×
García, F.; Melguizo Padial, M. A.

Paratingent derivative applied to the measure of the sensitivity in multiobjective differential programming. (English) Zbl 1470.90116

Abstr. Appl. Anal. 2013, Article ID 812125, 11 p. (2013).
Summary: We analyse the sensitivity of differential programs of the form \(\operatorname{Min} f(x)\) subject to \(g(x) = b, x \in D\) where \(f\) and \(g\) are \(\mathcal{C}^1\) maps whose respective images lie in ordered Banach spaces. Following previous works on multiobjective programming, the notion of \(T\)-optimal solution is used. The behaviour of some nonsingleton sets of \(T\)-optimal solutions according to changes of the parameter \(b\) in the problem is analysed. The main result of the work states that the sensitivity of the program is measured by a Lagrange multiplier plus a projection of its derivative. This sensitivity is measured by means of the paratingent derivative.

MSC:

90C29 Multi-objective and goal programming
49J52 Nonsmooth analysis
90C48 Programming in abstract spaces
PDF BibTeX XML Cite
\textit{F. García} and \textit{M. A. Melguizo Padial}, Abstr. Appl. Anal. 2013, Article ID 812125, 11 p. (2013; Zbl 1470.90116)
Full Text: DOI

References:

[1] Mordukhovich, B. S., Variational Analysis and Generalized Differentiation II. Applications. Variational Analysis and Generalized Differentiation II. Applications, A Series of Comprehensive Studies in Mathematics, 331 (2006), Berlin, Germany: Springer, Berlin, Germany
[2] Aubin, J.-P.; Frankowska, H., Set-Valued Analysis (1990), Boston, Mass, USA: Birkhäuser, Boston, Mass, USA
[3] Aubin, J.-P.; Frankowska, H., Controlability and Observability of Control Systems under Uncertainty (1989), Vienna, Austria: International Institute for Applied Systems Analysis, Vienna, Austria
[4] Balbás, A.; Jiménez Guerra, P., Sensitivity analysis for convex multiobjective programming in abstract spaces, Journal of Mathematical Analysis and Applications, 202, 2, 645-658 (1996) · Zbl 0856.90126
[5] Balbás, A.; Jiménez Guerra, P.; Caballero, R.; Ruiz, F., Sensitivity in multiobjective programming by differential equations methods. The case of homogeneous functions, Advances in Multiple Objective and Goal Programming. Advances in Multiple Objective and Goal Programming, Lecture Notes in Economics and Mathematical Systems, 455, 188-196 (1997), Berlin, Germany: Springer, Berlin, Germany
[6] Balbás, A.; Ballve, M. E.; Jiménez Guerra, P., Sensitivity and optimality conditions in the multiobjective differential programming, Indian Journal of Pure and Applied Mathematics, 29, 7, 671-680 (1998) · Zbl 0915.90225
[7] Balbás, A.; Ballvé, M.; Jiménez Guerra, P., Sensitivity in multiobjective programming under homogeneity assumptions, Journal of Multi-Criteria Decision Analysis, 8, 3, 133-138 (1999) · Zbl 0948.90128
[8] Balbás, A.; Fernández, F. J.; Jiménez Guerra, P., On the envolvent theorem in multiobjective programming, Indian Journal of Pure and Applied Mathematics, 26, 11, 1035-1047 (1995) · Zbl 0844.90072
[9] Gal, T.; Wolf, K., Stability in vector maximization. A survey, European Journal of Operational Research, 25, 2, 169-182 (1986) · Zbl 0589.90077
[10] García, F.; Melguizo, M. A., Sensitivity analysis in convex optimization through the circatangent derivative · Zbl 1327.90283
[11] Jiménez Guerra, P.; Melguizo, M. A.; Muñoz-Bouzo, M. J., Sensitivity analysis in convex programming, Computers & Mathematics with Applications, 58, 6, 1239-1246 (2009) · Zbl 1189.90166
