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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 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!!!

Future in Oz…

Melbourne

Also collaborate here: Melbourne University

Work here: Swinburne University

Maybe also collaborate here:Monash University

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

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.

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

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!!!

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