3rd International Conference onComplementarity Problems (ICCP 2002)

Titles and Abstracts of Papers(in alphabetical order of speakers)

Title: A complementarity-based time-stepping scheme for multibodydynamics for contact and friction

Speaker: Anitescu, MihaiArgonne National Laboratory, USA

Abstract: We present a time-stepping scheme for multibody dynamics withcontact and friction that solves one linear complementarity problem per timestep. The scheme includes constraint stabilization, accommodates stiff forcesand avoids the inconsistency issues of classical acceleration-force schemes. Weshow that the linear complementarity problem may have a nonconvex solu-tion for arbitrarily small friction. However, for sufficiently small friction, afixed point iteration that solves convex subproblems is shown to recover theunique velocity solution of the linear complementarity problem.

Title: Modelling of pressurized cracks in concrete damsand some related problems

Speaker: Bolzon, GabriellaPolitecnico di Milano, Italy

Co-author: Giuseppe Cocchetti

Abstract: Safety assessment against failure of hydraulic structures such as

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concrete dams requires the consideration of fracture possibly driven by waterpressure along natural and artificial joints and the dam-foundation interface.Computation-oriented modelling of such structural problems can be cast inthe format of (non-convex) complementarity problems (or non-convex con-strained optimization) involving space-discretization variables. The relateddifficulties, to be discussed in this talk, are at least twofold: (a) from thephysical standpoint, the role of water penetration is not yet fully under-stood, and uncertainties associated with its modelling, as well as with thepermeability and the mechanical features of the joints, suggest statisticalapproaches; (b) these approaches require extensive parametric studies and,therefore, a large number of computer simulations; the computational costof such discretized problems, involving thousands of variables, requires thedevelopment of efficient algorithms and solvers, interfaced with preprocessing(in particular discretization) and postprocessing (in particular visualization)tools.

Title: Fast simulation of hydraulic systems

Speaker: Borchsenius, FredrikTechnical University of Munich, Germany

Co-authors: Friedrich Pfeiffer

Abstract: The efficient design of hydraulic systems requires fast simulationmethods. In most simulation programmes, the hydraulic components arecoupled by compressible joints. Since the compressibility of oil is very small,this leads to stiff differential equations. To avoid these difficulties, stiff elasticcouplings can be replaced by algebraic or unilateral constraints. Examplesfor hydraulic components with unilateral behaviour are checkvalves, cylin-ders with stop limits and fluid volumes, in which cavitation can occur. Theresulting complementarity equations can be solved with a standard Lemkealgorithm. Compared to conventional methods, this leads to a significantreduction of computational effort.

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Title: Numerical solution of inelastic contact problems

Speaker: Christensen, Peter W.Linkoping University, Sweden

Abstract: In this talk we study the incremental, small–displacement, fric-tional contact problem for elastoplastic bodies. The von Mises yield functiontogether with an associated flow rule and linear hardening is used to describethe elastoplasticity. The contact problem is written as a set of unconstrainedsemismooth equations. These equations are then solved by using a Newtonmethod developed by J.–S. Pang. An example illustrating the efficiency ofthe algorithm is presented.

Title: Pricing American financial options by linear programming

Speaker: Dempster, MichaelUniversity of Cambridge, UK

Abstract: We investigate numerical solution of finite difference approxi-mations to American option pricing problems using a new direct numeri-cal method: simplex solution of a linear programming formulation. Thisapproach is based on an extension to the parabolic case of the abstractequivalence between linear order complementarity problems and linear pro-grammes (LPs) known for certain elliptic operators, which will be discussed.We then consider the numerical solution of American option pricing prob-lems using this novel formulation of the valuation problem as an LP. Byexploiting the structure of the constraint matrices derived from standardBlack-Scholes “vanilla” problems we obtain a fast and accurate revised sim-plex method which performs at most a linear number of pivots in the tem-poral discretization. When empirically compared with projected successiveoverrelaxation (PSOR) or a commercial LP solver the new method is or-ders of magnitude faster for vanilla problems. Utilising this method we nextvalue discretely sampled Asian and lookback American options and showthat path-dependent PDE problems – even allowing for the volatility smile– can be solved in ‘desktop’ solution times. We conclude that LP solutiontechniques, which are robust to parameter changes, can be tuned to pro-vide extremely fast efficient valuation methods for finite-difference approx-imations to many vanilla and exotic option valuation problems – they give

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lattice method speeds with PDE accuracy.

