1. 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 of Erjen Lefeber

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”

4. 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

5. 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 gezelich om te pratten…
Raising young kids in Eindhoven: HIGHLY RECOMMENEDED!!!

6. Pedaling to see the Low Lands``

7. Future in Oz…
Melbourne

8. Melbourne…

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

10. Swinburne University of Technology

11. Looking for Ph.D Students…

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

13. In the Middle East…

18. In the Netherlands

19. A slow death…

20. Australia must be a safe place….

21. Or is it?

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

30. Finally… The Linear Complementarity Problem (LCP)

31. Definition

32. It’s all about Choosing a Subset…

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

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

35. 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

36. LCP References And Resources
Linear Complementarity, Linear and Nonlinear Programming, Katta G. Murty, 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, , 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

37. Some Applications (and Sources) of LCP

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

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

40. Lekker!

41. 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

42. 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.

43. Our Application: Min-Linear Equations

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

45. 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

46. Traffic Equations

47. 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!!!

48. The End