Abstract
In general Banach space setting, we study the perturbed distance function \({d_S^J(\cdot)}\) determined by a closed subset S and a lower semicontinuous function J (·). In particular, we show that the Fréchet subdifferential and the proximal subdifferential of a perturbed distance function are representable by virtue of corresponding normal cones of S and subdifferentials of J (·).
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References
Baranger J.: Existence de solution pour des problemes d’optimisation nonconvexe. C. R. Acad. Sci. Paris. 274, 307–309 (1972) (in French)
Baranger J.: Existence de solutions pour des problems d’optimisation non-convexe. J. Math. Pures Appl. 52, 377–405 (1973) (in French)
Baranger J., Temam R.: Problemes d’optimisation non-convexe dependants d’un parametre. In: Aubin, J.P. (eds) Analyse Non-convexe et ses Applications, pp. 41–48. Springer-Verlag, Berlin/New York (1974) (in French)
Baranger J, Temam R.: Nonconvex optimization problems depending on a parameter. SIAM J. Control 13, 146–152 (1975)
Bidaut M.F.: Existence theorems for usual and approximate solutions of optimal control problem. J. Optim. Theory Appl. 15, 393–411 (1975)
Bounkhel M., Thibault L.: On various notions of regularity of sets in nonsmooth analysis. Nonlinear Anal. 48, 223–246 (2002)
Burke J.V.: A sequential quadratic programming method for potentially infeasible mathematical programs. J. Math. Anal. Appl. 139, 319–351 (1989)
Burke, J.V.: An Exact Penalization Viewpoint of Constrained Optimization. Technical Report ANL/MCS-TM-95, Mathematics and Computer Science Division, Argonne National Laboratories, Argonne, IL 60439 (1987)
Burke J.V., Ferris M.C., Qian M.: On the Clarke subdifferential of the distance function of a closed set. J. Math. Anal. Appl. 166, 199–213 (1992)
Burke J.V., Han S.P.: A Gauss-Newton approach to solving generalized inequalities. Math. Oper. Res. 11, 632–643 (1986)
Clarke F.H.: Optimization and Nonsmooth Analysis. Wiley, New York (1983)
Clarke F.H., Ledyaev Y.S., Stern R.J., Wolenski P.R.: Nonsmooth Analysis and Control Theory. Springer, New York (1998)
Clarke F.H., Stern R.J., Wolenski P.R.: Proximal smoothness and the lower-C 2 property. J. Convex. Anal. 2, 117–144 (1995)
Cobzas S.: Nonconvex optimization problems on weakly compact subsets of Banach spaces. Anal. Numér. Théor. Approx. 9, 19–25 (1980)
Cobzas S.: Generic existence of solutions for some perturbed optimization problems. J. Math. Anal. Appl. 243, 344–356 (2000)
Colombo G., Wolenski P.R.: Variational analysis for a class of minimal time functions in Hilbert spaces. J. Convex. Anal. 11, 335–361 (2004)
Colombo G., Wolenski P.R.: The subgradient formula for the minimal time function in the case of constant dynamics in Hilbert space. J. Global Optim. 28, 269–282 (2004)
Dontchev A.L., Zolezzi T.: Well Posed Optimization Problems. Lecure Notes in Math, Vol. 1543. Springer-Verlag, Berlin (1993)
He Y., Ng K.F.: Subdifferentials of a minimum time function in Banach spaces. J. Math. Anal. Appl. 321, 896–910 (2006)
Ioffe A.D.: Proximal analysis and approximate subdifferentials. J. Lond. Math. Soc. 2, 175–192 (1990)
Kruger, A.Y.: ε-semidifferentials and ε-normal elements, Depon. VINITI, No. 1331-81, Moscow 1981 (in Russian)
Lebourg G.: Perturbed optimization problems in Banach spaces. Bull. Soc. Math. Fr. 60, 95–111 (1979)
Li C., Peng L.H.: Porosity of perturbed optimization problems in Banach spaces. J. Math. Anal. Appl. 324, 751–761 (2006)
Mordukhovich B.S.: Variational Analysis and Generalized Differentiation. I: Basic Theory, Grundlehren Series (Fundamental Principles of Mathematical Sciences), Vol. 330. Springer-Verlag, Berlin (2006)
Mordukhovich B.S.: Variational Analysis and Generalized Differentiation. II: Applications, Grundlehren Series (Fundamental Principles of Mathematical Sciences), Vol. 331. Springer-Verlag, Berlin (2006)
Mordukhovich, B.S., Nam, N.M.: Limiting subgradients of minimal time functions in Banach spaces. J. Global Optim. (to appear)
Murat F.: Un contre-example pour le problem du controle dans les coefficients. C. R. Acad. Sci. Ser. A 273, 708–711 (1971) (in French)
Ni R.X.: Generic solutions for some perturbed optimization problem in non-reflexive Banach space. J. Math. Anal. Appl. 302, 417–424 (2005)
Peng L.H., Li C., Yao J.C.: Well-posedness of a class of perturbed optimization problems in Banach spaces. J. Math. Anal. Appl. 346, 384–394 (2008)
Stechkin S.B.: Approximative properties of sets in linear normed spaces. Rev. Math. Pures. Appl. 8, 5–18 (1963) (in Russian)
Wolenski P.R., Zhuang Y.: Proximal analysis and the minimal time function. SIAM J. Control Optim. 36, 1048–1072 (1998)
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Wang, JH., Li, C. & Xu, HK. Subdifferentials of perturbed distance functions in Banach spaces. J Glob Optim 46, 489–501 (2010). https://doi.org/10.1007/s10898-009-9433-z
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DOI: https://doi.org/10.1007/s10898-009-9433-z
Keywords
- Subdifferential
- Fréchet subdifferential
- Proximal subdifferential
- Perturbed optimization problem
- Well-posedness