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Mathematical Programs with Equilibrium Constraints: Theory and Numerical Methods

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Nonsmooth Mechanics of Solids

Part of the book series: CISM International Centre for Mechanical Sciences ((CISM,volume 485))

Abstract

The lecture notes deal with optimization problems, where a generalized equation (modeling an equilibrium) arises among the constraints. The main attention is paid to necessary optimality conditions and methods to the numerical solution of such problems. The applications come from continuum mechanics.

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References

  • A.M. Al-Fahed, Stavroulakis, G. & Panagiotopoulos, P. (1991), ‘Hard and soft fingered robot grippers. The linear complementarity approach’, ZAMM-Zeit. Angew. Math. Mech. 71, 257–265.

    Article  MATH  MathSciNet  Google Scholar 

  • Anitescu, M. (2000), On solving mathematical programs with complementarity constraints as nonlinear programs, Technical Report ANL/MCS-P864-1200, Argonne National Laboratory.

    Google Scholar 

  • Aubin, J. & Frankowska, H. (1990), Set-valued Analysis, Birkhäuser.

    Google Scholar 

  • Aubin, J.-P. (1993), Optima and Equilibria, Springer-Verlag.

    Google Scholar 

  • Beremlijski, R., Haslinger, J., Kočvara, M. & Outrata, J. (2002), ‘Shape optimization in contact problems with Coulomb friction’, SIAM J. Optimization 13, 561–587.

    Article  MATH  Google Scholar 

  • Clarke, F. (1983), Optimization and Nonsmooth Analysis, J. Wiley & Sons.

    Google Scholar 

  • Dempe, S. (2002), Foundation of Bilevel Programming, Kluwer Acad. Publ.

    Google Scholar 

  • Dempe, S. (2003), ‘Annotated bibliography on bilevel programming and mathematical programs with equilibrium constraints’, Optimization 52, 333–359.

    Article  MATH  MathSciNet  Google Scholar 

  • Dontchev, A. & Rockafellar, R. (1996), ‘Characterizations of strong regularity for variational inequalities over polyhedral convex sets’, SIAM J. Optimization 7, 1087–1105.

    Article  MathSciNet  Google Scholar 

  • Ekeland, I. & Temam, R. (1976), Convex Analysis and Variational Problems, North-Holland Publ. Comp.

    Google Scholar 

  • Flegel, M., Kanzow, C. & Outrata, J. (2004), Optimality conditions for disjunctive programs with application to mathematical programs with equilibrium constraints, Technical Report 256, Math. Inst, of Julius-Maximilian University Wuerzburg.

    Google Scholar 

  • Fletcher, R. & Leyffer, S. (2002), ‘Nonlinear programming without a penalty function’, Mathematical Programming 91, 239–269.

    Article  MATH  MathSciNet  Google Scholar 

  • Fletcher, R. & Leyffer, S. (2004), ‘Solving mathematical programs with complementarity constraints as nonlinear programs’, Optimization Methods and Software 19, 15–40.

    Article  MATH  MathSciNet  Google Scholar 

  • Fletcher, R., Leyffer, S., Ralph, D. & Scholtes, S. (2002), Local convergence of SQP methods for mathematical programs with equilibrium constraints, Technical Report NA/209, University of Dundee.

    Google Scholar 

  • Haslinger, J., Dostál, Z. & Kučera, R. (2002), ‘On a splitting type algorithm for the numerical realization of contact problems with Coulomb friction’, Comput. Methods Appl. Mech. Engrg. 191, 2261–2281.

    Article  MATH  MathSciNet  Google Scholar 

  • Haslinger, J. & Panagiotopoulos, R (1984), ‘The reciprocal variational approach to the Signorini problem with friction. Approximation results’, Proc. of the Royal Society of Edinburgh, Sect. A 98, 365–383.

    MATH  MathSciNet  Google Scholar 

  • Henrion, R., Jourani, A. & Outrata, J. (2002), ‘On the calmness of a class of multifunctions’, SIAM J. Optimization 13, 603–618.

