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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 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)

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  1. 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

  2. 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”

  3. 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

  4. 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!!!

  5. Pedaling to see the Low Lands ``

  6. Future in Oz… Melbourne

  7. Melbourne…

  8. Maybe live here Also collaborate here: Melbourne University Work here: Swinburne University Maybe also collaborate here:Monash University

  9. Swinburne University of Technology

  10. Looking for Ph.D Students…

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

  12. In the Middle East…

  13. In the Netherlands

  14. A slow death…

  15. Australia must be a safe place….

  16. Or is it?

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

  18. Finally…The Linear Complementarity Problem (LCP)

  19. Definition

  20. It’s all about Choosing a Subset…

  21. Illustration: n=2 Complementary cones: Immediate naïve algorithm with complexity

  22. Existence and Uniqueness Relation of P-matrixes to positive definite (PD) matrixes: P-Matrixes Symmetric Matrixes PD Matrixes

  23. 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

  24. 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

  25. Some Applications(and Sources) of LCP

  26. Linear Programming (LP) Primal-LP: Dual-LP: Theorem: Complementary slackness conditions

  27. The LCP of LP Find: Such that: And (complementary slackness):

  28. Lekker!

  29. 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

  30. 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.

  31. Our Application: Min-Linear Equations

  32. Open Jackson NetworksJackson 1957, Goodman & Massey 1984, Chen & Mandelbaum 1991 Problem Data: Assume: open, no “dead” nodes Traffic Equations:

  33. 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

  34. Traffic Equations

  35. 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!!!

  36. The End

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