a bit on the linear complementarity problem and a bit about me (since this is epps) yoni nazarathy...
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A bit on the Linear Complementarity Problemand a bit about me (since this is EPPS)
Yoni Nazarathy
EPPSEURANDOM
November 4, 2010
* Supported by NWO-VIDI Grant 639.072.072 of Erjen Lefeber
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Overview• Yoni Nazarathy (EPPS #2):
– Brief past, brief look at future…
• The Linear Complementarity Problem (LCP)– Definition– Basic Properties– Linear and Quadratic Programming– Min-Linear Equations– My Application: Queueing Networks
Just to be clear: Almost nothing in this presentation (except for pictures of my kids), is original work, it is rather a “reading seminar”
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Some Things From the Past
High School in USA
Primary School in Israel (Haifa)
Israeli Army
Israeli Army Reserve
Married
Divorced
Married AgainEmily Born
Kayley Born
Undergraduate Statistics/Economics
Masters in Applied Probability
Software Engineer in High-Tech
Industry
Ph.D with Gideon Weiss
Cycle Racing
Born 1974
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Netherlands (Feb 2009 – Nov 2010)
Collaborations: Matthieu, Yoav, Erjen, Johan, Ivo, Gideon, Stijn, Dieter, Michel, Bert, Ahmad, Koos, Harm, Oded, Ward, Rob,
Gerard, Florin…
Yarden Born!!!
Nederlands: Ik dank dat het is heel gezelich om te pratten…
Raising young kids in Eindhoven:
HIGHLY RECOMMENEDED!!!
EURANDOM / Mechanical Engineering / CWI Amsterdam
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Pedaling to see the Low Lands
``
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Future in Oz…
Melbourne
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Melbourne…
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Maybe live here
Work here: Swinburne University
Also collaborate here: Melbourne University
Maybe also collaborate here: Monash University
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Swinburne University of Technology
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Looking for Ph.D Students…
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What is driving my travels??Maybe fears of some things that can kill…
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In the Middle East…
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In the Netherlands
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A slow death…
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Australia must be a safe place….
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Or is it?
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In Summary…I hope to stay lucky, also in Oz…
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Finally…The Linear Complementarity Problem
(LCP)
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Definition,
( , ) :Find , such that,
,
0, 0,
' 0.
n n n
n
q M
LCP q M z w
w Mz q
w z
w z
The last (complemenatrity) condition reads:
0 0 and 0 0.i i i iw z z w
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It’s all about Choosing a Subset…For {1,..., } denote by ( ) a matrix with
collumns taken from (identity matrix)
and collumns {1,..., } \ taken from .
n B
I
n M
is about finding and 0
such that
( )
In this case:
LCP x
B x q
0, .
0i
i ii
ix iw z
x ii
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Illustration: n=2
1 0 11 20 1 2
1 12 11 20 22 2
011 11 2121 2
11 12 11 2
21 22 2
{1,2}:
{1}:
{2}:
:
qw w
q
m qw z
m q
m qz w
m q
m m qz z
m m q
1 11 12 1 1
2 21 22 2 2
1 0
0 1
w m m z q
w m m z q
{1,2}C
Complementary cones:
1
0
0
1
12
22
m
m
11
21
m
m
1
2
q
q
{1}C
{2}C
{ : ( ) , 0}C y y B u u
C
Immediate naïve algorithm with complexity 3 32 2n nn or n
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Existence and UniquenessDefinition: A matrix, is a P-matrix if the
determinants of all (2 1) principal submatrices are positive.
n n
n
M
Theorem (1958): ( , ) has a unique solution
for all if and only if is a P-matrix.n
LCP q M
q M
11 22 11 22 12 21e.g.for 2 : 0, 0, 0n m m m m m m
P-matrix means that the complementary cones "parition" n
P-Matrixes
Symmetric Matrixes PD Matrixes
Relation of P-matrixes to positive definite (PD) matrixes:
Reminder(PD) :
' 0 0x Mx x
Reminder(PSD) :
' 0x Mx x
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Computation (Algorithms)• Naive algorithm, runs on all subsets alpha• Generally, LCP is NP complete• Lemeke’s Algorithm, a bit like simplex• If M is PSD: polynomial time algs exists• PD LCP equivalent to QP• Special cases of M, linear number of iterations• For non-PD sub-class we (Stijn & Eren) have an
algorithm. Where does it fit in LCP theory?We still don’t know…
• Note: Checking for P-Matrix is NP complete, checking for PD is quick
2n
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LCP References And Resources• Linear Complementarity, Linear and Nonlinear
Programming, Katta G. Murty, 1988. Internet edition.• The Linear Complementarity Problem, Second Edition, Richard
W. Cottle, Jong-Shi Pang, Richard E. Stone. 1991, 2009.• Richard W. Cottle, George B. Dantzig, Complementary Pivot
Theory of Mathematical Programming, Linear Algebra and its Applications 1, 103-125, 1968.
