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
Just to be clear: Almost nothing in this presentation (except for pictures of my kids), is original work, it is rather a “reading seminar”
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
Cycle Racing
Israeli Army Reserve
Divorced
Emily Born
Married Again
Kayley Born
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!!!
``
Melbourne
Maybe live here
Also collaborate here: Melbourne University
Work here: Swinburne University
Maybe also collaborate here:Monash University
A slow death…
Complementary cones:
Immediate naïve algorithm with complexity
Relation of P-matrixes to positive definite (PD) matrixes:
P-Matrixes
Symmetric Matrixes
PD Matrixes
Primal-LP:
Dual-LP:
Theorem: Complementary slackness conditions
Find:
Such that:
And (complementary slackness):
Lekker!
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
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
Traffic Equations