Abstract
In this paper, we consider the Lagrangian dual problem of a class of convex optimization problems, which originates from multi-stage stochastic convex nonlinear programs. We study the Moreau–Yosida regularization of the Lagrangian-dual function and prove that the regularized function η is piecewise C 2, in addition to the known smoothness property. This property is then used to investigate the semismoothness of the gradient mapping of the regularized function. Finally, we show that the Clarke generalized Jacobian of the gradient mapping is BD-regular under some conditions.
Similar content being viewed by others
References
Chen X., Nashed Z., Qi L.: Smoothing methods and semismooth methods for nondifferentiable operator equations. SIAM J. Numer. Anal. 38, 1200–1216 (2000)
Clarke F.H.: Optimization and Nonsmooth Analysis. Wiley, New York (1983)
Dontchev A.L., Qi H.-D., Qi L.: Convergence of Newton method for convex best interpolation. Numer. Math. 87, 435–456 (2001)
Dontchev A.L., Qi H.-D., Qi L.: Quadratic convergence of Newton method for convex interpolation and smoothing. Constr. Approx. 19, 1230–143 (2003)
Fukushima M., Qi L.: A global and superlinear convergent algorithm for nonsmooth convex minimization. SIAM J. Optim. 6, 1106–1120 (1996)
Hiriart-Urruty J.B., Lemaréchal C.: Convex Analysis and Minimization Algorithms. Springer Verlag, Berlin (1993)
Janin R.: Directional derivative of the marginal function in nonlinear programming. Math. Program. Stud. 21, 110–126 (1984)
Meng F., Hao Y.: The property of piecewise smoothness of Moreau–Yosida approximation for a piecewise C 2 convex function. Adv. Math. (China) 30, 354–358 (2001)
Meng F., Sun D., Zhao G.: Semismoothness of solutions to generalized equations and the Moreau–Yosida regularization. Math. Program. 104, 561–581 (2005)
Meng F., Zhao G., Goh M., Souza R.D.: Lagrangian-dual functions and Moreau–Yosida regularization. SIAM J. Optim. 19, 39–61 (2008)
Mifflin R.: Semismooth and semiconvex functions in constrained optimization. SIAM J. Control Optim. 15, 957–972 (1977)
Mifflin R., Qi L., Sun D.: Properties of the Moreau–Yosida regularization of a piecewise C 2 convex function. Math. Program. 84, 269–281 (1999)
Moreau J.J.: Proximite et dualite dans un espace hilbertien. Bull. Soc. Math. France 93, 273–299 (1965)
Pang J.-S., Ralph D.: Piecewise smoothness, local invertibility, and parametric analysis of normal maps. Math. Oper. Res. 21, 401–426 (1996)
Pang J.-S., Sun D., Sun J.: Semismooth homeomorphisms and strong stability of semidefinite and Lorentz complementarity problems. Math. Oper. Res. 28, 39–63 (2003)
Qi L., Sun J.: A nonsmooth version of Newton’s method. Math. Program. 58, 353–367 (1993)
Qi L., Sun D., Zhou G.: A new look at smoothing Newton methods for nonlinear complementarity problems and box constrained variational inequalities. Math. Program. 87, 1–35 (2000)
Robinson S.M.: Generalized equations and their solutions, Part II: application to nonlinear programming. Math. Program. Stud. 19, 200–211 (1982)
Rockafellar R.T.: Convex Analysis. Princeton, New Jersey (1970)
Ruszczyński, A., Shapiro, A. (eds.): Stochastic Programming. Handbook in Operations Research and Management Science. Elsevier Science, Amsterdam (2003)
Scholtes S.: Introduction to Piecewise Smooth Equations. Habilitation thesis, University of Karlsruhe, Karlsruhe, Germany (1994)
Sun D., Han J.: On a conjecture in Moreau–Yosida regularization of a nonsmooth convex function. Chin. Sci. Bull. 42, 1140–1143 (1997)
Sun D., Sun J.: Semismooth matrix valued functions. Math. Oper. Res. 27, 150–169 (2002)
Yosida K.: Functional Analysis. Springer Verlag, Berlin (1964)
Zhao G.: A Lagrangian dual method with self-concordant barrier for multi-stage stochastic convex nonlinear programming. Math. Program. 102, 1–24 (2005)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Meng, F. Moreau–Yosida regularization of Lagrangian-dual functions for a class of convex optimization problems. J Glob Optim 44, 375–394 (2009). https://doi.org/10.1007/s10898-008-9333-7
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10898-008-9333-7