    Article  MATH  MathSciNet  Google Scholar 

  • Henrion, R. & Outrata, J. (to appear), ‘Calmness of constraint systems with applications’, Mathematical Programming.

    Google Scholar 

  • Hiriart-Urruty, J.-B. (1979), ‘Refinements of necessary optimality conditions in nondifferentiable programming I’, Appl. Math. Optim. 5, 63–82.

    Article  MATH  MathSciNet  Google Scholar 

  • Klatte, D. & Kummer, B. (2003), Strong Lipschitz stability of stationary solutions for nonlinear programs and non-polyhedral variational inequalities under MFCQ, Technical Report 2003-9, Humboldt-Universitaet zu Berlin.

    Google Scholar 

  • Kočvara, M., Mielke, A. & Roubíček, T. (n.d.), ‘A rate-independent approach to the delamination problem’, Mathematics and Mechanics of Solids (to appear).

    Google Scholar 

  • Kocvara, M. & Outrata, J. (2004), On the modeling and control of delamination processes, in J. Cagnol & J.-R Zolésio, eds, ‘Control and Boundary Analysis’, Marcel Dekker.

    Google Scholar 

  • Kummer, B. (1992), Newton’s method based on generalized derivatives for nonsmooth functions: Convergence analysis, in W. Oettli & D. Pallaschke, eds, ‘Advances in Optimization’, Springer-Verlag.

    Google Scholar 

  • Lemaréchal, C. (1981), A view of line searches, in W. Oettli & J. Stoer, eds, ‘Optimization and Optimal Control’, Springer-Verlag.

    Google Scholar 

  • Lemaréchal, C. (1989), Nondifferentiable optimization, in G. Nemhauser, A. Kan & M. Todd, eds, ‘Handbooks in Operations Research and Management Science, Volume I’, North-Holland Publ. Comp.

    Google Scholar 

  • Lemaréchal, C. & Sagastizábal, C. (1997), ‘Variable metric bundle method. From conceptual to implement able forms’, Mathematical Programming 76, 393–410.

    Article  MathSciNet  Google Scholar 

  • Lemaréchal, C, Strodiot, J.-J. & Bihain, A. (1981), On a bundle algorithm for nonsmooth optimization, in O. Mangasarian, R. Meyer & S. Robinson, eds, ‘Nonlinear Programming 4’, Academic Press.

    Google Scholar 

  • Lukšan, L. & Vlček, J. (1998), ‘A bundle Newton method for nonsmooth unconstrained minimization’, Mathematical Programming 83, 373–391.

    Article  MATH  MathSciNet  Google Scholar 

  • Luo, Z., Pang, J.-S. & Ralph, D. (1996), Mathematical Programs with Equilibrium Constraints, Cambridge Univ. Press.

    Google Scholar 

  • Mikhalevich, V., Gupal, A. & Norkin, V. (1987), Methods of Nonconvex Optimization (in Russian), Nauka.

    Google Scholar 

  • Mordukhovich, B. (1988), Approximation Methods in Problems of Optimization and Control (In Russian), Nauka.

    Google Scholar 

  • Mordukhovich, B. (1994), ‘Lipschitzian stability of constraint systems and generalized equations’, Nonl. Anal.: Theory, Methods & Appl. 22, 173–206.

    Article  MATH  MathSciNet  Google Scholar 

  • Mordukhovich, B. & Outrata, J. (2001), ‘On second-order subdifferentials and their applications’, SIAM J. Optimization 12, 139–169.

    Article  MATH  MathSciNet  Google Scholar 

  • Murty, K. (1988), Linear Complementarity, Linear and Nonlinear Programming, Heldermann.

    Google Scholar 

  • Outrata, J. (1999), ‘Optimality conditions for a class of mathematical programs with equilibrium constraints’, Mathematics of Operations Research 24, 627–644.

    MATH  MathSciNet  Google Scholar 

  • Outrata, J. (2000), ‘A generalized mathematical program with equilibrium constraints’, SIAM J. Control Optimization 38, 1623–1638.

