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Typical convex program is very well posed. (English) Zbl 1082.49030
Summary: In this paper we consider the collection of convex programming problems with inequality and equality constraints, in which every problem of the collection is obtained by linear perturbations of the cost function and right-hand side perturbation of the constraints, while the “core” cost function and the left-hand side constraint functions are kept fixed. The main result shows that the set of the problems which are not well-posed is $$\sigma$$-porous in a certain strong sense. Our results concern both the infinite and finite dimensional case. In the last case the conclusions are significantly sharper.

##### MSC:
 49K40 Sensitivity, stability, well-posedness 90C25 Convex programming 90C31 Sensitivity, stability, parametric optimization
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##### References:
 [1] Asplund, E., Rockafellar, R.T.: Gradients of convex functions. Trans. Amer. Math. Soc. 139, 443–467 (1969) · Zbl 0181.41901 · doi:10.1090/S0002-9947-1969-0240621-X [2] Borwein, J.M.: Partially monotone operators and generic differentiability of concave-convex and biconvex functions. Israel J. Math. 54, 42–50 (1986) · Zbl 0609.46018 · doi:10.1007/BF02764875 [3] Coban, M.M., Kenderov, P.S., Revalski, J.P.: Generic well-posedness of optimization problems in topological spaces. Mathematika 36, 301–324 (1989) · Zbl 0679.49010 · doi:10.1112/S0025579300013152 [4] De Blasi, F.S., Myjak, J., Papini, P.L.: Porous sets in best approximation theory. J. London Math. Soc. 44, 135–142 (1991) · Zbl 0786.41027 · doi:10.1112/jlms/s2-44.1.135 [5] Deville, R., Godefroy, G., Zizler, V.: Smoothness and renormings in Banach spaces. Pitman Monographs and Surveys in Pure and Appl. Math. Longman Scientific & Technical, 1993 · Zbl 0782.46019 [6] Deville, R., Revalski, J.P.: Porosity of ill-posed problems. Proc. Amer. Math. Soc. 128, 1117–1124 (2000) · Zbl 0956.46015 · doi:10.1090/S0002-9939-99-05091-1 [7] Dontchev, A.L., Zolezzi, T.: Well-posed optimization problems. Lectures Notes in Mathematics Vol. 1543, Springer-Verlag, Berlin, 1993 · Zbl 0797.49001 [8] Ioffe, A.D., Lucchetti, R.: Generic existence, uniqueness and stability in optimization, In: Nonlinear Optimization and Related Topics. G. Di Pillo and F. Giannessi (eds.), Kluwer Academic Publishers, Dordrecht, 2000, pp. 169–182 · Zbl 0972.90090 [9] Ioffe, A.D., Lucchetti, R., Revalski, J.P.: A variational principle for problems with functional constraints, SIAM J. Optimization 12, 461–478 (2001) · Zbl 1023.49021 [10] Ioffe, A.D., Zaslavski, A.J.: Variational principles and well-posedness in optimization and calculus of variations. SIAM J. Control Optimization 38, 566–581 (2000) · Zbl 0997.49023 · doi:10.1137/S0363012998335632 [11] Lucchetti, R., Zolezzi, T.: On well–posedness and stability analysis in optimization. In: Mathematical Programming with data perturbations, A.V. Fiacco Ed. M. Dekker, New York, 1997, pp. 223–252 · Zbl 0891.90149 [12] Mignot, F.: Contrôle dans les inéquations variationnelles elliptiques. J. Funct. Analysis 22, 130–185 (1976) · Zbl 0364.49003 · doi:10.1016/0022-1236(76)90017-3 [13] Phelps, R.R.: Convex Functions, Monotone Operators and Differentiability. Lectures Notes in Mathematics Vol. 1364, Springer-Verlag, Berlin, 1992 · Zbl 0921.46039 [14] Preiss, D., Zajíček, L.: Fréchet differentiation of convex functions in a Banach space with separable dual. Proc. Amer. Math. Soc. 91, 202–204 (1984) · Zbl 0521.46034 [15] Rockafellar, R.T.: Saddle points and convex analysis, in Differential games and related topics. H.W. Kuhn and G.P.Szegö eds. North Holland, 1971, pp. 109-128 [16] Spingarn, J.E., Rockafellar, R.T.: The generic nature of optimality conditions in nonlinear programming. Math. Oper. Res. 4, 425–430 (1979) · Zbl 0423.90071 · doi:10.1287/moor.4.4.425 [17] Zajíček, L.: Porosity and $$\sigma$$-porosity. Real Anal. Exchange 13, 314–350 (1987) · Zbl 0666.26003 [18] Zaslavski, A.: Well-posedness and porosity in convex programming. Nonlinear An. Forum 8, 101-110 (2003) · Zbl 1259.49034 [19] Zolezzi, T.: Well-posedness criteria in optimization with application to the Calculus of variations. Nonlinear Anal., TMA 25, 437–453 (1995) · Zbl 0841.49005 · doi:10.1016/0362-546X(94)00142-5 [20] Zolezzi, T.: Extended well-posedness of optimization problems. J. Optim. Theory and Appl. 91, 57–268 (1996) · Zbl 0873.90094 · doi:10.1007/BF02192292
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