t1, .•. ,tr }
denotes the nonsingular part of the Smith normalform of A (r=rank(A}) and
inv(A}=det(S+(A}} is the product of the invariant factors of A called the
invariant of A.
Theorem 1.1
Let A be an (m,n) integer matrix with rank m. Then the polyhedral
monoid
has a finite basis with at most
2(n-m+1} inv(A}m-1(d 1}-m
m-
2
- 3 -
elements. Here dm-1 denotes the greatest common divisor of all (m-1)
minors. Moreover there are matrices S,T and vectors v such that
n
M = {g(v)+Sw / V,WEZ+ v ~ h}
where
Proof:
We divide the proof into two parts:
Case 1) m=n
Let f(x)=-Ax and we get immediately:
n -1 n
f(M) {f(x) / XEZ Ax ~ o}= {y/-A YEZ , Y > o}
= n n
{ye:lR+ / y=.o mod A}={y€Z+ / UAy=o mod SeA)}
n -1
{YEZ+ / Z:O mod S(A),y=UA z}.
Let S(A) = diag (t1, ... ,tn) and T=diag(t1(n), .•• ,t (n» where
-1 n
ti ~ ti(n) ~ tn and ti/ti(n) such that UA T=TK holds with an
unimodular matrix K. Obviously such a matrix always exists setting
-1 -1
ti(n) = tn(i=l, ... ,n) and K=T ·UA ·T. Theorem 1.2 will show how to
choose ti(n) in order to get a span with minimal cardinality. Now it is
easy to get a representation of all solutions of the congruences
t:1 t.(n)z.:o mod t.(n) i=l, ••• ,n
~ ~ ~ ~
which are equivalent to
z.:o mod t. i=l, ..• ,n
~ ~
namely
z = S(A)v + Tw
o < v. < t:1. t.(n)
~ ~ ~
Let us denote by h=(h1, .•. ,hn)'
Re s ubst~·tu·t~n g y= UA-1 z we get
3
- 4 -
f(M) = {UA-1 S (A)v + Tw / 0 2 v 2 h, W€z n }
= {UA-1 S(A)v(=modT)+Tw / 0 2 v 2 h, W€Zn+ }
because o 2 U~1S(A)v(=modT)«t1(n), ••• ,tn(n»"
i -1
Setting x =UA S(A)v(=modT) for each 0 < v < h (V€Zn) in any order
i=1, ... ,m (m=h1 ... ·hn) and xm+i=(o, ... ,o,ti (n) ,0, .•• ,0)' i=1, ... ,n
we get a span for f(M) by f(M) = (x1)+ '" +(xm+n) and xi > 0 for all i.
Because
is a bounded set, it es easy (polynomially easy in a fraction of some
coordinates of some xj ) to check KiiJ. ¢. If Ki=¢ put x i into the basis
B. Let us prove that B is a basis for f(M). Clearly f(M) =.E (xi).
x~EB
Assume now (x i )=(y+z) for some y,z€f(M) (y+z). Because y= .E A .xj
xJEB J
and z= .E Vj xj we get x i = .E (A.+ v.)xj . But since xho (for
xJEB x? €B J J
all j) this implies A. + V· = 0 and therefore K. + ¢ in contradiction
~ .~ ~
to our construction. So x ~ is an integer extreme ray and B a basis.
It is easy to see that if B is a basis for f(M),f-1(B) is a basis for
M (because f is a linear bijection). Because ti(n) 2 tn we have
h. < t~l t and get h1' ..• ·h < det(A)-1(t )n. Therefore
~ - ~ n n-1 n - n
hi" .·hn 2 det(A) (dn_1)-n = (tn)n (det(A»-1
= inv(A)m-1 (d 1)-m
m-
< 2 inv(A)m-1(d 1)-m
m-
Case 2) m < n
Let K be a unimodular matrix which transforms A into Hermite normalform
H(A) = (H I 0 )
I
where H is an (m,m) integer matrix of full rank.
M = K-1({XEZn / Ax 2 o})
n-m
x Z •
4