A feasible directions algorithm for nonlinear complementarity problems and applications in mechanics Research Paper First Online: 12 April 2008 Received: 02 August 2007 Revised: 05 February 2008 Accepted: 09 February 2008 DOI :
10.1007/s00158-008-0252-5

Cite this article as: Herskovits, J. & Mazorche, S.R. Struct Multidisc Optim (2009) 37: 435. doi:10.1007/s00158-008-0252-5
3
Citations
143
Downloads
Abstract Complementarity problems are involved in mathematical models of several applications in engineering, economy and different branches of physics. We mention contact problems and dynamics of multiple bodies systems in solid mechanics. In this paper we present a new feasible direction algorithm for nonlinear complementarity problems. This one begins at an interior point, strictly satisfying the inequality conditions, and generates a sequence of interior points that converges to a solution of the problem. At each iteration, a feasible direction is obtained and a line search performed, looking for a new interior point with a lower value of an appropriate potential function. We prove global convergence of the present algorithm and present a theoretical study about the asymptotic convergence. Results obtained with several numerical test problems, and also application in mechanics, are described and compared with other well known techniques. All the examples were solved very efficiently with the present algorithm, employing always the same set of parameters.

Keywords Feasible direction algorithm Interior point algorithm Nonlinear complementarity problems Variational formulations in mechanics

References Arora JS (2004) Introduction to optimum design, 2nd edn. Academic, London

Google Scholar Baiocchi C, Capelo A (1984) Variational and quasivariational inequalities. Applications to free-boundary problems. Wiley, Chichester

MATH Google Scholar Bazaraa MS, Shetty CM (1979) Theory and algorithms. Nonlinear programming. Wiley, New York

Google Scholar Chen C, Mangasarian OL (1996) A class of smoothing functions for nonlinear and mixed complementarity problems. Comput Optim Appl 5:97–138

MATH CrossRef MathSciNet Google Scholar Chen X, Ye Y (2000) On smoothing methods for the P_0 matrix linear complementarity problem. SIAM J Optim 11:341–363

MATH CrossRef MathSciNet Google Scholar Christensen PW, Klarbring A , Pang JS, Stroömberg N (1998) Formulation and comparison of algorithms for frictional contact problems. Int J Num Meth Eng 42:145–173

MATH CrossRef MathSciNet Google Scholar Crank J (1984) Free and moving boundary problems. Oxford University Press, New York

MATH Google Scholar Dennis JE, Schnabel RB (1996) Numerical methods for unconstrained optimization and nonlinear equations. SIAM, Philadelphia

MATH Google Scholar Fathi Y (1979) Computational complexity of LCPs associated with positive definite symmetric matrices. Math Program 17:335–344

MATH CrossRef MathSciNet Google Scholar Ferris MC, Kanzow C (2002) Complementarity and related problems: a survey. Handbook of applied optimization. Oxford University Press, New York, pp 514–530

Google Scholar Ferris MC, Pang JS (1997) Engineering and economic applications of complementarity problems. SIAM Rev 39:669–713

MATH CrossRef MathSciNet Google Scholar Fischer A (1992) A special newton-type optimization method. Optim 24:269–284

MATH CrossRef Google Scholar Geiger C, Kanzow C (1996) On the resolution of monotone complementarity problems. Comput Optim Appl 5:155–173

MATH CrossRef MathSciNet Google Scholar Harker PT (1998) Accelerating the convergence of the diagonal and projection algorithms for finite-dimensional variational inequalities. Math Program 41:29–59

CrossRef MathSciNet Google Scholar Herskovits J (1982) Développement d’une méthode númerique pour l’Optimisation non linéaire. Dr. Ing. Thesis, Paris IX University, INRIA-Rocquencourt (in English)

Herskovits J (1986) A two-stage feasible directions algorithm for nonlinear constrained optimization. Math Program 36:19–38

MATH CrossRef MathSciNet Google Scholar Herskovits J (1995) A view on nonlinear optimizaton. In: Herskovits J (ed) Advances in structural optimization. Kluwer Academic, Dordrecht, pp 71–117

Google Scholar Herskovits J (1998) A feasible directions interior point technique for nonlinear optimization. J Optim Theory Appl 99(1):121–146

MATH CrossRef MathSciNet Google Scholar Herskovits J, Santos G (1998) Feasible arc interior point algorithm for nonlinear optimization. In: Fourth world congress on computational mechanics, (in CD-ROM), Buenos Aires, June–July 1998

Herskovits J, Leontiev A, Dias G, Santos G (2000) Contact shape optimization: a bilevel programming approach. Struct Multidisc Optim 20:214–221

