zbMATH — the first resource for mathematics

An interior point algorithm for mixed complementarity nonlinear problems. (English) Zbl 1376.90064
Summary: Nonlinear complementarity and mixed complementarity problems arise in mathematical models describing several applications in Engineering, Economics and different branches of physics. Previously, robust and efficient feasible directions interior point algorithm was presented for nonlinear complementarity problems. In this paper, it is extended to mixed nonlinear complementarity problems. At each iteration, the algorithm finds a feasible direction with respect to the region defined by the inequality conditions, which is also monotonic descent direction for the potential function. Then, an approximate line search along this direction is performed in order to define the next iteration. Global and asymptotic convergence for the algorithm is investigated. The proposed algorithm is tested on several benchmark problems. The results are in good agreement with the asymptotic analysis. Finally, the algorithm is applied to the elastic-plastic torsion problem encountered in the field of Solid Mechanics.

90C33 Complementarity and equilibrium problems and variational inequalities (finite dimensions) (aspects of mathematical programming)
90C51 Interior-point methods
74C05 Small-strain, rate-independent theories of plasticity (including rigid-plastic and elasto-plastic materials)
74P10 Optimization of other properties in solid mechanics
Full Text: DOI
[1] Leontiev, A; Huacasi, W; Herskovits, J, An optimization technique for the solution of the Signorini problem using the boundary element method, Struct. Multidiscip. Optim., 24, 72-77, (2002)
[2] Glowinski, R., Lions, J., Trémolières, R.: Numerical Analysis of Variational Inequalities, vol. 8. North-Holland, Amsterdam (1981) · Zbl 0463.65046
[3] Judice, J; Soares, M, Solution of some linear complementarity problems arising in variational models of mechanics, Investigaç. Operac., 18, 17-31, (1998)
[4] Chapiro, G; Gutierrez, A; Herskovits, J; Mazorche, S; Pereira, W, Numerical solution of a class of moving boundary problems with a nonlinear complementarity approach, J. Optim. Theory Appl., 168, 534-550, (2016) · Zbl 1337.65115
[5] Ramírez, A; Mazorche, S; Chapiro, G, Numerical simulation of an in-situ combustion model formulated as mixed complementarity problem, Revista Interdisciplinar de Pesquisa em Engenharia—RIPE, 2, 172-181, (2016)
[6] Dirkse, S; Ferris, M, Mcplib: a collection of nonlinear mixed complementarity problems, Optim. Method Softw., 5, 319-345, (1995)
[7] Simantiraki, E., Shanno, D.: An infeasible-interior-point algorithm for solving mixed complementarity problems. In: Complementarity and Variational Problems: State of the Art, MC Ferris and JS Pang, eds., Philadelphia, Pennsylvania, pp. 386-404 (1997) · Zbl 0886.90163
[8] Dirkse, S; Ferris, M, The path solver: a nommonotone stabilization scheme for mixed complementarity problems, Optim. Method Softw., 5, 123-156, (1995)
[9] Daryina, A; Izmailov, A; Solodov, M, A class of active-set Newton methods for mixed complementarityproblems, SIAM J. Optim., 15, 409-429, (2005) · Zbl 1077.90064
[10] Daryina, A; Izmailov, A; Solodov, M, Numerical results for a globalized active-set Newton method for mixed complementarity problems, Comput. Appl. Math., 24, 293-316, (2005) · Zbl 1213.90226
[11] Liu, C; Atluri, S, A fictitious time integration method (FTIM) for solving mixed complementarity problems with applications to non-linear optimization, Comput. Model. Eng. Sci., 34, 155-178, (2008) · Zbl 1232.90334
[12] Gabriel, S, A hybrid smoothing method for mixed nonlinear complementarity problems, Comput. Optim. Appl., 9, 153-173, (1998) · Zbl 0904.90163
[13] Kanzow, C, Global optimization techniques for mixed complementarity problems, J. Global Optim., 16, 1-21, (2000) · Zbl 1009.90119
[14] Herskovits, J, A feasible directions interior point technique for nonlinear optimization, J. Optim. Theory Appl., 99, 121-146, (1998) · Zbl 0911.90303
[15] Herskovits, J, A two-stage feasible directions algorithm for nonlinear constrained optimization, Math. Program, 36, 19-38, (1986) · Zbl 0623.90070
[16] Herskovits, J.: A two-stage feasible direction algorithm including variable metric techniques for non-linear optimization problems. Ph.D. thesis, Inria (1982) · Zbl 0214.11104
[17] Herskovits, J; Mazorche, S, A feasible directions algorithm for nonlinear complementarity problems and applications in mechanics, Struct. Multidiscip. Optim., 37, 435-446, (2009) · Zbl 1274.90427
[18] Mazorche, S.: Algoritmos para problemas de complementaridade não linear. Ph.D. thesis, Universidade Federal do Rio de Janeiro (2007) · Zbl 0159.55802
[19] Billups, S; Dirkse, S; Ferris, M, A comparison of large scale mixed complementarity problem solvers, Comput. Optim. Appl., 7, 3-25, (1997) · Zbl 0883.90116
[20] McCormick, G.P.: Second Order Conditions for Constrained Minima, pp. 259-270. Springer, Basel (2014)
[21] Bazaraa, M., Sherali, H., Shetty, C.: Nonlinear programming: theory and algorithms. Wiley, New York (2013) · Zbl 0774.90075
[22] Herskovits, J, A feasible direction interior-point technique for nonlinear optimization, J. Optim. Theory Appl., 99, 121-146, (1998) · Zbl 0911.90303
[23] Gabriel, S.A.: Solving discretely constrained mixed complementarity problems using a median function. Optim. Eng. 18(3), 631-658 (2017) · Zbl 1390.90537
[24] Rheinboldt, W.: Some nonlinear test problems. http://folk.uib.no/ssu029/Pdf_file/Testproblems/testprobRheinboldt03.pdf (2003) · Zbl 1232.90334
[25] Sun, D; Han, J, Newton and quasi-Newton methods for a class of nonsmooth equations and related problems, SIAM J. Optim., 7, 463-480, (1997) · Zbl 0872.90087
[26] Kojima, M., Shindoh, S.: Extensions of Newton and quasi-Newton methods to systems of PC 1 equations. J. Oper. Res. Soc. Jpn. 29(4), 352-374 (1986) · Zbl 1274.90427
[27] Herakovich, C; Hodge, P, Elastic-plastic torsion of hollow bars by quadratic programming, Int. J. Mech. Sci., 11, 53-63, (1969) · Zbl 0159.55802
[28] Wagner, W; Gruttmann, F, Finite element analysis of Saint-Venant torsion problem with exact integration of the elastic-plastic constitutive equations, Comput. Method Appl. Mech. Eng., 190, 3831-3848, (2001) · Zbl 0995.74077
[29] Stampacchia, G, Formes bilinéaires coercitives sur LES ensembles convexes, CR Acad. Sci. Paris, 258, 964, (1964) · Zbl 0124.06401
[30] Brezis, H; Sibony, M, Équivalence de deux inéquations variationnelles et applications, Arch. Ration. Mech. Anal., 41, 254-265, (1971) · Zbl 0214.11104
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.