×

Optimization and flow invariance via first and second order tangent cones. (English) Zbl 1243.49030

Summary: This is a survey paper on optimization problems via the technique of first and second order tangent cones to a nonempty subset of a Banach space X. Such a technique is also used in the study of the flow invariance of a closed set with respect to a second order differential equation (motion on a given orbit in a force field). Many of the known results in these areas are included here.

MSC:

49K27 Optimality conditions for problems in abstract spaces
90C48 Programming in abstract spaces
90C30 Nonlinear programming
PDFBibTeX XMLCite
Full Text: Euclid

References:

[1] V.M. Alekseev, V.M. Tikhomirov and S.V. Fomin, Optimal control , Consultants Bureau, New York 1987.
[2] J.P. Aubin and I. Ekeland, Applied nonlinear analysis , Wiley-Interscience, New York 1984. · Zbl 0641.47066
[3] J.P. Aubin and H. Frankowska, Set-valued analysis , Birkhäuser, Boston 1990.
[4] H. Brézis, On a characterization of flow-invariant sets. Commun. Pure Appl. Math. 23 (1970), 261-263. · Zbl 0191.38703 · doi:10.1002/cpa.3160230211
[5] V. Barbu, Optimal Control of Variational Inequalities , Pitman, Boston, London, Melbourne, Vol. 100, 1984. · Zbl 0574.49005
[6] D.P. Bertsekas, Nonlinear programming , Second Edition, Athena Scientific, Belmont Massachusetts 1995. · Zbl 0935.90037
[7] G. Bouligand, Sur les surfaces dépourvues de points hyperlimités. Ann. Soc. Polon. Math. , 9 (1930), 32-41.
[8] F.H. Clarke, Optimization and nonsmooth Analysis , John Wiley&Sons, New York 1983. · Zbl 0582.49001
[9] E. Constantin, Higher order necessary and sufficient sonditions for optimality. PanAmer. Math.J. 14 (3) (2004), 1-25. · Zbl 1064.49023
[10] E. Constantin, An application of higher order tangent cones to flow invariance. Int. J. Appl. Math. Sci. 2 (2005), 121-129. · Zbl 1095.34037
[11] E. Constantin, Second order necessary optimality conditions based on second order tangent cones. Math. Sci. Res. J. 10 (2006), 42-56. · Zbl 1267.68265 · doi:10.1007/978-3-642-10210-3_25
[12] E. Constantin, I. Raykov and N.H. Pavel, Optimization problems in some critical cases via tangential cones. Adv. Math. Sci. Appl. 18 (2008), 251-267. · Zbl 1154.49019
[13] E. Constantin, Higher order necessary conditions in smooth constrained optimization. Communicating Mathematics, Contemp. Math. 479 (2009), 41-49. · Zbl 1180.90307 · doi:10.1090/conm/479/09341
[14] M.G. Crandall, A generalization of Peano’s existence theorem and flow-invariance. Proc. Am. Math. Soc. 36 (1972), 151-155. · Zbl 0271.34084 · doi:10.2307/2039051
[15] G. Giorgi, A. Guerraggio and J. Thierfelder, Mathematics of optimization: smooth and nonsmooth case , Elsevier, Amsterdam 2004. · Zbl 1140.90003
[16] I.V. Girsanov, Lectures on mathematical theory of extremum problems , Spring-Verlag, Berlin\(\cdot \) Heidelberg\(\cdot \)New York 1972. · Zbl 0234.49016
[17] U. Ledzewicz and H. Schaettler, Second order conditions for extremum problems with nonregularity equality constraints. J. Optim. Theory Appl. 86 (1995), 113-144. · Zbl 0835.49016 · doi:10.1007/BF02193463
[18] U. Ledzewicz and H. Schaettler, A high-order generalization of the Lyusternik theorem. Nonlinear Anal. 34 (1998), 793-815. · Zbl 0954.49014 · doi:10.1016/S0362-546X(98)00001-7