Title: Complementarity solvers: Now and beyond

Speaker: Ferris, MichaelUniversity of Wisconsin, Madison, USA

Abstract: Complementarity solvers can routinely solve large scale problemsof appropriate sparsity and structure. We review the state-of-the-art and ex-plain recent developments in the area, including interplays between nonlinearoptimization and complementarity. We look at challenges for the future, andclaim that progress is needed on two fronts, namely for large scale modelsfor which a direct factorization is not possible, and for models that do nothave appropriate convexity structure. A class of examples arising from videogaming will be used as motivation.

Title: Error bounds and local behavior of algorithms

Speaker: Fischer, AndreasUniversity of Dortmund, Germany

Abstract: Error bounds, usually formulated by means of merit functions,are employed to analyze and to improve the local convergence behavior ofalgorithms for the solution of complementarity problems or, more generally,of generalized equations. In particular, two approaches are dealt with. Thefirst uses merit functions for stabilizing the subproblems of Newton-type al-gorithms. The second additionally exploits merit functions for identifyingall indices that are active at a certain solution. To this end, the growthbehavior of the gradient of a once differentiable merit function is explored.Both approaches lead to algorithms with local superlinear convergence underconditions which do not necessarily imply the isolatedness of a solution.

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Title: Self-regulating, infinite-dimensional variational inequalitiesand Dynamic Traffic Assignment

Speaker: Friesz, TerryGeorge Mason University, USA

Abstract: In this paper we briefly review the history of the dynamic trafficassignment problem and its central role in the prediction of road networkflows and design of real-time travel information systems. We then give aformal definition of a specific type of dynamic traffic assignment: dynamicuser equilibrium (DUE). We show how the DUE problem may be expressedas an infinite dimensional variational inequality that clearly distinguishes be-tween state and control variables yet does not involve an explicit extremalproblem; we name this type of problem a self-regulating variational inequality(SRVI). We explore how an extremal problem may be formed for an SRVIwithout imposing symmetry restrictions, when the state dynamics and sin-gular controls obey certain assumptions. We discuss how the DUE SRVImay be solved by multigrid methods and by descent in Hilbert space. Somenumerical examples are provided to illustrate the algorithms.

Title: A penalty approach to generalized Nash equilibria,with application to multi-leader-follower games

Speaker: Fukushima, MasaoKyoto University, Japan

Co-authors: Jong-Shi Pang

Abstract: The noncooperative multi-leader-follower game can be formu-lated as a generalized Nash equilibrium problem where each player solves anonconvex mathematical program with equilibrium constraints. Two majordeficiencies exist with such a formulation: one is that the resulting Nashequilibrium may not exist, due to the nonconvexity in each player’s problem;the other is that such a nonconvex Nash game is computationally intractable.In order to obtain a viable formulation that is amenable to practical solution,we introduce a relaxation of the problem that is formulated as a generalizedNash game and develop an iterative penalty method for computing an equi-librium of the latter game. We establish the convergence of the method,

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which involves solving a sequence of penalized Nash games, under a set ofmodest assumptions. We also discuss applications of the proposed approachto some oligopolistic competition models in electricity power.

Title: Semidefinite relaxations of linear complementarity problems

Speaker: Gowda, Seetharama M.University of Maryland, Baltimore County, USA

Abstract: Given a linear complementarity problem LCP(A, q), we con-sider a semidefinite linear complementarity problem SDLCP(L, Diag(q))where L : Sn → Sn is a linear transformation satisfying the conditionL(D) = Diag(Adiag(D)) for all diagonal matrices D. In this paper, wedescribe some connections between (a) the solution sets of these two prob-lems, and (b) the complementarity properties of A and L. We show, forexample, that under certain conditions, the solution sets of these two prob-lems are (essentially) the same, and that L has the P-property if and only ifA is a P-matrix.

