Abstract
The proximal method is a standard regularization approach in optimization. Practical implementations of this algorithm require (i) an algorithm to compute the proximal point, (ii) a rule to stop this algorithm, (iii) an update formula for the proximal parameter. In this work we focus on (ii), when smoothness is present—so that Newton-like methods can be used for (i): we aim at giving adequate stopping rules to reach overall efficiency of the method.
Roughly speaking, usual rules consist in stopping inner iterations when the current iterate is close to the proximal point. By contrast, we use the standard paradigm of numerical optimization: the basis for our stopping test is a “sufficient” decrease of the objective function, namely a fraction of the ideal decrease. We establish convergence of the algorithm thus obtained and we illustrate it on some ill-conditioned problems. The experiments show that combining the proposed inexact proximal scheme with a standard smooth optimization algorithm improves the numerical behaviour of the latter for those ill-conditioned problems.
Similar content being viewed by others
References
Armijo, L.: Minimization of functions having Lipschitz continuous first partial derivatives. Pac. J. Math. 16, 1–3 (1966)
Bellman, R.E., Kalaba, R.E., Lockett, J.: Numerical Inversion of the Laplace Transform. Elsevier, New York (1966), pp. 143–144
Bongartz, I., Conn, A.R., Gould, N., Toint, Ph.L.: CUTE: Constrained and Unconstrained Testing Environment. ACM Trans. Math. Softw. 21(1), 123–260 (1995)
Correa, R., Lemaréchal, C.: Convergence of some algorithms for convex minimization. Math. Program. 62, 261–275 (1993)
Dennis, J.E., Schnabel, R.B.: Numerical Methods for Unconstrained Optimization and Nonlinear Equations. Prentice-Hall, Englewood Cliffs (1983)
Gilbert, J.C., Lemaréchal, C.: Some numerical experiments with variable-storage quasi-Newton algorithms. Math. Program. 45, 407–435 (1989)
Goldstein, A.A., Price, J.F.: An effective algorithm for minimization. Numer. Math. 10, 184–189 (1967)
Hager, W.W., Zhang, H.: Self-adaptive inexact proximal point methods. Comput. Optim. Appl. 39(2), 161–181 (2008)
Humes, C., Silva, P.J.S.: Inexact proximal point algorithms and descent methods in optimization. Optim. Eng. 6, 257–271 (2005)
Lemaréchal, C., Sagastizábal, C.: Variable metric bundle methods: from conceptual to implementable forms. Math. Program. 76(3), 393–410 (1997)
Levenberg, K.: A method for the solution of certain nonlinear problems in least squares. Q. Appl. Math. 2, 164–168 (1944)
Maréchal, P., Rondepierre, A.: A proximal approach to the inversion of ill-conditioned matrices. C. R. Acad. Sci. Paris 347(23–24), 1435–1438 (2009)
Marquardt, D.W.: An algorithm for least-squares estimation of nonlinear parameters. J. Soc. Ind. Appl. Math. 11, 431–441 (1963)
Martinet, B.: Régularisation d’inéquations variationelles par approximation successives. Rev. Fr. Inform. Rech. Oper. R3, 154–158 (1970)
Moré, J.J.: Recent developments in algorithms and software for trust region methods. In: Bachem, A., Grötschel, M., Korte, B. (eds.) Mathematical Programming, the State of the Art, pp. 258–287. Springer, Berlin (1983)
Nocedal, J.: Updating quasi-Newton matrices with limited storage. Math. Comput. 35, 773–782 (1980)
Qi, L., Sun, J.: nonsmooth version of Newton’s method. Math. Program. (1993)
Rockafellar, R.T.: Augmented Lagrangians and applications of the proximal point algorithm in convex programming. Math. Oper. Res. 1, 97–116 (1976)
Solodov, M.V., Svaiter, B.F.: A hybrid projection-proximal point algorithm. J. Convex Anal. 6(1), 059–070 (1999)
Solodov, M.V., Svaiter, B.F.: A unified framework for some inexact proximal point algorithms. Numer. Funct. Anal. Optim. 22, 1013–1035 (2001)
Todd, M.: On convergence properties of algorithms for unconstrained minimization. IMA J. Numer. Anal. 9, 435–441 (1989)
Wolfe, P.: Convergence conditions for ascent methods. SIAM Rev. 11, 226–235 (1969)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Fuentes, M., Malick, J. & Lemaréchal, C. Descentwise inexact proximal algorithms for smooth optimization. Comput Optim Appl 53, 755–769 (2012). https://doi.org/10.1007/s10589-012-9461-3
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10589-012-9461-3