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Variational pairs and applications to stability in nonsmooth analysis. (English) Zbl 1035.49014
In the paper, it is pointed out that many of the basic results of nonsmooth analysis (e.g., subdifferential calculus, mean-value inequalities and optimality conditions) can be derived directly by general variational principles. In this manner, the authors provide a general definition of a variational pair \((X,\delta)\) where \(X\) is a complete metric space and \(\delta\) associates to any lower semicontinuous function \(f:X\to \mathbb{R}\cup \{+\infty\}\) and any point \(x\in X\), a nonnegative extended real number \(\delta f(x)\). The main property which is required on \(\delta\) is that for every \(\overline{x}\in X\) and for every numbers \(\sigma,r>0\) with \[ f(\overline{x})< \inf_{d(x,\overline{x})< r}f(x)+ \sigma r \] one can find some \(x\in X\) with \(d(x,\overline{x})< r\) and \(\delta f(x)< \sigma\). It is shown that with \[ \delta_1f(x)= \begin{cases} 0, &\text{ if \(x\) is a local minimizer of \(f\)},\\ \limsup_{y\to x} \frac{f(y)-f(x)} {y-x}, &\text{ otherwise}, \end{cases} \] and \[ \delta_2f(x)= d(0,\partial f(x)) \] (where \(\partial\) is a suitable subdifferential operator), \((X,\delta_1)\) and \((X,\delta_2)\) are examples of a variational pair.
The both main theorems of the paper are consequences of the notion of variational pair. The first one can be seen as a general (global) metric regularity result for a lower semicontinuous function. The second one is basically a local and parametric refinement of the previous one, dealing with the local solvability of an inequality of the type \(f(x,p)\leq 0\) with parameter \(p\) of a topological space.
The results are applied for the characterization of so-called asymptotically well-behaved functions, for the discussion of metric regularity properties for parametrized families of multifunctions and for the derivation of a stability assertion for parametrized optimization problems assuming an extended Mangasarian-Fromowitz type constraint qualification.

49J52 Nonsmooth analysis
49J40 Variational inequalities
Full Text: DOI
[1] Attouch, H., Variational convergence for functions and operators, Applicable mathematics series, (1984), Pitman London · Zbl 0561.49012
[2] Attouch, H.; Lucchetti, R.; Wets, R.J.-B., The topology of the ρ-Hausdorff distance, Ann. mat. pura appl., 160, 303-320, (1991) · Zbl 0769.54009
[3] Aubin, J.-P., Lipschitz behavior of solutions to convex minimization problems, Math. oper. res., 9, 87-111, (1984) · Zbl 0539.90085
[4] Aubin, J.-P.; Wets, R.J.-B., Stable approximations of set-valued maps, Ann. inst. H. Poincaré, anal. non linéaire, 5, 519-535, (1988) · Zbl 0681.54012
[5] Auslender, A., Stability in mathematical programming with nondifferentiable data, SIAM J. control optim., 22, 239-254, (1984) · Zbl 0538.49020
[6] Auslender, A.; Crouzeix, J.-P., Well behaved asymptotical convex functions, Ann. inst. H. Poincaré, anal. non linéaire, 6, 101-121, (1989) · Zbl 0675.90070
[7] Aussel, D.; Corvellec, J.-N.; Lassonde, M., Mean value property and subdifferential criteria for lower semicontinuous functions, Trans. amer. math. soc., 347, 4147-4161, (1995) · Zbl 0849.49016
[8] Aussel, D.; Corvellec, J.-N.; Lassonde, M., Nonsmooth constrained optimization and multidirectional Mean value inequalities, SIAM J. optim., 9, 690-706, (1999) · Zbl 0981.49014
[9] Azé, D.; Chou, C.C., On a Newton type iterative method for solving inclusions, Math. oper. res., 20, 790-800, (1995) · Zbl 0860.41030
[10] Azé, D.; Chou, C.C.; Penot, J.-P., Subtraction theorems and approximate openness for multifunctions: topological and infinitesimal viewpoints, J. math. anal. appl., 221, 33-58, (1998) · Zbl 0943.47036
[11] D. Azé, L. Michel, Weak Palais-Smale conditions for convex composite functions with unbounded sublevel sets, preprint 1997.
