## Abstract and Applied Analysis

### Paratingent Derivative Applied to the Measure of the Sensitivity in Multiobjective Differential Programming

#### Abstract

We analyse the sensitivity of differential programs of the form $\text{Min}f(x)$ subject to $g(x)=b,x\in D$ where $f$ and $g$ are ${\mathrm{\scr C}}^{1}$ 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.

#### Article information

Source
Abstr. Appl. Anal., Volume 2013, Special Issue (2012), Article ID 812125, 11 pages.

Dates
First available in Project Euclid: 26 February 2014

https://projecteuclid.org/euclid.aaa/1393442738

Digital Object Identifier
doi:10.1155/2013/812125

Mathematical Reviews number (MathSciNet)
MR3068869

Zentralblatt MATH identifier
07095380

#### Citation

García, F.; Melguizo Padial, M. A. Paratingent Derivative Applied to the Measure of the Sensitivity in Multiobjective Differential Programming. Abstr. Appl. Anal. 2013, Special Issue (2012), Article ID 812125, 11 pages. doi:10.1155/2013/812125. https://projecteuclid.org/euclid.aaa/1393442738

#### References

• B. S. Mordukhovich, Variational Analysis and Generalized Differentiation II. Applications, vol. 331 of A Series of Comprehensive Studies in Mathematics, Springer, Berlin, Germany, 2006.
• J.-P. Aubin and H. Frankowska, Set-Valued Analysis, Birkhäuser, Boston, Mass, USA, 1990.
• J.-P. Aubin and H. Frankowska, Controlability and Observability of Control Systems under Uncertainty, International Institute for Applied Systems Analysis, Vienna, Austria, 1989.
• A. Balbás and P. Jiménez Guerra, “Sensitivity analysis for convex multiobjective programming in abstract spaces,” Journal of Mathematical Analysis and Applications, vol. 202, no. 2, pp. 645–658, 1996.
• A. Balbás and P. Jiménez Guerra, “Sensitivity in multiobjective programming by differential equations methods. The case of homogeneous functions,” in Advances in Multiple Objective and Goal Programming, R. Caballero and F. Ruiz, Eds., vol. 455 of Lecture Notes in Economics and Mathematical Systems, pp. 188–196, Springer, Berlin, Germany, 1997.
• A. Balbás, M. E. Ballve, and P. Jiménez Guerra, “Sensitivity and optimality conditions in the multiobjective differential programming,” Indian Journal of Pure and Applied Mathematics, vol. 29, no. 7, pp. 671–680, 1998.
• A. Balbás, M. Ballvé, P. Jiménez Guerra et al., “Sensitivity in multiobjective programming under homogeneity assumptions,” Journal of Multi-Criteria Decision Analysis, vol. 8, no. 3, pp. 133–138, 1999.
• A. Balbás, F. J. Fernández, and P. Jiménez Guerra, “On the envolvent theorem in multiobjective programming,” Indian Journal of Pure and Applied Mathematics, vol. 26, no. 11, pp. 1035–1047, 1995.
• T. Gal and K. Wolf, “Stability in vector maximization. A survey,” European Journal of Operational Research, vol. 25, no. 2, pp. 169–182, 1986.
• F. García and M. A. Melguizo, “Sensitivity analysis in convex optimization through the circatangent derivative,” submitted to Journal of Optimization Theory and Applications.
• P. Jiménez Guerra, M. A. Melguizo, and M. J. Muñoz-Bouzo, “Sensitivity analysis in convex programming,” Computers & Mathematics with Applications, vol. 58, no. 6, pp. 1239–1246, 2009.
• T. Tanino, “Sensitivity analysis in multiobjective optimization,” Journal of Optimization Theory and Applications, vol. 56, no. 3, pp. 479–499, 1988.
• T. Tanino, “Stability and sensitivity analysis in convex vector optimization,” SIAM Journal on Control and Optimization, vol. 26, no. 3, pp. 521–536, 1988.
• T. Tanino, “Sensitivity analysis in MCDM,” in Multicriteria Decision Making: Advances in MCDM Models, Algorithms, Theory, and Applications, T. Gal, T. Stewart, and T. Hanne, Eds., pp. 7.1–7.29, Kluwer Academic, Boston, Mass, USA, 1999.
• T. D. Chuong and J. C. Yao, “Generalized Clarke epiderivatives of parametric vector optimization problems,” Journal of Optimization Theory and Applications, vol. 146, no. 1, pp. 77–94, 2010.
• T. D. Chuong, J.-C. Yao, and N. D. Yen, “Further results on the lower semicontinuity of efficient point multifunctions,” Pacific Journal of Optimization, vol. 6, no. 2, pp. 405–422, 2010.
• T. D. Chuong, “Clarke coderivatives of efficient point multifunctions in parametric vector optimization,” Nonlinear Analysis. Theory, Methods & Applications, vol. 74, no. 1, pp. 273–285, 2011.
• P. Jiménez Guerra and M. A. Melguizo Padial, “Sensitivity analysis in differential programming through the Clarke derivative,” Mediterranean Journal of Mathematics, vol. 9, no. 3, pp. 537–550, 2012.
• P. Jiménez Guerra, M. A. Melguizo, and M. J. Muñoz-Bouzo, “Sensitivity analysis in multiobjective differential programming,” Computers & Mathematics with Applications, vol. 52, no. 1-2, pp. 109–120, 2006.
• L. Zemin, “The optimality conditions of differentiable vector optimization problems,” Journal of Mathematical Analysis and Applications, vol. 201, no. 1, pp. 35–43, 1996.
• J.-P. Aubin and A. Cellina, Differential Inclusions Set-Valued Maps and Viability Theory, Springer, Berlin, Germany, 1984.
• L. Boudjenah, “Existence of solutions to a paratingent equation with delayed argument,” Electronic Journal of Differential Equations, vol. 2005, no. 14, pp. 1–8, 2005.
• A. Gorre, “Evolutions of tubes under operability constraints,” Journal of Mathematical Analysis and Applications, vol. 216, no. 1, pp. 1–22, 1997.
• M.-C. Arnaud, “The link between the shape of the irrational Aubry-Mather sets and their Lyapunov exponents,” Annals of Mathematics, vol. 174, no. 3, pp. 1571–1601, 2011.
• G. Tierno, “The paratingent space and a characterization of ${C}^{1}$-maps defined on arbitrary sets,” Journal of Nonlinear and Convex Analysis, vol. 1, no. 2, pp. 129–154, 2000.
• S. Z. Shi, “Choquet theorem and nonsmooth analysis,” Journal de Mathématiques Pures et Appliquées, vol. 67, no. 4, pp. 411–432, 1988.
• A. Balbás, M. Ballvé, and P. Jiménez Guerra, “Density theorems for ideal points in vector optimization,” European Journal of Operational Research, vol. 133, no. 2, pp. 260–266, 2001.
• W. Rudin, Functional Analysis, McGraw-Hill, New York, NY, USA, 2nd edition, 1991.
• J. M. Borwein and A. S. Lewis, Convex Analysis and Nonlinear Optimization. Theory and Examples, Canadian Mathematical Society, Springer, New York, NY, USA, 2nd edition, 2006.
• D. T. Luc and P. H. Dien, “Differentiable selection of optimal solutions in parametric linear programming,” Proceedings of the American Mathematical Society, vol. 125, no. 3, pp. 883–892, 1997.