×

zbMATH — the first resource for mathematics

Minimizing and stationary sequences of convex constrained minimization problems. (English) Zbl 0987.90067
Summary: In the asymptotic analysis of the minimization problem for a nonsmooth convex function on a closed convex set \(X\) in \(\mathbb{R}^n\), one can consider the corresponding problem of minimizing a smooth convex function \(F\) on \(\mathbb{R}^n\), where \(F\) denotes the Moreau-Yosida regularization of \(f\). We study the interrelationship between the minimizing/stationary sequence for \(f\) and that for \(F\). An algorithm is given to generate iteratively a possibly unbounded sequence, which is shown to be a minimizing sequence of \(f\) under certain regularity and uniform continuity assumptions.

MSC:
90C25 Convex programming
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Fukushima, M., and Qi, L., A Globally and Superlinearly Convergent Algorithm for Nonsmooth Convex Minimization, SIAM Journal on Optimization, Vol. 6, pp. 1106–1120, 1996. · Zbl 0868.90109 · doi:10.1137/S1052623494278839
[2] Mifflin, R., Sun, D., and Qi, L., Quasi-Newton Bundle-Type Methods for Nondifferentiable Convex Optimization, SIAM Journal on Optimization, Vol. 8, pp. 583–603, 1998. · Zbl 0927.65074 · doi:10.1137/S1052623496303329
[3] Rockafellar, R. T., Monotone Operators and the Proximal Point Algorithm, SIAM Journal on Control and Optimization, Vol. 14, pp. 877–898, 1976. · Zbl 0358.90053 · doi:10.1137/0314056
[4] Chou, C. C., Ng, K. F., and Pang, J. S., Minimizing and Stationary Sequences of Constrained Optimization Problems, SIAM Journal on Control and Optimization, Vol. 36, pp. 1908–1936, 1998. · Zbl 0909.90235 · doi:10.1137/S0363012995292548
[5] Hiriart-Urruty, J. B., and LemarÉchal, C., Convex Analysis and Minimization Algorithms, Vols. 1 and 2, Springer Verlag, Berlin, Germany, 1993. · Zbl 0795.49001
[6] Pang, J. S., Error Bounds in Mathematical Programming, Mathematical Programming, Vol. 79, pp. 299–332, 1997. · Zbl 0887.90165
[7] Auslender, A., Numerical Methods for Nondifferentiable Convex Optimization, Mathematical Programming Study, Vol. 30, pp. 102–126, 1987. · Zbl 0616.90052 · doi:10.1007/BFb0121157
[8] Correa, R., and LemarÉchal, C., Convergence of Some Algorithms for Convex Minimization, Mathematical Programming, Vol. 62, pp. 261–275, 1993. · Zbl 0805.90083 · doi:10.1007/BF01585170
[9] Fukushima, M., A Descent Algorithm for Nonsmooth Convex Optimization, Mathematical Programming, Vol. 30, pp. 163–175, 1984. · Zbl 0545.90082 · doi:10.1007/BF02591883
[10] Wei, Z., and Qi, L., Convergence Analysis of a Proximal Newton Method, Numerical Functional Analysis and Optimization, Vol. 17, pp. 463–472, 1996. · Zbl 0884.90123 · doi:10.1080/01630569608816705
[11] Polyak, B. T., Introduction to Optimization, Optimization Software Publications Division, New York, NY, 1987.
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.