[12] Tanino, T., Sensitivity analysis in multiobjective optimization, Journal of Optimization Theory and Applications, 56, 3, 479-499 (1988) · Zbl 0619.90073
[13] Tanino, T., Stability and sensitivity analysis in convex vector optimization, SIAM Journal on Control and Optimization, 26, 3, 521-536 (1988) · Zbl 0654.49011
[14] Tanino, T.; Gal, T.; Stewart, T.; Hanne, T., Sensitivity analysis in MCDM, Multicriteria Decision Making: Advances in MCDM Models, Algorithms, Theory, and Applications, 7.1-7.29 (1999), Boston, Mass, USA: Kluwer Academic, Boston, Mass, USA · Zbl 0946.90081
[15] Chuong, T. D.; Yao, J. C., Generalized Clarke epiderivatives of parametric vector optimization problems, Journal of Optimization Theory and Applications, 146, 1, 77-94 (2010) · Zbl 1206.90158
[16] Chuong, T. D.; Yao, J.-C.; Yen, N. D., Further results on the lower semicontinuity of efficient point multifunctions, Pacific Journal of Optimization, 6, 2, 405-422 (2010) · Zbl 1357.49098
[17] Chuong, T. D., Clarke coderivatives of efficient point multifunctions in parametric vector optimization, Nonlinear Analysis. Theory, Methods & Applications, 74, 1, 273-285 (2011) · Zbl 1207.49031
[18] Jiménez Guerra, P.; Melguizo Padial, M. A., Sensitivity analysis in differential programming through the Clarke derivative, Mediterranean Journal of Mathematics, 9, 3, 537-550 (2012) · Zbl 1274.90350
[19] Jiménez Guerra, P.; Melguizo, M. A.; Muñoz-Bouzo, M. J., Sensitivity analysis in multiobjective differential programming, Computers & Mathematics with Applications, 52, 1-2, 109-120 (2006) · Zbl 1190.90188
[20] Zemin, L., The optimality conditions of differentiable vector optimization problems, Journal of Mathematical Analysis and Applications, 201, 1, 35-43 (1996) · Zbl 0851.90105
[21] Aubin, J.-P.; Cellina, A., Differential Inclusions Set-Valued Maps and Viability Theory (1984), Berlin, Germany: Springer, Berlin, Germany · Zbl 0538.34007
[22] Boudjenah, L., Existence of solutions to a paratingent equation with delayed argument, Electronic Journal of Differential Equations, 2005, 14, 1-8 (2005) · Zbl 1075.34054
[23] Gorre, A., Evolutions of tubes under operability constraints, Journal of Mathematical Analysis and Applications, 216, 1, 1-22 (1997) · Zbl 1008.49004
[24] Arnaud, M.-C., The link between the shape of the irrational Aubry-Mather sets and their Lyapunov exponents, Annals of Mathematics, 174, 3, 1571-1601 (2011) · Zbl 1257.37028
[25] Tierno, G., The paratingent space and a characterization of \(C^1\)-maps defined on arbitrary sets, Journal of Nonlinear and Convex Analysis, 1, 2, 129-154 (2000) · Zbl 0970.58007
[26] Shi, S. Z., Choquet theorem and nonsmooth analysis, Journal de Mathématiques Pures et Appliquées, 67, 4, 411-432 (1988) · Zbl 0679.49015
[27] Balbás, A.; Ballvé, M.; Jiménez Guerra, P., Density theorems for ideal points in vector optimization, European Journal of Operational Research, 133, 2, 260-266 (2001) · Zbl 1001.90060
[28] Rudin, W., Functional Analysis (1991), New York, NY, USA: McGraw-Hill, New York, NY, USA · Zbl 0867.46001
[29] Borwein, J. M.; Lewis, A. S., Convex Analysis and Nonlinear Optimization. Theory and Examples, Canadian Mathematical Society (2006), New York, NY, USA: Springer, New York, NY, USA · Zbl 1116.90001
[30] Luc, D. T.; Dien, P. H., Differentiable selection of optimal solutions in parametric linear programming, Proceedings of the American Mathematical Society, 125, 3, 883-892 (1997) · Zbl 0861.90117
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.