Title: On the second-best toll pricing problem with elasticdemand

Speaker: Hearn, DonaldUniversity of Florida, USA

Co-authors: Siriphong Lawphongapanich, Mehmet B. Yildirim

Abstract: In this talk, we address the second best toll pricing problem withelastic demand. We give three equivalent MPEC formulations, establish ex-istence conditions for optimal toll vectors, and relate these to marginal socialcost pricing tolls. In addition, we present a cutting plane algorithm and aframework for determining a second best toll vector that both induces a userequilibrium flow and achieves a desired (secondary) goal, such as minimizingthe maximum toll.

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Title: MPECS in function spaces: Optimality conditions andnumerical treatment

Speaker: Hintermuller, MichaelUniversity of Graz, Austria

Abstract: MPECS arising from optimal control problems for a class of vari-ational inequalities are considered. In an appropriate function space setting,first order conditions are discussed. Since the proof is constructive, some ofthe key ingredients are highlighted. Based on this first order characteriza-tion of optimal solutions, in the discretized setting a numerical algorithm isintroduced which, typically, has to deal with thousands of variables. It es-sentially combines semi-smooth Newton techniques and an interior point liketreatment of a part of the variables with nested iteration techniques. Finally,a report on numerical test runs emphasizes the efficiency of the algorithm.

Title: How can equilibrium models contribute to powermarket design and monitoring?

Speaker: Hobbs, BenThe Johns Hopkins University, USA

Abstract: Electricity provision is transitioning from a regime in which priceand quantity decisions are determined in regulatory proceedings or by pub-lic authorities, to more market-based decision making. This does not meanthat the public sector no longer has a role to play; competent market de-sign and surveillance is crucial to successful restructuring. These functionsrequire models for understanding how markets might respond to changes intheir design, and for defining competitive benchmarks for monitoring theirperformance.These models are often complex because power supply systems themselvesare complex. The complexities include the need for simultaneous balancingof supply and demand at every location in the network; the governance ofnetwork flows by Kirchhoff’s laws and so-called FACTS devices; the needfor ”ancillary services” such as operating reserves and capacity markets toensure reliable supply; interactions with other commodity markets, includ-ing fuels and emissions allowances; and public concerns over environmental

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impacts of construction and operation of generation, transmission, and dis-tribution facilities. Because individual market participants are often largerelative to these systems and interactions naturally lend themselves to beingmodeled as convex optimization or equilibrium processes in which marketparticipants each pursue their objectives. Thus, complementarity is a nat-ural way to model these market processes. Because regulators and marketdesigners must take the reactions of market participants into account, theirproblems can often be cast as MPECs.In this talk, I will present an overview of power market operation, invest-ment, regulation, and design problems. I will then suggest ways in whichcomplementarity and MPEC methods might be used for gaining insights onsystem behavior and for making decisions. The potential usefulness of a newclass of models (EPECs: equilibrium problems with equilibrium constraints)will also be explored. Summaries of selected results from CP, MPEC, andEPEC analyses of North American and EU power markets will be providedas illustrations.

Title: Optimality conditions for mathematical programs withequilibrium constraints

Speaker: Kanzow, ChristianUniversity of Wurzburg, Germany

Co-authors: Michael L. Flegel

Abstract: Mathematical programs with equilibrium constraints (MPECsfor short) are difficult optimization problems which do not satisfy standardconstraint qualifications like the Mangasarian-Fromovitz condition. Hencethe standard KKT theory cannot be applied to MPECs. It has thereforebeen the subject of intense research to develop special MPEC constraintqualifications together with suitable optimality conditions. The aim of thistalk is to give a new MPEC constraint qualification and a new optimalitycondition. Both the new constraint qualification and the new optimality con-dition will be related to existing ones.

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Title: Solving MPECs:Implicit programming and NLP methods face-to-face

Speaker: Kocvara, MichalUniversity of Erlangen, Germany

Abstract: We will present two rather different approaches to solving Math-ematical Programs with Equilibrium Constraints (MPEC). One is based onthe implicit function theorem and is suitable for problems with unique solu-tion of the equilibrium problem. The other one sees MPEC as a standardNLP problem and tries to apply NLP solvers. We will show situations whereone of the methods cannot be used in principle, and other, where both meth-ods can be used but one is far superior to the other. We will also introducea new NLP code PENNON, based on a generalized augmented Lagrangianalgorithm, which is rather robust when solving MPCC-NLP problems, andcompare it to SQP solvers. All presented ideas will be illustrated by variousnonacademic problems of mechanical design.

