×

zbMATH — the first resource for mathematics

From scalar to vector optimization. (English) Zbl 1164.90399
Summary: Initially, second-order necessary optimality conditions and sufficient optimality conditions in terms of Hadamard type derivatives for the unconstrained scalar optimization problem \(\varphi (x)\to \min \), \(x\in \mathbb R^m\), are given. These conditions work with arbitrary functions \(\varphi :\mathbb R^m \to \overline {\mathbb R}\), but they show inconsistency with the classical derivatives. This is a base to pose the question whether the formulated optimality conditions remain true when the “inconsistent” Hadamard derivatives are replaced with the “consistent” Dini derivatives. It is shown that the answer is affirmative if \(\varphi \) is of class \({\mathcal C}^{1,1}\) (i.e.,  differentiable with locally Lipschitz derivative).
Further, considering \({\mathcal C}^{1,1}\) functions, the discussion is raised to unconstrained vector optimization problems. Using the so called “oriented distance” from a point to a set, we generalize to an arbitrary ordering cone some second-order necessary conditions and sufficient conditions given by L. Liu, P. Neittaanmäki and M. Křížek for a polyhedral cone. Furthermore, we show that the conditions obtained are sufficient not only for efficiency but also for strict efficiency.

MSC:
90C29 Multi-objective and goal programming
90C30 Nonlinear programming
49J52 Nonsmooth analysis
PDF BibTeX XML Cite
Full Text: DOI EuDML Link
References:
[1] B. Aghezzaf: Second-order necessary conditions of the Kuhn-Tucker type in multiobjective programming problems. Control Cybern. 28 (1999), 213–224. · Zbl 0946.90075
[2] V. M. Alekseev, V. M. Tikhomirov, S. V. Fomin: Optimal Control. Consultants Bureau, New York, 1987 (Russian original: Optimal’noe upravlenie. Publ. Nauka, Moscow, 1979).
[3] T. Amahroq, A. Taa: On Lagrange-Kuhn-Tucker multipliers for multiobjective optimization problems. Optimization 41 (1997), 159–172. · Zbl 0882.90114
[4] A. Auslender: Stability in mathematical programming with nondifferentiable data. SIAM J. Control Optimization 22 (1984), 239–254. · Zbl 0538.49020
[5] E. Bednarczuk, W. Song: PC points and their application to vector optimization. PLISKA, Stud. Math. Bulg. 12 (1998), 21–30. · Zbl 0949.90085
[6] S. Bolintineanu, M. El Maghri: Second-order efficiency conditions and sensitivity of efficient points. J. Optimization Theory Appl. 98 (1998), 569–592. · Zbl 0915.90226
[7] M. Ciligot-Travain: On Lagrange-Kuhn-Tucker multipliers for Pareto optimization problems. Numer. Funct. Anal. Optimization 15 (1994), 689–693. · Zbl 0831.49021
[8] G. P. Crespi, I. Ginchev, M. Rocca: Minty variational inequality, efficiency and proper efficiency. Vietnam J. Math. 32 (2004), 95–107. · Zbl 1056.49009
[9] V. F. Demyanov, A. M. Rubinov: Constructive Nonsmooth Analysis. Peter Lang, Frankfurt am Main, 1995. · Zbl 0887.49014
[10] I. Ginchev:: Higher order optimality conditions in nonsmooth optimization. Optimization 51 (2002), 47–72. · Zbl 1011.49014
[11] I. Ginchev, A. Guerraggio, M. Rocca: Equivalence on (n+1)-th order Peano and usual derivatives for n-convex functions. Real Anal. Exch. 25 (2000), 513–520. · Zbl 1009.26008
[12] I. Ginchev, A. Guerraggio, M. Rocca: On second-order conditions in vector optimization. Preprint 2002/32, Universita dell’Insubria, Facolta di Economia, Varese 2002, http://eco.uninsubria.it/dipeco/Quaderni/files/QF2002_32.pdf.
