Mathematical Programming at Oberwolfach II

257 Pages·1984·3.542 MB·English

by  M. L. Balinski, Andrew Russakoff (auth.), Bernhard Korte, Klaus Ritter (eds.)

MATHEMATICAL PROGRAMMING STUDIES
Editor-in-Chief
R.W. COTTLE, Department of Operations Research, Stanford University, Stanford, CA 94305, U.S.A.
Co-Editors
L.C.W. DIXON, Numerical Optimisation Centre, The Hatfield Polytechnic, College Lane, Hatfield,
Hertfordshire ALl0 9AB, England
B. KORTE, Institut f~ir Okonometrie und Operations Research, Universit/it Bonn, Nassestrasse 2,
D-5300 Bonn I, W. Germany
M.J. TODD, School of Operations Research and Industrial Engineering, Upson Hall, Cornell Universi-
ty, Ithaca, NY 14853, U.S.A.
Associate Editors
E.L. ALLGOWER, Colorado State University, Fort Collins, CO, U.S.A.
W.H. CUNNINGHAM, Carleton University, Ottawa, Ontario, Canada
J.E. DENNIS, Jr., Rice University, Houston, TX, U.S.A.
B.C. EAVES, Stanford University, CA, U.S.A.
R. FLETCHER, University of Dundee, Dundee, Scotland
D. GOLDFARB, Columbia University, New York, USA
J.-B. HIRIART-URRUTY, Universit6 Paul Sabatier, Toulouse, France
M. IRI, University of Tokyo, Tokyo, Japan
R.G. JEROSLOW, Georgia Institute of Technology, Atlanta, GA, U.S.A.
D.S. JOHNSON, Bell Telephone Laboratories, Murray Hill, N J, U.S.A.
C. LEMARECHAL, INRIA-Laboria, Le Chesnay, France
L. LOVASZ, University of Szeged, Szeged, Hungary
L. MCLINDEN, University of Illinois, Urbana, IL, U.S.A.
M.J.D. POWELL, University of Cambridge, Cambridge, England
W.R. PULLEYBLANK, University of Waterloo, Waterloo, Ontario, Canada
A.H.G. RINNOOY KAN, Erasmus University, Rotterdam, The Netherlands
K. RITTER, Technische Universit/it Mianchen, Mfinchen, W. Germany
R.W.H. SARGENT, Imperial College, London, England
D.F. SHANNO, University of California, Davis, CA, U.S.A.
L.E. TROTTER, Jr., Cornell University, Ithaca, NY, U.S.A.
H. T UY, Institute of Mathematics, Hanoi, Socialist Republic of Vietnam
R.J.B. WETS, University of Kentucky, Lexington, KY, U.S.A.
Senior Editors
E.M.L. BEALE, Scicon Computer Services Ltd., Milton Keynes, England
G.B. DANTZIG, Stanford University, Stanford, CA, U.S.A.
L.V. KANTOROVICH, Academy of Sciences, Moscow, U.S.S.R.
T.C. KOOPMANS, Yale University, New Haven, CT, U.S.A.
A.W. TUCKER, Princeton University, Princeton, N J, U.S.A.
P. WOLFE, IBM Research Center, Yorktown Heights, NY, U.S.A.
MATHEMATICAL
P R O G R A M M I N G
STUDY22
A PUBLICATION FO
THE MATHEMATICAL PROGRAMMING SOCIETY
Mathematical Programming
at Oberwolfach II
Edited by B. KORTE and K. RITTER
rebmeceD 4891
NORTH-HOLLAND - AMSTERDAM
© The Mathematical Programming Society, Inc. - 1984
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This STUDY is also available to non-subscribers in a book edition.
Printed in The Netherlands
PREFACE
From January 9 to ,51 1983 the third conference on "Mathematische Optimie-
rung" was held at the Mathematisches Forschungsinstitut Oberwolfach. The first
Oberwolfach meeting devoted to the subject of mathematical programming took
place in 1979. Part of its results has been documented in the Mathematical Program-
ming Study No. 14 entitled "Mathematical Programming at Oberwolfach".
The second Oberwolfach conference in 1981 had no proceedings volume. We feel
that mandatory proceedings announced in advance do not contribute to the spon-
taneuos workshop atmosphere of an Oberwolfach week.