Title: Chain rules and transformations for generalized derivativesof Lipschitz functions

Speaker: Kummer, BerndHumboldt University Berlin, Germany

Co-authors: Diethard Klatte

Abstract: The talk is based on Ref. 1-6 and deals with interrelations be-tween generalized derivatives of (locally) Lipschitz functions in finite dimen-sion (Generalized Jacobians, Contingent-, Thibault-, Co- derivatives and oth-ers) and with chain rules of equation type for composed functions as well. Wewill point out that particular classes of Lipschitz functions, called pseudo-smooth (= C1 on a dense subset of the domain), ”simple” and locally PC1,and special forms of the composition, e.g., f(x,y) = P(x) Q(y), play a ba-sic role and are sufficiently general for various applications in the context ofnonsmooth functions and generalized equations. In particular, we considersets of limits of Jacobians taken at C1 points which are obviouosly impor-tant for Newton-type solution methods and do not coincide with so-calledB-derivatives.

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References:1. J.-P. Aubin and I. Ekeland, Applied Nonlinear Analysis, Wiley 1984, NewYork2. F.H. Clarke, Optimization and Nonsmooth Analysis, Wiley 1983, NewYork3. D. Klatte, B. Kummer, Nonsmooth Equations in Optimization - Regular-ity, Calculus, Methods and Applications, Kluwer Academic Publ., Ser. Non-convex Optimization and Its Applications, to appear May 2002, Dordrecht-Boston-London4. B. Kummer, Generalized Newton and NCP-methods: Convergence, regu-larity, actions, Discussiones Mathematicae- Differential Inclusions, Vol. 20,no. 2 (2000), 209-2445. B. Kummer, Lipschitzian inverse functions, directional derivatives andapplication in C1,1 optimization, JOTA, Vol. 70 (1991), 559-5806. B.S. Mordukhovich, Complete characterization of openness, metric reg-ularity and Lipschitzian properties of multifunctions. Transactions of theAmerican Mathematical Society, Vol. 340 (1993), 1-35

Title: The hitch-hikers guide to solving MPECs

Speaker: Leyffer, SvenUniversity of Dundee, UK

Co-authors: Roger Fletcher, Jorge Nocedal, Jose Luis MoralesDanny Ralph and Stefan Scholtes

Abstract: Many practical optimization problems contain equilibrium con-straints. Over recent years, it has been argued that equilibrium constraintscannot be solved satisfactorily with standard techniques for Nonlinear Pro-gramming (NLP). We re-examine this assertion and give some startling nu-merical and theoretical evidence to show that many equilibrium constraintscan be solved using standard solvers. We also comment on the suitability ofinterior point methods for solving MPECs.

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Title: On linear complementarity problems in structuralmechanics and engineering

Speaker: Maier, GiulioPolitecnico di Milano, Italy

Co-authors: R.Ardito, G.Cocchetti

Abstract: Linear complementarity problems (LCP) and mathematical pro-gramming problems under ”equilibrium”, here LCP, constraints (MPEC)arise in a variety of structural mechanics and structural engineering situa-tions. This lecture is intended to briefly present four representative of suchreal-life situations met by the speaker and his co-workers, and to conciselydescribe the formulation of the mechanical and mathematical models andthe numerical solutions achieved to practical purposes. In mechanical termsand in terms of engineering situations, the outlined issues can be specifiedas follows: (a) unilateral contact (LCP) and minimum cost design undercontact constraints (MPEC): the Transmediterranean Pipeline design (1977-79); (b) quasi-brittle fracture (LCP) and fracture parameter identification(MPEC): syntactic foam for future ships (Surface Piercing Wings); concretefor large dams (1999-2001); (c) ”exact” integration (LCP) and limited de-formation analysis combined (MPEC) in piecewiselinear plasticity and poro-plasticity: concrete, masonry and earth dams (2002); (d) piecewise-linear in-terface and cohesive crack models (LCP) and design optimization (MPEC):concrete dams (current research). A few closing remarks will concern futureresearch results in mathematical programming advocated and expected bysome areas of structural engineering.

