×

Quantitative stability of variational systems. III: \(\varepsilon\)- approximate solutions. (English) Zbl 0802.49009

Starting from the usual problem of solving numerically convex optimization problems the authors analyse data dependence of the algorithms. Continuity with respect to suitable distance between objective functions and sets is considered.

MSC:

49J45 Methods involving semicontinuity and convergence; relaxation
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] H. Attouch,Variational Convergence for Functions and Operators. Applicable Mathematics Series (Pitman, London, 1984). · Zbl 0561.49012
[2] H. Attouch, R. Lucchetti and R.J.-B. Wets ”The topology of the{\(\rho\)}-Hausdorff distance,”Annali di Matematica pura ed applicata CLX (1991), 303–320. · Zbl 0769.54009
[3] H. Attouch and R.J.-B. Wets, ”Isometries for the Legendre–Fenchel transform,”Transactions of the American Mathematical Society 296 (1986) 33–60. · Zbl 0607.49009
[4] H. Attouch and R.J.-B. Wets, ”Another isometry for the Legendre–Frenchel transform,”Journal of Mathematical Analysis and Applications 131 (1988) 404–411. · Zbl 0644.49010
[5] H. Attouch and R.J.-B. Wets, ”Quantitative stability of variational systems: I. The epigraphical distance,”Transactions of the American Mathematical Society 328 (1991) 695–729. · Zbl 0753.49007
[6] H. Attouch and R.J.-B. Wets, ”Quantitative stability of variational systems: II. A framework for nonlinear conditioning,” to appear in:SIAM Journal on Optimization. · Zbl 0793.49005
[7] D. Azé and J.-P. Penot, ”Recent quantitative results about the convergence of convex sets and functions,” in: P.L. Papini, ed.,Functional Analysis and Approximations. Proceedings of the International Conference Bagni di Lucca. May 1988 (Pitagora Editrice, Bologna, 1990) pp. 90–110.
[8] G. Beer and R. Lucchetti, ”Convex optimization and the epi-distance topology,” to appear in:Transactions of the American Mathematical Society (1992). · Zbl 0767.49011
[9] G. Beer and R. Lucchetti, ”The epi-distance topology: continuity and stability results with applications to convex optimization,”Mathematics of Operations Research 17 (1992), 715–726. · Zbl 0767.49011
[10] Y. Ermoliev and A. Gaivoronski, ”Simultaneous nonstationary optimization, estimation and approximation procedures,” IIASA CP-82-16 (Laxenburg, Austria, 1982).
[11] A. Gaivoronski,Study of Nonstationary Stochastic Programming Problems (Institute of Cybernetics Press, Kiev, 1979).
[12] J.-B. Hiriat-Urruty, ”Lipschitz r-continuity of the approximate subdifferential of a convex function,”Matematica Scandinavia 47 (1980) 123–134. · Zbl 0426.26005
[13] E. Nurminskii, ”Continuity of{\(\epsilon\)}-subgradient mappings,”Kibernetika (Kiev)5 (1978) 148–149. · Zbl 0392.26008
[14] R.T. Rockafellar and R.J.-B. Wets, ”Variational systems, an introduction,” in: G. Salinetti, ed.,Multifunctions and Integrands: Stochastic Analysis, Approximation and Optimization. Lecture Notes in Mathematics No. 1091 (Springer, Berlin, 1984) pp. 1–54.
[15] G. Salinetti and R.J.-B. Wets, ”On the convergence of sequence of convex sets in finite dimensions,”SIAM Review 21 (1979) 16–33. · Zbl 0421.52003
[16] P. Shunmugaraj, ”On stability aspects in optimization,” Doctoral Thesis, Indian Institute of Technology (Bombay, 1990).
[17] P. Shunmugaraj and D.V. Pai, ”On stability of approximate solutions of minimization problems,” Manuscript, Indian Institute of Technology (Bombay, 1990). · Zbl 0772.49011
[18] D. Walkup and R.J.-B. Wets, ”A Lipschitzian characterization of convex polyhedra,”Proceedings American Mathematical Society 23 (1969) 167–173. · Zbl 0182.25003
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.