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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 problem and a bit about me since this is epps

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
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
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
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!!!



Future in oz
Future in Oz…

Melbourne



A bit on the linear complementarity problem and a bit about me since this is epps

Maybe live here

Also collaborate here: Melbourne University

Work here: Swinburne University

Maybe also collaborate here:Monash University



Looking for ph d students
Looking for Ph.D Students…


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







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


Finally the linear complementarity problem lcp
Finally…The Linear Complementarity Problem (LCP)




Illustration n 2
Illustration: n=2

Complementary cones:

Immediate naïve algorithm with complexity


Existence and uniqueness
Existence and Uniqueness

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

P-Matrixes

Symmetric Matrixes

PD Matrixes


Computation algorithms
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
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
Some Applications(and Sources) of LCP


Linear programming lp
Linear Programming (LP)

Primal-LP:

Dual-LP:

Theorem: Complementary slackness conditions


The lcp of lp
The LCP of LP

Find:

Such that:

And (complementary slackness):



Quadratic programming
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
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.



A bit on the linear complementarity problem and a bit about me since this is epps

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

Problem Data:

Assume: open, no “dead” nodes

Traffic Equations:


A bit on the linear complementarity problem and a bit about me since this is epps

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


A bit on the linear complementarity problem and a bit about me since this is epps

Traffic Equations


Wrapping up
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!!!