Title: A strategic model of traffic assignment: Formulation andproperties

Speaker: Marcotte, PatriceUniversity of Montreal, Canada

Co-authors: Younes Hamdouch, Sang Nguyen

Abstract: Traditional traffic assignment models, either static or dynamic,assume that commodity flows are assigned to paths of the underlying trans-

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portation network. In contrast with this approach, we claim that car driversbehave strategically, i.e., choices are made en route, depending on currenttraffic conditions. In the strategic model, congestion is induced by a virtualqueueing process that takes place at the entrance of capacitated arcs. Al-though the resulting variational formulation involves a non-monotone point-to-set map, we prove that the set of equilibria is nonempty and proposeiterative algorithms for finding an equilibrium.

Title: Identifying and exploiting variational inequality structurein PATH

Speaker: Munson, ToddArgonne National Laboratory, USA

Abstract: PATH can identify affine variational inequality structure in agiven mixed complementarity problem during the presolve phase. In thistalk, we discuss the use of this additional information to improve the relia-bility and performance of PATH on such problems.

Title: On shape optimization in contact problems with Coulombfriction

Speaker: Outrata, JiriCzech Academy of Sciences, Czech Republic

Abstract: The contribution deals with a discretized problem of shape opti-mization of an elastic body in unilateral contact with a rigid obstacle, wherethe Coulomb friction law is considered. In the resulting mathematical pro-gram with equilibrium constraints, the control-state mapping is single-valuedand locally Lipschitz, which enables to apply the so-called implicit program-ming approach to its numerical solution. In the two-dimensional case thepiecewise continuously differentiable nature of the control-state map can beused to compute a subgradient information, required by the applied solver (abundle method). In the three-dimensional case, however, we propose a differ-ent approach, based on the generalized differential calculus of Mordukhovich.

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Title: Some new paradigms in VICPs

Speaker: Pang, Jong-ShiThe Johns Hopkins University, USA

Abstract: After discussing some mathematics in a unified treatment of sev-eral electricity market models, we highlight some of the mathematical andcomputational challenges that are inspired by new applications of comple-mentarity. The discussion will touch on DCPs, EPECs, GNEs, MPECs,QVIs, and VICONs, which are topics covered throughout the conference.The goal of this presentation is to underscore the need for collaborationsbetween the modelers, the analysts, and the algorithm designers in solvingimportant applications of VICPs.

Title: Sensitivity analysis of traffic equilibria revisited

Speaker: Patriksson, MichaelChalmers University of Technology, Sweden

Abstract: The presence of the complementarity conditions in the descrip-tion of a Wardrop equilibrium creates a problem not only when devisingMPEC algorithms for traffic management, but also when concerned with thesensitivity analysis of traffic equilibria. The focus of the talk is a descriptionof when a directional derivative of the equilibrium solution exists, when thedirectional derivative generates a gradient or a subgradient, and also a de-scription of the computations leading to them. The analysis is for the mostpart a characterization, and includes both traditional problem representa-tions, a link to SUE models, and the sensitivity of flows, demands and costs.Time permitting, it is also explained how the analysis in a rather straight-forward manner extends to more general models.

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Title: Modelling bilevel games in electricity markets

Speaker: Ralph, DannyUniversity of Cambridge, UK

Co-authors: Xinmin Hu, Eric K Ralph

Abstract: Electricity markets facilitate pricing and delivery of wholesalepower. Generators submit bids to an Independent System Operator (ISO)to indicate how much power they can produce depending on price. TheISO takes these bids with demand forecasts and minimizes the total costof power production subject to feasibility of distribution in the electricalnetwork. Each generator can optimise its bid using a bilevel program ormathematical program with equilibrium constraints, with the ISO’s problemat the lower level. A Nash equilibrium is a list of bids that is simultaneouslyoptimal for all generators. We present some theory and numerical examples.