[12] Beer, G., Topologies on closed and closed convex sets, Mathematics and its applications, Vol. 268, (1993), Kluwer Dordrecht · Zbl 0792.54008
[13] Borwein, J.M., Stability and regular points of inequality systems, J. optim. theory appl., 48, 9-52, (1986) · Zbl 0557.49020
[14] Borwein, J.M.; Preiss, D., A smooth variational principle with applications to subdifferentiability and to differentiability of convex functions, Trans. amer. math. soc., 303, 517-527, (1987) · Zbl 0632.49008
[15] Borwein, J.M.; Zhang, D.M., Verifiable necessary and sufficient conditions for openness and regularity of set-valued and single valued maps, J. math. anal. appl., 134, 441-459, (1988) · Zbl 0654.49004
[16] Borwein, J.M.; Zhu, Q.J., Viscosity solutions and viscosity subderivatives in smooth Banach spaces with applications to metric regularity, SIAM J. control optim., 34, 1576-1591, (1996) · Zbl 0882.49020
[17] J.M. Borwein, Q.J. Zhu, A survey of subdifferential calculus with applications, Nonlinear Analysis, TMA 38A (1999) 687-773. · Zbl 0933.49006
[18] F.H. Clarke, Optimization and Nonsmooth Analysis, 2nd Edition, Classics Applied Mathematics, Vol. 5, SIAM, Philadelphia, PA, 1990 (originally published by Wiley-Interscience, New York, 1983). · Zbl 0582.49001
[19] Clarke, F.H.; Ledyaev, Yu.S., Mean value inequalities in Hilbert space, Trans. amer. math. soc., 344, 307-324, (1994) · Zbl 0803.49018
[20] Clarke, F.H.; Ledyaev, Yu.S.; Stern, R.J.; Wolenski, P.R., Nonsmooth analysis and control theory, (1998), Springer New York · Zbl 0951.49003
[21] Combari, C.; Elhilali Alaoui, A.; Levy, A.; Poliquin, R.; Thibault, L., Convex composite functions in Banach spaces and the primal lower-Nice property, Proc. amer. math. soc., 126, 3701-3708, (1998) · Zbl 0918.58008
[22] Corvellec, J.-N., A note on coercivity of lower semicontinuous functions and nonsmooth critical point theory, Serdica math. J., 22, 57-68, (1996) · Zbl 0863.58013
[23] Dal Maso, G., An introduction to γ-convergence, (1993), Birkhäuser Basel · Zbl 0816.49001
[24] De Giorgi, E.; Marino, A.; Tosques, M., Problemi di evoluzione in spazi metrici e curve di massima pendenza, Atti accad. naz. lincei rend. cl. sci. fis. mat. natur., 68, 180-187, (1980) · Zbl 0465.47041
[25] Degiovanni, M.; Marino, A.; Tosques, M., Evolution equations with lack of convexity, Nonlinear anal. theory methods appl., 9, 1401-1443, (1985) · Zbl 0545.46029
[26] Degiovanni, M.; Marzocchi, M., A critical point theory for nonsmooth functionals, Ann. mat. pura appl., 167, 73-100, (1994) · Zbl 0828.58006
[27] Deville, R.; Godefroy, G.; Zizler, V., A smooth variational principle with applications to Hamilton-Jacobi equations in infinite dimensions, J. funct. anal., 111, 197-212, (1993) · Zbl 0774.49021
[28] Deville, R.; Godefroy, G.; Zizler, V., Smoothness and renormings in Banach spaces, (1993), Longman New York · Zbl 0782.46019
[29] Dolecki, S.; Angleraud, P., When a well behaving function is well-conditioned, J. southeast Asian bull. math., 20, 59-63, (1996) · Zbl 0855.49007
[30] Ekeland, I., Nonconvex minimization problems, Bull. amer. math. soc., 1, 443-474, (1979) · Zbl 0441.49011
[31] Fabian, M., Subdifferentiability and trustworthyness in the light of the new variational principle of Borwein and preiss, Acta univ. carolinae, 30, 51-56, (1989) · Zbl 0714.49022
[32] Hiriart-Urruty, J.-B., Refinements of necessary optimality conditions in non differentiable programming I, Appl. math. optim., 5, 63-82, (1979) · Zbl 0389.90088
