Nonlinear Complementarity Problems and Extensions Michael C. Ferris Computer Sciences Department University of Wisconsin 1210 West Dayton Street Madison, WI 53706 [email protected]http://www.cs.wisc.edu/∼ferris INFORMS Practice Meeting, Baltimore, April 14, 2008 Michael Ferris (University of Wisconsin) Nonlinear Complementarity April 14, 2008 1 / 24
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Nonlinear Complementarity Problems and Extensionspages.cs.wisc.edu/~ferris/talks/baltimore.pdfAbstract While optimizers are familiar with complementary slackness within the optimality
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Nonlinear Complementarity Problems and Extensions
Michael C. Ferris
Computer Sciences DepartmentUniversity of Wisconsin
INFORMS Practice Meeting, Baltimore, April 14, 2008
Michael Ferris (University of Wisconsin) Nonlinear Complementarity April 14, 2008 1 / 24
Abstract
While optimizers are familiar with complementary slackness within theoptimality conditions of linear and nonlinear programming, there arenumerous complementarity problems arising naturally in many practicalapplications from engineering and economics. These include appliedgeneral equilibrium modeling, traffic network design, structural engineeringand finance. Several examples will be outlined, together with an overviewof modern, practical modeling and solution techniques within this field.Since complementarity allows for competition among players, optimizationproblems that involve complementarity constraints, and models withembedded complementarities are becoming increasingly important withinapplications. We outline these new ideas, highlight several computationalschemes and explain their utility by application.
Michael Ferris (University of Wisconsin) Nonlinear Complementarity April 14, 2008 2 / 24
Modeling languages: state-of-the-art
Optimization models improve understanding of underlying systemsand facilitate operational/strategic improvements
Key link to applications, prototyping of optimization capability
Widely used in:I engineering - operation/designI economics - policy/energy modelingI military - operations/planningI finance, medical treatment, supply chain management, etc.
Interface to solutions: facilitates automatic differentiation, separationof data, model and solver
Modeling languages no longer novel: typically represent another toolfor use within a solution process.
Michael Ferris (University of Wisconsin) Nonlinear Complementarity April 14, 2008 3 / 24
Some constraints can be reformulated easily, others not!
Michael Ferris (University of Wisconsin) Nonlinear Complementarity April 14, 2008 19 / 24
CVaR constraints: mean excess dose (radiotherapy)VaR, CVaR, CVaR+ and CVaR-
Loss
Fre
qu
en
cy
1111 −−−−αααα
VaR
CVaR
Probability
Maximumloss
Move mean of tail to the left!
Michael Ferris (University of Wisconsin) Nonlinear Complementarity April 14, 2008 20 / 24
EMP: Extended nonlinear programs
minx∈X
f0(x)+θ(f1(x), . . . , fm(x))
Examples of different θ
least squares, absolute value, Huber functionSolution reformulations are very differentHuber function used in robust statistics.
Michael Ferris (University of Wisconsin) Nonlinear Complementarity April 14, 2008 21 / 24
More general θ functions
In general any piecewise linear penalty function can be used: (differentupside/downside costs).General form:
θ(u) = supy∈Y{yTu − k(y)}
First order conditions for optimality are an MCP!
Michael Ferris (University of Wisconsin) Nonlinear Complementarity April 14, 2008 22 / 24
EMP: Heirarchical models
Bilevel programs:
minx ,y
f (x , y)
s.t. g(x , y) ≤ 0,y solves min
sv(x , s) s.t. h(x , s) ≤ 0
Model as:model bilev /deff,defg,defv,defh/;plus empinfo: bilevel y min v defh
EMP tool automatically creates the MPEC
Michael Ferris (University of Wisconsin) Nonlinear Complementarity April 14, 2008 23 / 24
Conclusions
Large scale complementarity problems reliably solvable
Complementarity constraints within optimization problems
Extended Mathematical Programming available within a modelingsystem
System can easily formulate and solve second order cone programs,robust optimization, soft constraints via piecewise linear penalization(with strong supporting theory)