After the third Oberwolfach conference we believed that many new results and
ideas presented there would justify a collection of them which eventually led to this
Mathematical Programming Study entitled "Mathematical Programming at Ober-
wolfach I1".
This Oberwolfach meeting was attended by 63 scholars from 31 different coun-
tries. Not too surprisingly, almost all invitations were accepted. Thus, the total size
was somewhat beyond the scope of the Oberwolfach Institute. However, 25 par-
ticipants came from overseas and especially from North America. They made sub-
stantial travel expenditures from their own resources in order to spend one week in
the stimulating environment of Oberwolfach. This indicates strongly that the
mathematical programming community has developed a very positive attitude
toward Oberwolfach and considers this meeting to be one of the highlights in the
field.
The 81 papers of this volume reflect the different areas of mathematical program-
ming as well as the interaction of its two main streams, namely continuous and non-
linear optimization on one side and discrete and combinatorial optimization on the
other. It contains papers about special algorithms for nonlinear programming,
polyhedral theory, submodularity, min-max and duality relations, complexity,
quasi-Newton and approximation methods, network algorithms and other numeri-
cal aspects of optimization methods.
We appreciate very much the help of numerous referees. An acknowledgement of
their work will appear with the next referee listing of Mathematical Programming.
Finally, we are very much indebted to the staff of the Oberwolfach Institute whose
excellent support made this conference a success. Our most sincere thanks go to the
director of the Mathematisches Forschungsinstitut, Professor Dr. M. Barner, for
hosting the meeting at this unique conference center.
Bernhard Korte
Klaus Ritter
CONTENTS
Preface v
M.L. Balinski and A. Russakoff, Faces of dual transportation polyhedra 1
F. Barahona and W.H. Cunningham, A submodular network simplex
method 9
I. Barany, T. Van Roy and L.A. Wolsey, Uncapacitated lot-sizing: The
convex hull of solutions 23
J.-M. Bourjolly, P.L. Hammer and .B Simeone, Node-weighted graphs
having the K6nig-Egervfiry property 44
W. Cook, L. Lov~isz and A. Schrijver, A polynomial-time test for total
dual integrality in fixed dimension 64
J.E. Dennis and H.F. Walker, Inaccuracy in quasi-Newton methods: Local
improvement theorems 70
J. Fonlupt and M. Raco, Orientation of matrices 86
A. Frank, A. Seb6 and E. Tardos, Covering directed and odd cuts 99
S. Fujishige, Submodular systems and related topics 311
B. Gollan, Inner estimates for the generalized gradient of the optimal value
function in nonlinear programming 231
M.D. Grigoriadis and T. Hsu, Numerical methods for basic solutions of
generalized flow networks 741
R.G. Jeroslow and J.K. Lowe, Modelling with integer variables 761
P.V. Kamesam and R.R. Meyer, Multipoint methods for separable
nonlinear networks 581
O.L. Mangasarian, Normal solutions of linear programs 206
S.M. Robinson, Local structure of feasible sets in nonlinear programming,
part I II: Nondegeneracy 217
J.B. Rosen, Performance of approximate algorithms for global
minimization 132
K. Zimmermann, Some optimization problems with extremal operations 237
Mathematical Programming Study 22 (1984) 1-8
North-Holland
FACES OF DUAL TRANSPORTATION POLYHEDRA
M.L. BALINSKI
C.N.R.S., eriotarobaL eirtdmonocE'd ed I'Ecole ,euqinhcetyloP ,siraP ecnarF
Andrew RUSSAKOFF
egelloC of ssenisuB ,noitartsinimdA St John's ,ytisrevinU ,aciamaJ NY, USA
Received 26 November 2891
Revised manuscript received 82 February 1984
The extreme points of any nondegenerate dual transportation polyhedron are characterized by
the m-partitions (or n-partitions) of m + n- I. This is used to show that all such polyhedra have
exactly the same number of r-dimensional faces, 0~- r~ < m +n -2, and to count them.
Key :sdrow Polyhedra, Transportation Problem, }:aces.