[33] Ioffe, A., Regular points of Lipschitz functions, Trans. amer. math. soc., 251, 61-69, (1979) · Zbl 0427.58008
[34] Ioffe, A., Nonsmooth analysis: differential calculus of nondifferentiable mappings, Trans. amer. math. soc., 266, 1-56, (1981) · Zbl 0651.58007
[35] Ioffe, A., On the local surjection property, Nonlinear anal. theory methods appl., 11, 565-592, (1987) · Zbl 0642.49010
[36] Ioffe, A., Approximate subdifferential and applications. III: the metric theory, Mathematika, 36, 1-38, (1989) · Zbl 0713.49022
[37] A. Ioffe, Codirectional compactness, metric regularity and subdifferential calculus, preprint, 1996. · Zbl 0966.49014
[38] Ioffe, A., Fuzzy principles and characterization of trustworthiness, Set-valued anal., 6, 265-276, (1998) · Zbl 0927.49009
[39] A. Ioffe, Variational Methods in Local and Global Nonsmooth Analysis, Lecture Notes NATO ASI & Séminaire de Mathématiques Supérieures, “Nonlinear Analysis, differential equations and control”, Montreal, Canada, 1998, Kluwer. Dordrecht, NATO ASI Ser., Ser. C, Math. Phys. Sci. 528 (1999) 447-502.
[40] Ioffe, A.; Penot, J.-P., Subdifferentials of performance functions and calculus of coderivatives of set-valued mappings, Serdica math. J., 22, 257-282, (1996)
[41] Jourani, A.; Thibault, L., Verifiable conditions for openness and regularity of multivalued mappings, Trans. amer. math. soc., 347, 1255-1268, (1995) · Zbl 0827.54013
[42] M. Lassonde, First order rules for nonsmooth constrained optimization, Nonlinear Anal., Theory Methods Appl. 44 (8) (2001) 1031-1056. · Zbl 1033.49030
[43] Yu.S. Ledyaev, Q.J. Zhu, Implicit multifunction theorems, preprint, 1998.
[44] Mordukhovich, B.S., Metric approximations and necessary optimality conditions for general classes of nonsmooth extremal problems, Sov. math. dokl., 22, 526-530, (1980) · Zbl 0491.49011
[45] Mordukhovich, B.S.; Shao, Y., Differential characterizations of covering, metric regularity and Lipschitzian properties of multifunctions, Nonlinear anal. theory methods appl., 25, 1401-1428, (1995) · Zbl 0863.47030
[46] Mordukhovich, B.S.; Shao, Y., Nonsmooth sequential analysis in asplund spaces, Trans. amer. math. soc., 34, 1235-1280, (1996) · Zbl 0881.49009
[47] Mordukhovich, B.S.; Shao, Y., Stability of set-valued mappings in infinite dimensions: point criteria and applications, SIAM J. control optim., 35, 295-314, (1997) · Zbl 0895.49011
[48] Penot, J.-P., Metric regularity, openness and Lipschitzian behavior of multifunctions, Nonlinear anal. theory methods appl., 13, 629-643, (1989) · Zbl 0687.54015
[49] Penot, J.-P., Well-behavior, well-posedness and nonsmooth analysis, Pliska stud. math. bulgar., 12, 141-190, (1998) · Zbl 0946.49019
[50] Rockafellar, R.T., Generalized directional derivatives and subgradients of nonconvex functions, Canad. J. math, 32, 257-280, (1980) · Zbl 0447.49009
[51] Rockafellar, R.T., Proximal subgradients, marginal values and augmented Lagrangians in nonconvex optimization, Math. oper. res., 6, 424-436, (1981) · Zbl 0492.90073
[52] Zhu, Q.J., Clarke – ledyaev Mean value inequalities in smooth Banach spaces, Nonlinear anal. theory methods appl., 32, 315-324, (1998) · Zbl 0952.49014
[53] Zhu, Q.J., The equivalence of several basic theorems on subdifferentials, Set-valued anal., 6, 171-185, (1998) · Zbl 0917.49017
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