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.


90C25 Convex programming
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[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
[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
[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
[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
[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
[8] Correa, R., and LemarÉchal, C., Convergence of Some Algorithms for Convex Minimization, Mathematical Programming, Vol. 62, pp. 261–275, 1993. · Zbl 0805.90083
[9] Fukushima, M., A Descent Algorithm for Nonsmooth Convex Optimization, Mathematical Programming, Vol. 30, pp. 163–175, 1984. · Zbl 0545.90082
[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
[11] Polyak, B. T., Introduction to Optimization, Optimization Software Publications Division, New York, NY, 1987.
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