A concept of inner prederivative for set-valued mappings and its applications

Geoffroy, Michel H.; Marcelin, Yvesner

ESAIM: Control, Optimisation and Calculus of Variations, Tome 24 (2018) no. 3, pp. 1059-1074.

We introduce a class of positively homogeneous set-valued mappings, called inner prederivatives, serving as first order approximants to set-valued mappings. We prove an inverse mapping theorem involving such prederivatives and study their stability with respect to variational perturbations. Then, taking advantage of their properties we establish necessary optimality conditions for the existence of several kind of minimizers in set-valued optimization. As an application of these last results, we consider the problem of finding optimal allocations in welfare economics. Finally, to emphasize the interest of our approach, we compare the notion of inner prederivative to the related concepts of set-valued differentiation commonly used in the literature.

[1] J.-P. Aubin, Contingent derivatives of set-valued maps and existence of solutions to nonlinear inclusions and differential inclusions. Mathematical analysis and applications, Part A, Advances in Mathematics Supplementary Studies 7A. Academic Press, New York-London (1981) 159–229 | MR | Zbl

[2] J.-P. Aubin, H. Frankowska, On inverse function theorems for set-valued maps. J. Math. Pures Appl. 66 (1987) 71–89 | MR | Zbl

[3] J.-P. Aubin and H. Frankowska, Set-valued analysis. Systems & Control: Foundations & Applications, 2. Birkhäuser Boston, Inc., Boston, MA (1990) | MR | Zbl

[4] D. Azé, An inversion theorem for set-valued maps. Bull. Austral. Math. Soc. 37 (1988) 411–414 | DOI | MR | Zbl

[5] T.Q. Bao and B.S. Mordukhovich, Relative Pareto minimizers for multiobjective problems: existence and optimality conditions. Math. Program. 122 (2010) 301–347 | DOI | MR | Zbl

[6] T.Q. Bao, B.S. Mordukhovich, Set-valued optimization in welfare economics. In Vol. 13 of Advances in Mathematical Economics. Springer (2010) 113–153 | DOI | MR | Zbl

[7] J. Borwein and R. Goebel, Notions of relative interior in Banach spaces, Optimization and related topics. J. Math. Sci. (NY) 115 (2003) 2542–2553 | DOI | MR | Zbl

[8] J. Borwein and A.S. Lewis, Partially finite convex programming. II. Explicit lattice models. Math. Program. 57 (1992) 49–83 | DOI | MR | Zbl

[9] H.W. Corley, Optimality conditions for maximizations of set-valued functions. J. Optim. Theory Appl. 58 (1988) 1–10 | DOI | MR | Zbl

[10] A. Daniilidis and C.H. Pang, Continuity of set-valued maps revisited in the light of tame geometry. J. Lond. Math. Soc. 83 (2011) 637–658 | DOI | MR | Zbl

[11] U. Dini, Analisi infinitesimale, Lezioni dettate nella R. Universitá di Pisa (1877/78) | JFM

[12] A.L. Dontchev and R.T. Rockafellar, Implicit functions and solution mappings. A view from variational analysis. Springer Monographs in Mathematics. Springer, Dordrecht (2009) | DOI | MR | Zbl

[13] A.L. Dontchev, M. Quincampoix and N. Zlateva, Aubin criterion for metric regularity. J. Convex Anal. 13 (2006) 281–297 | MR | Zbl

[14] B. Fisher, Common fixed points of mappings and set-valued mappings. Rostock Math. Kolloq. 18 (1981) 69–77 | MR | Zbl

[15] H. Frankowska, Inclusions adjointes associées aux trajectoires minimales d’inclusions différentielles. C. R. Acad. Sci. Paris Sr. I Math. 297 (1983) 461–464 | MR | Zbl

[16] M. Gaydu, M.H. Geoffroy and C. Jean-Alexis, An inverse mapping theorem for H-differentiable set-valued maps. J. Math. Anal. Appl. 421 (2015) 298–313 | DOI | MR | Zbl

[17] M. Gaydu, M.H. Geoffroy and Y. Marcelin, Prederivatives of convex set-valued maps and applications to set optimization problems. J. Global Optim. 64 (2016) 141–158 | DOI | MR | Zbl

[18] E. Goursat, Sur la théorie des fonctions implicites. Bull. Soc. Math. France 31 (1903) 184–192 | DOI | JFM | Numdam | MR

[19] R.B. Holmes, Geometric Functional Analysis and Its Applications. Graduate Texts in Mathematics. Springer-Verlag, New-York (1975) | DOI | MR | Zbl

[20] A. Ioffe, Nonsmooth analysis: differential calculus of nondifferentiable mappings. Trans. Amer. Math. Soc. 266 (1981) 1–56 | DOI | MR | Zbl

[21] J. Jahn and R. Rauh, Contingent epiderivatives and set-valued optimization. Math. Methods Oper. Res. 46 (1997) 193–211 | DOI | MR | Zbl

[22] D.T. Luc, Theory of Vector Optimization. In Vol. 319 of Lecture Notes in Economics and Mathematical Systems. Springer (1989) | DOI | MR | Zbl

[23] D.T. Luc, Contingent derivatives of set-valued maps and applications to vector optimization. Math. Programming 50 (1991) 99–111 | DOI | MR | Zbl

[24] B. Mordukhovich, Variational analysis and generalized differentiation. I. Basic theory. In Vol. 330 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer-Verlag, Berlin (2006) | MR | Zbl

[25] K. Nachi and J.-P. Penot, Inversion of multifunctions and differential inclusions. Control Cybernet 34 (2005) 871–901 | MR | Zbl

[26] J.W. Nieuwenhuis, Separating sets with relative interior in Fréchet spaces. Appl. Math. Optim. 3 (1976/77) 373–376 | DOI | MR

[27] Z. Páles, Linear selections for set-valued functions and extension of bilinear forms. Arch. Math. (Basel) 62 (1994) 427–432 | DOI | MR | Zbl

[28] C.H.J. Pang, Generalized differentiation with positively homogeneous maps: applications in set-valued analysis and metric regularity. Math. Oper. Res. 36 (2011) 377–397 | DOI | MR | Zbl

[29] J. Ponstein, Approaches to the theory of optimization. In Vol. 77 of Cambridge Tracts in Mathematics. Cambridge University Press, Cambridge-New York (1980) | MR | Zbl

[30] H. Radström, An embedding theorem for spaces of convex sets. Proc. Amer. Math. Soc. 3 (1952) 165–169 | DOI | MR | Zbl

[31] R.T. Rockafellar and R. Wets, Variational analysis. In Vol. 317 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer-Verlag, Berlin (1998) | DOI | MR | Zbl

[32] A. Smajdor and W. Smajdor, Affine selections of convex set-valued functions. Aequationes Math. 51 (1996) 12–20 | DOI | MR | Zbl