    Article  MATH  MathSciNet  Google Scholar 

  • Outrata, J., Kočvara, M. & Zowe, J. (1998), Nonsmooth Approach to Optimization Problems with Equilibrium Constraints, Kluwer Acad. Publ.

    Google Scholar 

  • Panagiotopoulos, P. (1993), Hemivariational Inequalities. Applications in Mechanics and Engineering, Springer Verlag.

    Google Scholar 

  • Qi, L. (1993), ‘Convergence analysis of some algorithms for solving nonsmooth equations’, Mathematics of Operations Research 17, 227–244.

    Google Scholar 

  • Qi, L. & Sun, J. (1993), ‘A nonsmooth version of Newton’s method’, Mathematical Programming 58, 353–368.

    Article  MathSciNet  Google Scholar 

  • Robinson, S. (1980), ‘Strongly regular generalized equations’, Mathematics of Operations Research 5, 43–62.

    MATH  MathSciNet  Google Scholar 

  • Robinson, S. (1981), ‘Some continuity properties of polyhedral multifunctions’, Mathematical Programming Study 14, 206–214.

    MATH  Google Scholar 

  • Robinson, S. (1991), ‘An implicit-function theorem for a class of nonsmooth functions’, Math. Oper. Res. 16, 292–309.

    Article  MATH  MathSciNet  Google Scholar 

  • Rockafellar, R. & Wets, R. (1998), Variational Analysis, Springer-Verlag.

    Google Scholar 

  • Scheel, H. & Scholtes, S. (2000), ‘Mathematical programs with complementarity constraints: Stationarity, optimality, and sensitivity’, Mathematics of Operations Research 25, 1–22.

    Article  MATH  MathSciNet  Google Scholar 

  • Scholtes, S. (1994), Introduction to piecewise differentiable equations, Technical Report (Habilitation Thesis), Universitaet Karlsruhe.

    Google Scholar 

  • Scholtes, S. (2001), ‘Convergence properties of a regularization scheme for mathematical programs with complementarity constraints’, SIAM J. Optimization 11, 918–936.

    Article  MATH  MathSciNet  Google Scholar 

  • Schramm, H. & Zowe, J. (1992), ‘A version of the bundle idea for minimizing a nonsmooth function: conceptual idea, convergence analysis, numerical results’, SIAM J. Optimization 2, 121–152.

    Article  MATH  MathSciNet  Google Scholar 

  • Shapiro, A. (1990), ‘On concepts of directional differentiability’, J. Optim. Theory Appl. 66, 477–487.

    Article  MATH  MathSciNet  Google Scholar 

  • von Stackelberg, H. (1934), Marktform und Gleichgewicht, Springer-Verlag.

    Google Scholar 

  • Ye, J. (2000), ‘Constraint qualifications and necessary optimality conditions for optimization problems with variational inequality constraints’, SIAM J. Optimization 10, 943–962.

    Article  MATH  Google Scholar 

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Author information

Authors and Affiliations

  1. Institute of Information Theory and Automation, Czech Academy of Science, Prague, Czech Republic

    Jiří V. Outrata

Authors
  1. Jiří V. Outrata
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Editor information

Editors and Affiliations

  1. Charles University, Prague, Czech Republic

    Jaroslav Haslinger

  2. Technical University of Crete, Chania, Greece

    Georgios E. Stavroulakis

  3. Technical University of Braunschweig, Germany

    Georgios E. Stavroulakis

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© 2006 CISM

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Outrata, J.V. (2006). Mathematical Programs with Equilibrium Constraints: Theory and Numerical Methods. In: Haslinger, J., Stavroulakis, G.E. (eds) Nonsmooth Mechanics of Solids. CISM International Centre for Mechanical Sciences, vol 485. Springer, Vienna. https://doi.org/10.1007/978-3-211-48243-8_5

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  • DOI: https://doi.org/10.1007/978-3-211-48243-8_5

  • Publisher Name: Springer, Vienna

  • Print ISBN: 978-3-211-48241-4

  • Online ISBN: 978-3-211-48243-8

  • eBook Packages: EngineeringEngineering (R0)

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