A bit on the linear complementarity problem and a bit about me since this is epps
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A bit on the Linear Complementarity Problem and a bit about me (since this is EPPS). Yoni Nazarathy. EPPS EURANDOM November 4, 2010. * Supported by NWO-VIDI Grant 639.072.072 of Erjen Lefeber. Overview. Yoni Nazarathy ( EPPS #2): Brief past, brief look at future…

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A bit on the Linear Complementarity Problemand a bit about me (since this is EPPS)

Yoni Nazarathy

EPPS

EURANDOM

November 4, 2010

* Supported by NWO-VIDI Grant 639.072.072 of ErjenLefeber


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”


Some Things From the Past

Israeli Army

Masters in Applied Probability

High School in USA

Software Engineer in High-Tech Industry

Born 1974

Primary School in Israel (Haifa)

Ph.D with Gideon Weiss

Married

  • Undergraduate Statistics/Economics

Cycle Racing

Israeli Army Reserve

Divorced

Emily Born

Married Again

Kayley Born


Netherlands (Feb 2009 – Nov 2010)

EURANDOM / Mechanical Engineering / CWI Amsterdam

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 gezelichomtepratten…

Raising young kids in Eindhoven: HIGHLY RECOMMENEDED!!!


Pedaling to see the Low Lands

``


Future in Oz…

Melbourne


Melbourne…


Maybe live here

Also collaborate here: Melbourne University

Work here: Swinburne University

Maybe also collaborate here:Monash University


Swinburne University of Technology


Looking for Ph.D Students…


What is driving my travels??Maybe fears of some things that can kill…


In the Middle East…


In the Netherlands


A slow death…


Australia must be a safe place….


Or is it?


In Summary…I hope to stay lucky, also in Oz…


Finally…The Linear Complementarity Problem (LCP)


Definition


It’s all about Choosing a Subset…


Illustration: n=2

Complementary cones:

Immediate naïve algorithm with complexity


Existence and Uniqueness

Relation of P-matrixes to positive definite (PD) matrixes:

P-Matrixes

Symmetric Matrixes

PD Matrixes


Computation (Algorithms)

  • Naive algorithm, runs on all subsets alpha

  • Generally, LCP is NP complete

  • Lemeke’sAlgorithm, 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


LCP References And Resources

  • Linear Complementarity, Linear and Nonlinear Programming, Katta G. Murty, 1988. Internet edition.

  • The Linear ComplementarityProblem, 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), AviMandelbaum, 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


Some Applications(and Sources) of LCP


Linear Programming (LP)

Primal-LP:

Dual-LP:

Theorem: Complementary slackness conditions


The LCP of LP

Find:

Such that:

And (complementary slackness):


Lekker!


Quadratic Programming

QP:

Lemma: An optimizer, , of the QP also optimizes

QP-LP:

Proof:

QP-LP gives a necessary condition for optimality of QP in terms of an checking optimality of an LP


The Resulting LCP of QP

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

Proof: Write down KKT conditions and check.

Corollary: If D is PSD then x solving the LCP optimizes QP.

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.


Our Application: Min-Linear Equations


Open Jackson NetworksJackson 1957, Goodman & Massey 1984, Chen & Mandelbaum 1991

Problem Data:

Assume: open, no “dead” nodes

Traffic Equations:


Modification: Finite Buffers and Overflows Wolff, 1988, Chapter 7 & references there in & after

Problem Data:

Assume: open, no “dead” nodes, no “jam” (open overflows)

Explicit Stochastic Stationary Solutions:

Generally No

Exact Traffic Equations for Stochastic System:

Generally No

Traffic Equations for Fluid System

Yes


Traffic Equations


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 StijnFleuren and ErjenLefeber, “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!!!


The End


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