CrossRef Google Scholar Herskovits J, Mappa P, Goulart E, Mota Soares CM (2005) Mathematical programming models and algorithms for engineering design optimization. Comput Methods Appl Mech Eng 194(30–33):3244–3268

MATH CrossRef Google Scholar Hock W, Schittkowski K (1981) Test example for nonlinear programming codes. Springer, Berlin Heidelberg New York

Google Scholar Jiang H, Qi L (1997) A new nonsmooth equations approach to nonlinear complementarity problems. SIAM J Control Optim 35(1):178–193

MATH CrossRef MathSciNet Google Scholar Kanzow C (1994) Some equation-based methods for the nonlinear complementarity problem. Optim Methods Softw 3:327–340

CrossRef Google Scholar Kanzow C (1996) Nonlinear complementarity as unconstrained optimization. J Optim Theory Appl 88:139–155

MATH CrossRef MathSciNet Google Scholar Kinderlehrer D, Stampacchia G (1984) An introduction to variational. Oxford University Press, New York

Google Scholar Leontiev A, Huacasi W (2001) Mathematical programming approach for unconfined seepage flow problem. Eng Anal Bound Elem 25:49–56

MATH CrossRef Google Scholar Leontiev A, Huacasi W, Herskovits J (2002) An optimization technique for the solution of the signorini problem using the boundary element method. Struct Multidisc Optim 24:72–77

CrossRef Google Scholar Mangasarian OL (1973) Equivalence of the complementarity problem to a system of nonlinear equations. SIAM J Appl Math 31:89–92

CrossRef MathSciNet Google Scholar Mangasarian OL, Solodov MV (1993) Nonlinear complementarity as unconstrainede and constrained minimization. Math Program (Serie B) 62:277–297

CrossRef MathSciNet Google Scholar Murphy FH, Sherali HD, Soyster AL (1982) A mathematical programming approach for determining oligopolistic market equilibrium. Math Program 24:92–106

MATH CrossRef MathSciNet Google Scholar Murty KG (1988) Limear complementarity, linear and nonlinear programming. Sigma series in applied mathematics, vol 3. Heldermann, Berlin

Google Scholar Petersson J (1995) Behaviourally constrained contact force optimization. Struct Multidisc Optim 9:189–193

Google Scholar Qi L, Sun D (1998) Nonsmooth equations and smoothing newton methods. Technical Report, School of Mathematics, University of New South Wales, Sydney

Subramanian PK (1993) Gauss-Newton methods for the complementarity problem. J Optim Theory Appl 77:467–482

MATH CrossRef MathSciNet Google Scholar Tanoh G, Renard Y, Noll D (2004) Computational experience with an interior point algorithm for large scale contact problems. Optimization Online

Tin-Loi F (1999a) On the numerical solution of a class of unilateral contact structural optimization problems. Struct Multidisc Optim 17:155–161

Google Scholar Tin-Loi F (1999b) A smoothing scheme for a minimum weight problem in structural plasticity. Struct Multidisc Optim 17:279–285

Google Scholar Tseng P (1997) An infeasible path-following method for mono tone complementarity problems. SIAM J Optim 7:386–402

MATH CrossRef MathSciNet Google Scholar Vanderplaats G (1999) Numerical optimization techniques for engineering design, 3rd edn. VR&D, Colorado Springs

Google Scholar Wright SJ (1997) Primal-dual interior-point methods. SIAM, Philadelphia

MATH Google Scholar Yamashita N, Fukushima M (1995) On stationary points of the implicit lagrangian for nonlinear complementarity problems. J Optim Theory Appl 84:653–663

MATH CrossRef MathSciNet Google Scholar Yamashita N, Dan H, Fukushima M (2004) On the identification of degenerate indices in the nonlinear complementarity problem with the proximal point algorithm. Math Programming 99:377–397

MATH CrossRef MathSciNet Google Scholar Xu S (2000) The global linear convergence of an infeasible non-interior path-following algorithm for complemenarity problems with uniform P-functions. Math Program 87:501–517

MATH CrossRef MathSciNet Google Scholar Zouain N, Herskovits J, Borges LA, Feijóo RA (1993) An iterative algorithm for limit analysis with nonlinear yield functions. Int J Solids Struct 30(10):1397–1417

MATH CrossRef Google Scholar Authors and Affiliations 1. COPPE, Mechanical Eng. Prog. Federal University of Rio de Janeiro Rio de Janeiro Brazil 2. Department of Mathematics, UFJF, ICE Campus Universitário Federal University of Juiz de Fora Juiz de Fora-MG Brazil