Title: Modelling competition among electric generating firmsincluding emission limitations: A complementarityprogramming approach

Speaker: Rogers, ScottUniversity of Toronto, Canada

Co-authors: Karen E. Lew

Abstract: This paper describes a complementarity programming model ofthe strategic competition among electric generating firms subject to a system-wide emissions cap. Each generating firm seeks to maximize its profits usingits mix of hydro, oil, coal and nuclear generating assets and competes bymaking price and quantity ”offers” to a Market Operator. The MO choosesthe lowest cost set of offers that meet the demand whose level is set by thesystem marginal cost. There are markets for energy by blocks of a load du-ration curve. We report on initial results for SO2 emissions in the provinceof Ontario in Canada.

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Title: tba

Speaker: Rutherford, TomUniversity of Colorado, Bolder, USA

Abstract: tba

Title: On the existence and computation of EPEC solutions

Speaker: Scholtes, StefanUniversity of Cambridge, UK

Co-authors: Andreas Ehrenmann

Abstract: This paper seeks to contribute to the ongoing discussion of twounresolved issues in connection with equilibrium problems with equilibriumconstraints (EPECs): Existence and computation of solutions. It is well-known that EPECs may not have solutions, even if the data is linear; themultilevel nature of the underlying market dynamics can lead to cyclic ormore complicated limiting behaviour. This is also true in the simpler case offinite multi-stage games. In this case, however, Kuhn’s theorem guaranteesthe existence of mixed strategy solutions. We explain how this concept andKuhn’s theorem can be extended to EPECs, thus guaranteeing the existenceof mixed-strategy equilibria under appropriate assumptions. In the secondpart, we focus on the computation of local equilibrium solutions, providedthey exist. We examine an extension of an MPEC-regularization procedurewhose limit points, if existent, are B-stationary solutions of the EPEC undersuitable assumptions. The method is simple and can be readily implementedin conjunction with a complementarity solver such as PATH.

Title: A system-theoretic approach to dynamic complementarityproblems

Speaker: Schumacher, J.M. (Hans)Tilburg University, The Netherlands

Abstract: In many applications ranging from mechanics to finance, comple-

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mentarity problems arise naturally in conjunction with differential equations.Variational inequalities which involve partial differential equations have beenstudied extensively, but also systems of ordinary differential equations cou-pled to finitely many complementarity conditions are of substantial interest.It is possible to look at the latter systems as input-output dynamical systemsin which inputs and outputs have been connected through the complementar-ity relation. In this way, one can relate dynamic complementarity problemsto the results of mathematical system theory. Well known special classes ofi/o dynamical systems give rise to types of dynamic complementarity prob-lems that are of particular interest. The connection between mathematicalsystem theory and complementarity theory can be used fruitfully for instancein the definition of dynamics, in the application of transform methods, andin the study of well-posedness of complementarity systems in the dynamicalsense.

Title: Modelling market power and regulation within the FlorenceRegulatory Forum framework

Speaker: Smeers, YvesUniversite Catholique de Louvain, Belgium

Co-authors: Olivier Daxhelet

Abstract: The Florence Regulatory Forum is in charge of determining theorganisation of cross border trade of electricity in the European Union. Itrecently produced recommendations in terms of the allocation of the fixedcharges of the network and the management of congestion on the interconnec-tions. We propose a model of the European electricity market that followsthe recommendations of the Florence Regulatory Forum. The model is atwo-stage game: the second stage simulates the operations of the differentagents on the markets subject to the two regulations imposed by the FlorenceRegulatory Forum and according to various assumptions of market power.It is formulated as a complementarity problem. Some of these regulationsresult from network charge parameters decided by national regulators. Wemodel this decision process as a game between regulators. The entire modelis thus an equilibrium problem subject to equilibrium constraints where theequilibrium constraints are formulated as a complementarity problem. The

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model has little mathematical properties that one can take advantage of. Oursolution method therefore contains a lot of brute force. We show numericalresults obtained on a stylised model.

