Abstract
Hemivariational inequalities can be considered as a generalization of variational inequalities. Their origin is in nonsmooth mechanics of solid, especially in nonmonotone contact problems. The solution of a hemivariational inequality proves to be a substationary point of some functional, and thus can be found by the nonsmooth and nonconvex optimization methods. We consider two type of bundle methods in order to solve hemivariational inequalities numerically: proximal bundle and bundle-Newton methods. Proximal bundle method is based on first order polyhedral approximation of the locally Lipschitz continuous objective function. To obtain better convergence rate bundle-Newton method contains also some second order information of the objective function in the form of approximate Hessian. Since the optimization problem arising in the hemivariational inequalities has a dominated quadratic part the second order method should be a good choice. The main question in the functioning of the methods is how remarkable is the advantage of the possible better convergence rate of bundle-Newton method when compared to the increased calculation demand.
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References
Auslander, A. (1987), Numerical methods for nondifferentiable convex optimization, Mathematical Programming Study 30: 102–126.
Bihain, A. (1984), Optimization of upper semidifferentiable functions, Journal of Optimization Theory and Applications 4: 545–568.
Bonnans, J.F., Gilbert, J.C., Lemaréchal, C. and Sagastizábal, C. (1995), A family of variable metric proximal methods, Mathematical Programming 68: 15–47.
Cheney, E.W. and Golstein, A.A. (1959), Newton's method for convex programming and Tchebycheff approximation, Numerische Mathematik 1: 253–268.
Clarke, F.H. (1983), Optimization and Nonsmooth Analysis. New York: J. Wiley & Sons.
Gaudioso, M. and Monaco, M.F. (1992), Variants to the cutting plane approach for convex nondifferentiable optimization, Optimization 25: 65–75.
Kelley, J.E. (1960), The cutting plane method for solving convex programs, SIAM J. 8: 703–712.
Kiwiel, K.C. (1985), Methods of Descent for Nondifferentiable Optimization (Lecture Notes in Mathematics 1133). Berlin/New York: Springer Verlag.
Kiwiel, K.C. (1986), A method for solving certain quadratic programming problems arising in nonsmooth optimization, IMA Journal of Numerical Analysis 6: 137–152.
Kiwiel, K.C. (1990), Proximity control in bundle methods for convex nondifferentiable optimization, Mathematical Programming 46: 105–122.
Lemaréchal, C. (1978), Nonsmooth Optimization and Descent Methods. IIASA-report, Laxemburg, Austria.
Lemaréchal, C. (1982), Numerical experiments in nonsmooth optimization, in E.A. Nurminski, (ed.), Progress in Nondifferentiable Optimization, (pp. 61–84). IIASA-report, Laxemburg, Austria.
Lemaréchal, C. and Sagastizábal, C. (1994), An approach to variable metric bundle methods, in J. Henry and J. P. Yvor (eds.), Lecture Notes in Control and Information Sciences, vol. 197, pp. 144–162. Berlin/New York: Springer-Verlag.
Lukšan, L. (1984), Dual method for solving a special problem of quadratic programming as a subproblem at linearly constrained nonlinear minmax approximation, Kybernetika 20: 445–457.
Lukšan, L. and Vlčcek, J. (1995), A Bundle-Newton Method for Nonsmooth Unconstrained Minimization. Technical report, No. 654, Institute of Computer Science, Academy of Sciences of the Czech Republic.
Mäkelä, M.M. and Neittanmäki, P. (1992), Nonsmooth Optimization. Analysis and Algorithms with Applications to Optimal Control. World Scientific, Singapore.
Miettinen, M. and Haslinger, J. (1995), Approximation of non-monotone multivalued differential inclusions, IMA Journal of Numerical Analysis 15: 475–503.
Miettinen, M. and Haslinger, J. (1997), Finite element approximation of vector-valued hemivariational inequalities, Journal of Global Optimization 10: 17–35.
Miettinen, M., Mäkelä, M.M. and Haslinger, J. (1995), On numerical solution of hemivariational inequalities by nonsmooth optimization methods, Journal of Global Optimization 6: 401–425.
Mifflin, R. (1996), A Quasi-second-order proximal bundle algorithm, Mathematical Programming 73: 51–72.
Mistakidis, E.S., Baniotopoulos, C.C. and Panagiotopoulos, P.D. (1995), On the numerical treatment of the delamination problem in laminated composites under cleavage loading, Composite Structures 30: 453–466.
Naniewicz, Z. and Panagiotopoulos, P.D. (1995), Mathematical Theory of Hemivariational Inequalities and Applications. Basel/New York: Marcel Dekker.
Panagiotopoulos, P.D. (1993), Hemivariational Inequalities. Berlin/New York: Springer Verlag.
Rockafellar, R.T. (1976), Monotone operators and the proximal point algorithm, SIAM Journal on Optimal Control and Optimization 14: 877–898.
Schramm, H. and Zowe, J. (1992), A version of the bundle idea for minimizing a nonsmooth functions: Conceptual idea, convergence analysis, numerical results, SIAM Journal on Optimization 2: 121–152.
Tzaferopoulos, M. Ap., Mistakidis, E.S., Bisbos, C.D. and Panagiotopoulos, P.D. (1995), Comparison of two methods for the solution of a class of nonconvex energy problems using convex minimization algorithms, Computers and Structures 57(6): 959–971.
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MÄKELÄ, M., Miettinen, M., LUKŠAN, L. et al. Comparing Nonsmooth Nonconvex Bundle Methods in Solving Hemivariational Inequalities. Journal of Global Optimization 14, 117–135 (1999). https://doi.org/10.1023/A:1008282922372
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DOI: https://doi.org/10.1023/A:1008282922372