Abstract
We study the local behavior of a primal-dual inexact interior point methods for solving nonlinear systems arising from the solution of nonlinear optimization problems or more generally from nonlinear complementarity problems. The algorithm is based on the Newton method applied to a sequence of perturbed systems that follows by perturbation of the complementarity equations of the original system. In case of an exact solution of the Newton system, it has been shown that the sequence of iterates is asymptotically tangent to the central path (Armand and Benoist in Math. Program. 115:199–222, 2008). The purpose of the present paper is to extend this result to an inexact solution of the Newton system. We give quite general conditions on the different parameters of the algorithm, so that this asymptotic property is satisfied. Some numerical tests are reported to illustrate our theoretical results.
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References
Al-Jeiroudi, G., Gondzio, J.: Convergence analysis of the inexact feasible interior-point method for linear optimization. J. Optim. Theory Appl. 141, 231–247 (2009)
Armand, P., Benoist, J.: A local convergence property of primal-dual methods for nonlinear programming. Math. Program. 115, 199–222 (2008)
Armand, P., Benoist, J., Orban, D.: Dynamic updates of the barrier parameter in primal-dual methods for nonlinear programming. Comput. Optim. Appl. 41, 1–25 (2008)
Bellavia, S.: Inexact interior-point method. J. Optim. Theory Appl. 96, 109–121 (1998)
Curtis, F.E., Schenk, O., Wächter, A.: An interior-point algorithm for large-scale nonlinear optimization with inexact step computations. SIAM J. Sci. Comput. 32, 3447–3475 (2010)
Dembo, R.S., Eisenstat, S.C., Steihaug, T.: Inexact newton methods. SIAM J. Numer. Anal. 19, 400–408 (1982)
Dembo, R.S., Steihaug, T.: Truncated-newton algorithms for large-scale unconstrained optimization. Math. Program. 26, 190–230 (1983)
Dolan, E.D., Moré, J.J.: Benchmarking optimization software with performance profiles. Math. Program. 91, 201–213 (2002)
Facchinei, F., Pang, J.S.: Finite-Dimensional Variational Inequalities and Complementarity Problems. Springer Series in Operations Research, vol. I. Springer, New York (2003)
Facchinei, F., Pang, J.S.: Finite-Dimensional Variational Inequalities and Complementarity Problems. Springer Series in Operations Research, vol. II. Springer, New York (2003)
Ferris, M.C., Wathen, A.J., Armand, P.: Limited memory solution of bound constrained convex quadratic problems arising in video games. RAIRO. Rech. Opér. 41, 19–34 (2007)
Fiacco, A.V., McCormick, G.P.: Nonlinear Programming: Sequential Unconstrained Minimization Techniques. Wiley, New York (1968)
Freund, R.W., Jarre, F., Mizuno, S.: Convergence of a class of inexact interior-point algorithms for linear programs. Math. Oper. Res. 24, 50–71 (1999)
Gondzio, J., Toraldo, G.: Linear algebra issues arizing in interior point methods. Comput. Optim. Appl. 36, 137–341 (2007) (Special issue)
Kelley, C.T.: Iterative Methods for Linear and Nonlinear Equations. Frontiers in Applied Mathematics, vol. 16. Society for Industrial and Applied Mathematics (SIAM), Philadelphia (1995)
Nash, S.G.: A survey of truncated-Newton methods. J. Comput. Appl. Math. 124, 45–59 (2000)
Wächter, A., Biegler, L.T.: On the implementation of an interior-point filter line-search algorithm for large-scale nonlinear programming. Math. Program. 106, 25–57 (2006)
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Armand, P., Benoist, J. & Dussault, JP. Local path-following property of inexact interior methods in nonlinear programming. Comput Optim Appl 52, 209–238 (2012). https://doi.org/10.1007/s10589-011-9406-2
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DOI: https://doi.org/10.1007/s10589-011-9406-2