• Related (to queueing networks): Unpublished paper (~1989), Avi Mandelbaum, The Dynamic Complementarity Problem.
• Open problems in LCP…. I am now not an expert (but a user) .... So I don’t know…
• Gideon Weiss, working on relations to SCLP
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Some Applications(and Sources) of LCP
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Linear Programming (LP)
min '
. .
0
c x
s t Ax b
x
max '
. . '
0
b y
s t A y c
y
Primal-LP: Dual-LP:
Theorem: Complementary slackness conditions
min '
. .
, 0
c x
s t Ax b v
x v
max '
. . '
, 0
b y
s t u c A y
y u
Assume , , , are feasible for primaland dual:
0, 0 Theyareoptimalsolutionsi i i i
x v y u
x u y v
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0 ',
0
c ALCP
b A
0 '
0
u A x c
v A y b
, , , 0u v x y
' 0u x ' 0v y
The LCP of LPFind:
Such that:
And (complementary slackness):
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Lekker!
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Quadratic Programming1
min ( ) ' '2
. .
0
Q x c x x D x
s t Ax b
x
Lemma: An optimizer, , of the QP also optimizesmin ( ) '
. .
0
c Dx x
s t Ax b
x
Proof:( )x x x x
( ) ( ) 0Q x Q x ( ' ) '( ) ( ) ' ( )
2c Dx x x x x D x x
x
QP-LP:
QP-LP gives a necessary condition for optimality of QP in terms of an checking optimality of an LP
QP:
0 1, Let be feasible.x
( ' ) '( ) 0c Dx x x
( ' ) ' ( ' ) 'c Dx x c Dx x
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The Resulting LCP of QP
',
0
c D ALCP
b A
Allows to find “suspect” points that satisfy the necessary conditions: QP-LP
Theorem: Solutions of this LCP are KKT (Karush-Kuhn-Tucker) points for the QP
Corollary: If D is PSD then x solving the LCP optimizes QP.
Proof: Write down KKT conditions and check.
Note: When D is PSD then M is PSD. In this case it can be shown that the LCP is equivalent to a QP (solved in polynomial time). Similarly, every PSD LCP can be formulated as a PSD QP.
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Our Application: Min-Linear Equations( )B
0
0
( ) '( ) 0
B
,w z ( ) ( )
0, 0
' 0
u I B v I B
z w
w z
( ( ) , )LCP I B I B
Find :
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Open Jackson NetworksJackson 1957, Goodman & Massey 1984, Chen & Mandelbaum 1991
, ,M M M MP
1
'
( ') , ( ')
M
i i j j j ij
p
P
LCP I P I P
i
i
Traffic Equations:
i jp
1
M
1
1M
i jij
p p
Problem Data:
Assume: open, no “dead” nodes
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Modification: Finite Buffers and Overflows Wolff, 1988, Chapter 7 & references there in & after
ii
Exact Traffic Equations for Stochastic System:
i jp
M
1
1M
i jij
p p
Problem Data:
, , , ,M M M M M M MP K Q
Explicit Stochastic Stationary Solutions:
Generally NoiK
MK1
1M
i jij
q q
i jq
11K
Generally No
Assume: open, no “dead” nodes, no “jam” (open overflows)
Traffic Equations for Fluid System
Yes
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Traffic Equations
1 1
M M
i i j j ji j j jij j
p q
out rate
overflow rate ( ) ( )
1 1( ') ( ( ') ) , ( ') ( ')LCP I Q I P I Q I P
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Wrapping Up
• LCP: Appears in several places (we didn’t show game-theory)
• Would like to fully understand the relation of our limiting traffic equations and LCP
• In progress paper with Stijn Fleuren and Erjen Lefeber, “Single Class Fluid Networks with Overflows” makes use of LCP theory (existence and uniqueness)
• I will miss EURANDOM and the Netherlands very much!• Visit me in Melbourne!!!
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The End