Title: KKT Systems: Regularity conditions, error bounds anda class of Newton-type methods

Speaker: Solodov, MikhailInstituto de Matematica Pura e Aplicada, Brazil

Co-authors: Alexey Izmailov

Abstract: We discuss error bounds and Newton-type methods for KKT sys-tems. We obtain a new error bound under an assumption which we show tobe strictly weaker than assumptions previously used for KKT systems, suchas quasi-regularity or semistability (equivalently, the R0-property). Errorbounds are useful, among other things, for identifying active constraints anddeveloping efficient local algorithms. We propose a family of local Newton-type algorithms. This family contains some known active-set Newton meth-ods, as well as some new methods. Regularity conditions required for localsuperlinear convergence compare favorably with convergence conditions ofnonsmooth Newton methods and SQP methods.

Title: Convex and nonconvex models for interfaces and contact

Speaker: Stavroulakis, GeorgiosUniversity of Ioannina, Greece

Abstract: Linear and nonlinear interaction problems in mechanics play asignificant role in applications. A systematic way to treat the interfaces isby using monotone and nonmonotone, possibly multivalued relations. Thecorresponding potentials are convex or nonconvex, possibly nondifferentiable,respectively. This provides a link between optimization and mechanics, withobvious importance for both the theoretical analysis and the numerical com-putations. Unilateral contact and frictional contact effects, as well as adhe-sion effects are among the most known interface laws which can be treated

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with this procedure. The recent experience of the author with finite andboundary elements in statics and dynamics will be discussed in the presen-tation.

Title: Lemke’s algorithm with exact arithmeticfor solving two-person games

Speaker: von Stengel, BernhardLondon School of Economics, UK

Co-authors: Antoon van den Elzen, Dolf Talman

Abstract: Lemke’s LCP solution algorithm is implemented in exact arith-metic using “integer pivoting”, which uses arbitrary-precision integers andavoids fractions and their costly cancellations. The algorithm is applied tofinding Nash equilibria of two-person games in normal form and in sequenceform, the latter being an efficient method for solving game trees. Variationsof the starting point allow to find more than equilibrium, but cannot guaran-tee to find all LCP solutions. Degeneracies arise but are resolved, error-free,with a lexicographic rule. On average, the observed number of pivoting stepsis proportional to the matrix dimension.

Title: Convolution complementarity problems

Speaker: Stewart, DavidUniversity of Iowa, USA

Abstract: Convolution complementarity problems (CCP’s) are convolu-tion problems of the form: Given the functions k : [0, T ] → Rn×n andq : [0, T ] → Rn, find u : [0, T ] → Rn where

0 ≤ u(t) ⊥ (k ∗ u)(t) + q(t) ≥ 0

for (almost) all t. This is a dynamic complementarity system that is formallya generalization of the Linear Complementarity Systems (LCS’s) of Heemels,Schumacher & Weiland. In fact, we need to impose some other (fairly mild)restrictions to obtain existence, uniqueness, and convergence results. Appli-cations of CCP’s to a frictionless impact problem with an elastic body are

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shown, where existence & uniqueness of solutions, conservation of energy,and convergence of numerical solutions are shown.

Title: Some recent advances in inverse and implicit functiontheorems for semismooth functions

Speaker: Sun, DefengNational University of Singapore, Republic of Singapore

Abstract: In this talk we summarize some recent advances in inverse andimplicit function theorems for semismooth functions and their applicationsto the approximation of nonsmooth functions and the Moreau-Yosida regu-larization of convex functions.

Title: Semismooth semidefinite optimization

Speaker: Sun, JieNational University of Singapore, Singapore

Co-authors: Jong-Shi Pang, Defeng Sun

Abstract: Recent interest in optimization problems of symmetric matrixvariables has led to better understanding on semismooth functions and non-smooth Newton’s methods. The talk will report advances in related researchand their applications to semidefinite complementarity and symmetric eigen-value problems.

Title: Identification of fracture parameters as an MPEC:Some computational experiences

Speaker: Tin-Loi, FrancisUniversity of New South Wales, Australia

Abstract: In view of the difficulty of directly obtaining the required infor-mation by experimental test, we formulate and solve an important parameteridentification problem as an MPEC. The parameters represent key values of

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a discrete crack model, expressed in complementarity format. The presen-tation focuses on solution of this MPEC as an NLP. Computational resultsfor various methods (modeled within the GAMS environment) of tacklingthe complementarity constraints are briefly compared for MPECs involvingactual experimental data. We also present some preliminary computationalfindings on carrying out the identification with a simple genetic-based algo-rithm, which uses PATH to obtain the necessary fitness values.

