Abstract
In this chapter we study several classes of symmetric optimization problems which are identified with the corresponding spaces of objective functions, equipped with appropriate complete metrics. Using the Baire category approach, for any of these classes, we show the existence of subset of the space of functions, which is a countable intersection of open and everywhere dense sets, such that for every objective function from this intersection the corresponding symmetric optimization problem possesses a solution. These results are obtained as realizations of a general variational principle which is established in this chapter. We extend these results for certain classes of symmetric optimization problems using a porosity notion. We identify a class of symmetric minimization problems with a certain complete metric space of functions, study the set of all functions for which the corresponding minimization problem has a solution, and show that the complement of this set is not only of the first category but also a σ-porous set.
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References
Aubin JP, Ekeland I (1984) Applied nonlinear analysis. Wiley Interscience, New York
Boulos W, Reich S (2015) Porosity results for two-set nearest and farthest point problems. Rend Circ Mat Palermo 2:493–507
Ioffe AD, Zaslavski AJ (2000) Variational principles and well-posedness in optimization and calculus of variations. SIAM J Control Optim 38:566–581
Li C (2000) On well posed generalized best approximation problems. J Approx Theory 107:96–108
Marcus M, Zaslavski AJ (1999) The structure of extremals of a class of second order variational problems. Ann Inst H Poincaré Anal Non Linéaire 16:593–629
Mizel VJ, Zaslavski AJ (2004) Anisotropic functions: a genericity result with crystallographic implications. ESAIM Control Optim Calc Var 10:624–633
Peng L, Li C (2014) Porosity and fixed points of nonexpansive set-valued maps. Set Valued Var Anal 22:333–348
Peng L, Li C, Yao JC (2015) Porosity results on fixed points for nonexpansive set-valued maps in hyperbolic spaces. J Math Anal Appl 428:989–1004
Planiden C, Wang X (2016) Most convex functions have unique minimizers. J Convex Anal 23:877–892
Planiden C, Wang X (2016) Strongly convex functions, Moreau envelopes, and the generic nature of convex functions with strong minimizers. SIAM J Optim 26:1341–1364
Reich S, Zaslavski AJ (2014) Genericity in nonlinear analysis. Springer, New York
Vanderwerff J (2020) On the residuality of certain classes of convex functions. Pure Appl Funct Anal 5:791–806
Wang X (2013) Most maximally monotone operators have a unique zero and a super-regular resolvent. Nonlinear Anal 87:69–82
Zaslavski AJ (1995) Optimal programs on infinite horizon 1. SIAM J Control Optim 33:1643–1660
Zaslavski AJ (1995) Optimal programs on infinite horizon 2. SIAM J Control Optim 33:1661–1686
Zaslavski AJ (2006) Turnpike properties in the calculus of variations and optimal control. Springer, New York
Zaslavski AJ (2010) Optimization on metric and normed spaces. Springer, New York
Zaslavski AJ (2013) Nonconvex optimal control and variational problems. Springer Optimization and Its Applications, New York
Zaslavski AJ (2020) Generic existence of solutions of symmetric optimization problems. Symmetry 12(12):2004. https://doi.org/10.3390/sym12122004
Zaslavski AJ (2021) Generic well-posedness of symmetric minimization problems. Appl Anal Optim 5:343–356
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Zaslavski, A. (2022). Symmetric Optimization Problems. In: Turnpike Phenomenon and Symmetric Optimization Problems. Springer Optimization and Its Applications, vol 190. Springer, Cham. https://doi.org/10.1007/978-3-030-96973-8_2
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DOI: https://doi.org/10.1007/978-3-030-96973-8_2
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