[19] H. Lee and N. H. Pavel, Higher Order Optimality Conditions and Applications. PanAmerican Math. J , 14 (4) (2004), 11-24. · Zbl 1070.90128
[20] R. H. Jr. Martin, Differential equations on closed subsets of a Banach space. Trans. Am. Math. Soc. 179 (1973), 399-414. · Zbl 0293.34092 · doi:10.2307/1996511
[21] D. Motreanu and N.H. Pavel, Flow-invariance for second order differential equations on manifolds and orbital motions. Boll. U.M.I. 1-B (1987), 943-964. · Zbl 0641.58044
[22] D. Motreanu and N. H. Pavel, Tangency, flow invariance for differential equations and optimization problems , Monographs and Textbooks in Pure and Appl. Mathematics, Vol.219 Marcel Dekker, New York- Basel 1999. · Zbl 0937.34035
[23] M. Nagumo, Überdie dage der integralkurven gevöhnlicherö differentialgleichungen. Proc. Phys. Math. Soc. Japan 24 (1942), 551-559. · Zbl 0061.17204
[24] Z. Páles and V.M. Zeidan, Nonsmooth optimum problems with constraints. SIAM J. Control Optim. 32 (1994), 1476-1502. · Zbl 0821.49020 · doi:10.1137/S0363012992229653
[25] F. Pang, Finite-dimensional variational inequalities and complementarity problems , Vol.I, Springer Series in Operational Research 2003. · Zbl 1062.90001 · doi:10.1007/b97543
[26] N.H. Pavel and F. Iacob, Invariant sets for a class of perturbed differential equations of retarted type. Israel J. Math. 28 (1977), 254-264. · Zbl 0366.34057 · doi:10.1007/BF02759812
[27] N.H.Pavel, Second order differential equations on closed sets of a Banach space. Boll. U.M.I. 12 (1975), 348-353.
[28] N.H. Pavel, Invariant sets for a class of semilinear equations of evolution. Nonlin. Anal. TMA 1 (1977), 187-196.
[29] N.H. Pavel and C. Ursescu, Flow-invariant sets for autonomous second order differential equations an applications in Mechanics. Nonlin. Anal. TMA 6 (1982), 35-74. · Zbl 0478.34039 · doi:10.1016/0362-546X(82)90100-6
[30] N.H. Pavel, Differential Equations, flow-invariance and applications. Research Notes in Math. No. 113, Pitman, Boston, London, Melbourne 1984. · Zbl 0593.34003
[31] N.H. Pavel, J.K. Huang and J.K. Kim, Higher order necessary conditions for optimization. Libertas Math. 14 (1994), 41-50. · Zbl 0819.49016
[32] N.H. Pavel and F. Potra, Flow invariance and nonlinear optimization problems via first and second order generalized tangential cones. Analele Stiintifice ale Universitatii ”AL.I.CUZA” IASI Tomul LI, s.I, Matematica, 2005, f.2, 281-292. · Zbl 1164.90419
[33] P. Michel and J.-P. Penot, Calcul sous-différentiel pour des fonctions lipschitziennes et non lipschitziennes. C. R. Acad. Sci. Paris Sér. I Math. 12 (1984), 269-272. · Zbl 0567.49008
[34] C. Ursescu, Tangent sets’ calculus and necessary conditions for extramality. SIAM J. Control Optim. 20 (1982), 563-574. · Zbl 0488.49009 · doi:10.1137/0320041
[35] A.A. Tret’yakov, Necessary and sufficient conditions for optimality of \(p\)-th order. U.S.S.R. Comput. Maths. Math. Phys. 24 (1984), 123-127. · Zbl 0546.53036
[36] J.J. Ye, Nondifferentiable multiplier rules for optimization and bilevel optimization problems. SIAM J. Optim. 15 (2004), 252-274. · Zbl 1077.90077 · doi:10.1137/S1052623403424193
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.