Title: Rigid body dynamics with friction: Time-steppingand applications

Speaker: Trinkle, JeffSandia National Laboratories, USA

Abstract: The behavior of many engineered mechanical systems is stronglyaffected by intermittent contact and dry friction. Examples include dexter-ous manipulation by robotic systems, fixture design and product assembly,vehicles traversing uneven terrain, and the operation of ratchet mechanisms.The dynamic models of these systems are most naturally and rigorously for-mulated as complementarity problems. In order to analyze such systemsbefore building them, Sandia National Laboratories is developing simulationsoftware designed to work with three-dimensional bodies of arbitrary geom-etry (as produced by common computer-aided design tools), fast inter-bodydistance-computation software, and the PATH algorithm (Munson and Fer-ris) for solving complementarity problems.I will first review of the basic instantaneous dynamic model with unilateraland bilateral constraints and dry friction that underlies our software, thenderive the ”linearly implicit trapezoidal” time- stepping method (Anitescuand Potra) used to advance our simulations, and finally discuss recent appli-cation and current difficulties.

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Title: Recent progress on smoothing and nonsmoothmethods for second-order cone complementarityproblems.

Speaker: Tseng, PaulUniversity of Washington, USA

Co-authors: Jason Chen

Abstract: The Fischer-Burmeister (FB) function for NCP can be extendedto second-order cone complementarity problems (SOCCP) by using the as-sociated Jordan product. We show that the square of the 2-norm of the FBfunction is continuously differentiable and, for monotone problems, its sta-tionary points are solutions of SOCCP. This allows for a SOC program tobe transformed into an unconstrained differentiable minimization problem.Preliminary numerical experience with this approach will (hopefully) be re-ported. Progress on proving semismoothness of the FB function for SOCCPwill also be reported. Other nonsmooth equation formulations of SOCCP,such as one based on projection onto the SOC, will be discussed, if timepermits.

Title: An S-stage IRK method for the rigid bodycontact problem with spatial Coulomb friction

Speaker: Tzitzouris, JamesThe Johns Hopkins University, USA

Co-authors: Jong-Shi Pang

Abstract: Multi-rigid-body systems acting under Coulomb friction can bemodeled by an implicit system composed of an ODE, NCP and an NLP.Such a system is known as a differential complementarity problem (DCP).We present an S-stage implicit Runge-Kutta (IRK) scheme to simulate nu-merically the trajectories of such a DCP over a prescribed time interval. Ateach stage in the IRK scheme, we solve an NCP rather than a system of non-linear equations. To solve each stage NCP, we employ the B-differentiableNewton method. Numerical results from the simulation of some realistic me-chanical systems are presented.

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Title: Multilevel methods for large-scale complementarityproblems

Speaker: Ulbrich, MichaelTU Munich and University of Hamburg, Germany

Abstract: The efficient solution of large-scale (mixed) complementaritysystems arising from PDE-constrained optimization problems is considered.Hereby, reformulation techniques are used to develop a superlinearly con-vergent inexact Newton method. It is shown that in many situations thestructure of the problem can be exploited to develop highly efficient inexactmultigrid-preconditioned iterative solvers for the subproblems. Applicationsof the approach to optimal control problems and obstacle problems are con-sidered and numerical results are presented.

Title: Obtaining robust solutions to general variationalinequalities

Speaker: Wynter, LauraINRIA and Universite de Versailles, France

Co-authors: Michael Patriksson and Mats Werme

Abstract: This work is concerned with developing algorithms for generalvariational inequality problems (VIP) that arise frequently in network equi-librium models. In such models, the cost operator associated with the generalVIP is not everywhere monotone, and in this setting, some solutions to theVIP may not be robust. We define a notion of robustness in this sense thatallows us to adapt a method from non-convex optimization to this class ofproblems. Numerical results with different variants of the method are pre-sented.

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