Introduction
The transportation polytope ~ is
Pm,,(a,b)={x=(xij); .~ xij=a~, ~) x~j=bj, x~j~O, ic:_M,j~ N}
jcN iEM
where ai > 0 and bj > 0 for all i ~ M = {1,..., m} and j E N = { 1 ..... n}, and ~M ai =
~N bj. We assume throughout that m <~ n. Primal methods for solving the transporta-
tion problem travel on the extreme points and edges of I'm., so it is not surprising
that Pro,, for varying a, b has been extensively studied 1,2, 6, 7, 8. The precise
number of extreme points of P for certain choices of a, b is known. It has been
proven that for fixed m, n relatively prime, a = (n .... , n) and b = (m ..... m) gives
the P,,., having the maximum number of extreme points over all choices of a, b 7.
The number itself has simple expressions only for the cases n = km + 1 when it is
n"-2n!/(k!) ~' and n = km +m- 1 when it is m'-2n!/(k!) " 2. For the general case
n = km +c, it is (n!/(k!)")p(m, c, k) where p(m, c, k) is a polynomial in k (with m
and c fixed) with integer coefficients and highest term mm-2k "-c-~ i. For fixed
m, n the minimum number of extreme points of Pro,, is n "-~ when there is no
degeneracy, and n!/(n-m + I)! if degeneracy is allowed.
The dual transportation polyhedron ~ is
Dr,,.,,(c)={u=(u,), v = (vj), ui + vj~ ,uc is M,j~ ,N ul :=0}.
Arguments a, ,b m, n and c will be dropped unless necessary in the sequel.
2 M.L. Balinski and A. Russakoff/ Dual transportation polyhedra
It is unbounded. Fixing ~u =0 is an arbitrary choice that rules out cylinders of
solutions (u~+8), (v~-8). Dual methods for solving the transportation problem
travel on the extreme points and bounded edges of Din,, and so it is surprising that
relatively little is known about it, although folklore had it that dual methods required
fewer iterations.
The fact is that the number of r-dimensional faces of any nondegenerate D,,n(c)
is always the same, as is the number of bounded (and so also of unbounded) faces.
It will be proved that the maximum number of extreme points of any Din, n is at
most zt m ,. ,~ n--2~ ~ ), a number that is very much smaller than the minimum possible number
for P,,... This last fact we subsequently learned was shown in a much earlier but
somewhat obscure and involved paper of Zhu 9. The principal tool for proving
these results is a one-to-one correspondence between extreme points of nondegener-
ate D~.~ and m-partitions of m +n- .1
The folklore had it that dual methods for the transportation problem were better.
This may have something to do with the sheer number of extreme points of primal
as versus dual polyhedra. The comparisons are dramatic. For example, for m = 8,
n=9, the primal polytope has a minimum of 181 440 extreme points allowing
degeneracy and of 4 782 969 not allowing degeneracy, and a maximum number
of 19285x10J~; in contrast, the dual polyhedron has a minimum of 1 allowing
degneracy, and a maximum of 6435.
The characterization of the extreme points of nondegenerate dual transportation
polyhedra by partitions turns out to be quite useful 4. On the one hand, it may
be used to develop an algorithm for the assignment problem requiring at most
(n - l)(n -2)/2 pivots 3; on the other hand it may be used to show that the diameter
of D,,, is at most (m - l)(n - )1 5.
1. Partitions
An extreme point of Dm.,(c) is a (u, v) that is a unique solution to
ut=0, ui+vj=c 0 for(i,j)ciTcM(cid:141)
ui+vj<~c,j for(i,j)~T
where T corresponds to some m + n - 1 linearly independent equations ~u + jv = c .o
The extreme point is nondegenerate ifu~ t jv < jic for all (i,j)~ T, and D is nondegen-
erate if all of its extreme points are nondegenerate.
A natural model for extreme points and faces of D is a graph. Let C(M, N) be
the complete bi-partite graph having disjoint node sets M and N and all edges
(i,j), i~ M,j~ N. To each (u, v)~ D,.,, associate the subgraph G(u, v) of C(M, N)
defined as containing edge (i,j) if and only if ~u +vj = .j,c
Lemma .1 (u, v) c D is an extreme point of D if and only if G(u, v) contains a spanning
tree .T
M.L. Balinski and A. Russakoff / Dual transportation polyhedra 3
This is well known and easily established. If (u, v) is nondegenerate, G(u, v) is a
spanning tree. A spanning tree of C(M, N) must contain precisely m + n - 1 edges.