[13] I. Ginchev, A. Guerraggio, M. Rocca: First-order conditions for C 0,1 constrained vector optimization. In: Variational Analysis and Applications (F. Giannessi, A. Maugeri, eds.). Springer-Verlag, New York, 2005, pp. 427–450. · Zbl 1148.90011
[14] I. Ginchev, A. Hoffmann: Approximation of set-valued functions by single-valued one. Discuss. Math., Differ. Incl., Control Optim. 22 (2002), 33–66. · Zbl 1039.90051
[15] A. Guerraggio, D. T. Luc: Optimality conditions for C 1,1 vector optimization problems. J. Optimization Theory Appl. 109 (2001), 615–629. · Zbl 1038.49027
[16] J.-B. Hiriart-Urruty, J.-J. Strodiot, V. Hien Nguen: Generalized Hessian matrix and second order optimality conditions for problems with C 1,1 data. Appl. Math. Optimization 11 (1984), 43–56. · Zbl 0542.49011
[17] J.-B. Hiriart-Urruty: New concepts in nondifferentiable programming. Analyse non convexe, Bull. Soc. Math. France 60 (1979), 57–85. · Zbl 0469.90071
[18] J.-B. Hiriart-Urruty: Tangent cones, generalized gradients and mathematical programming in Banach spaces. Math. Oper. Res. 4 (1979), 79–97. · Zbl 0409.90086
[19] D. Klatte, K. Tammer: On second order sufficient optimality conditions for C 1,1-optimization problems. Optimization 19 (1988), 169–179. · Zbl 0647.49014
[20] L. Liu: The second-order conditions of nondominated solutions for C 1,1 generalized multiobjective mathematical programming. Syst. Sci. Math. Sci. 4 (1991), 128–138. · Zbl 0734.90078
[21] L. Liu, M. Krizek: The second order optimality conditions for nonlinear mathematical programming with C 1,1 data. Appl. Math. 42 (1997), 311–320. · Zbl 0903.90152
[22] L. Liu, P. Neittaanmaki, M. Krizek: Second-order optimality conditions for nondominated solutions of multiobjective programming with C 1,1 data. Appl. Math. 45 (2000), 381–397. · Zbl 0995.90085
[23] D. T. Luc: Theory of Vector Optimization. Springer Verlag, Berlin, 1988. · Zbl 0654.90082
[24] D. T. Luc: Taylor’s formula for C k,1 functions. SIAM J. Optim. 5 (1995), 659–669. · Zbl 0852.49012
[25] E. Miglierina: Characterization of solutions of multiobjective optimization problems. Rendiconti Circ. Mat. Palermo 50 (2001), 153–164. · Zbl 1072.90041
[26] E. Miglierina, E. Molho: Scalarization and stability in vector optimization. J. Optimization Theory Appl. 114 (2002), 657–670. · Zbl 1026.90080
[27] G. Peano: Sulla formola di Taylor. Atti Accad. Sci. Torino 27 (1891), 40–46. · JFM 23.0253.01
[28] R. T. Rockafellar: Convex Analysis. Princeton University Press, Princeton, 1970.
[29] X. Yang: Second-order conditions in C 1,1 optimization with applications. Numer. Funct. Anal. Optimization 14 (1993), 621–632. · Zbl 0804.90114
[30] X. Q. Yang: Generalized second-order derivatives and optimality conditions. Nonlinear Anal. 23 (1994), 767–784. · Zbl 0816.49008
[31] X. Q. Yang, V. Jeyakumar: Generalized second-order directional derivatives and optimization with C 1,1 functions. Optimization 26 (1992), 165–185. · Zbl 0814.49012
[32] A. Zaffaroni: Degrees of efficiency and degrees of minimality. SIAM J. Control Optimization 42 (2003), 1071–1086. · Zbl 1046.90084
[33] C. Zalinescu: On two notions of proper efficiency. In: Optimization in Mathematical Physics, Pap. 11th Conf. Methods Techniques Math. Phys., Oberwolfach (Brokowski and Martensen, eds.). Peter Lang, Frankfurt am Main, 1987.
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.