If (u, v) is degenerate then G(u, v) must contain a cyucle, say (see Figure 1), (i,,j,),
(i2,jj), (i2,j2) ..... (ik,jk), (i,,jk). Call the edges of type (il, jt) odd, the others even.
Then ~oOd OC = ~.Y ,~U( + Vj,)= ~ .... ,j~C characterizing how degeneracy occurs.
i
1 2
kJ
I 2J 3J
.giF I
A row partition is any a = (a,,..., a,,), a~/> ,1 ~ ~a = m + n - 1, corresponding to
an assignment of valences to nodes of M; and a column partition is any b=
(b,,..., b,), bj/> 1, .Y bj = m + n - I, corresponding to an assignment of valencies to
nodes of N. To each spanning tree T contained in the graph G(u, v) of any extreme
point (u, v) there corresponds a unique row partition which represents the valencies
of the nodes M and a unique column partition which represents the valencies of
the nodes N.
,emma 2. To one row (equivalently, column) partition there corresponds at most one
extreme point of a nondegenerate Dm,~( e).
Proof. Suppose, instead, that the graphs G(u, v) and G(u', v') of two different
extreme points contained different spanning trees T and T' having the same row
valencies (a, .... a,,).
There is some arc (i,j)c T' but ~ T, call it (i,,jO. Since T is a spanning tree there
exists a unique path joining J, and i, in .T Let (i2,j,)e T be the first edge on that
path and (i2,j2)c T'. There must be such an edge since the valence of 2i in T is at
least two and so the valence of 2i in T' is also at least two. Now, take (i3,j2)c T to
be the unique edge on the path joining J2 to i, in T and continue. At some point a
node previously visited must be encountered (see Fig. 2). But, then, calling edges
(i;,j;)c T' for I= s to t of the cycle odd and the others in the cycle which belong
to T even, we have
=~
X j,c (u,,+v~,)< , E j,c
odd s even
and
L-
(u,, + vj,) < ~ c,j.
E c,j -
even s odd
The inequalities are strict by nondegeneracy, giving a contradiction.
M.L. Balinski and A. Russakoff / Dual transportation po(vhedra
i i i i
2 S S+1 t
i t
i i
I I i i
I I
I I
J J J J J
1 2 s s+l t
Fig. .2 Solid edges in ,T dashed edges ni .'T
In fact the lemma is true for all D,,.,(c), degenerate or not 5. To one degenerate
extreme point there corresponds more than one partition, because degeneracy implies
G(u, v) contains more than one spanning tree.
Theorem 1. For nondegenerate D,,.,,( c) there exist one to one correspondences between
row partitions, extreme points and column partitions.
Proof. The proof is by induction on m +n. For m +n = 1,2 or3 the result is obvious.
Suppose it is true for all m +n < k.
Recall that m~< n. Any column partition (b,, b2,..., b,) must have at least one
component equal to 1, say b, = I. By the inductive hypothesis (b 2 .... , b,) determines
a unique extreme point ul~)=(u~ I),...,u,~'~')'), v~l)-(v~ ~,- ...,v~,, ))2 of D ...... ,(c~
where c ~) is c with the first column dropped. Define ui = u~/), VJ---v~// and v, =
min~(c~- u~)= Ok,- .kU By nondegeneracy the k is unique. This is an extreme point
of D,,.,(c) since it is feasible and its graph forms a spanning tree. This shows that
to each column partition b there corresponds an extreme point. These extreme
points are all different, for otherwise, one would have two different partitions,
m+n ~2-
implying degeneracy. There are x ~_,, z column partitions, so the same number of
extreme points. To each extreme point there corresponds a unique row partition,
and since all must be different and there are (,,,~,~-2) of them, all are accounted for,
which completes the proof.
2. Examples
The three examples below show that polyhedra D,,,.,,(c) for different c are not
necessarily combinatorially equivalent (in the sense that there does not exist a
one-to-one correspondence between faces that is inclusion preserving). Since m = n =
3 in each, their dimensions are all 5; however, we draw only the extreme points
and bounded extreme edges. (Since each is nondegenerate, every extreme point
should have 5 adjacent extreme edges: the missing ones are unbounded).