On nonsmooth Optimality theorems for robust optimization problems
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In this paper, we prove a necessary optimality theorem for a nonsmooth optimization problem in the face of data uncertainty, which is called a robust optimization problem. Recently, the robust optimization problems have been intensively studied by many authors. Moreover, we give examples showing that the convexity of the uncertain sets and the concavity of the constraint functions are essential in the optimality theorem. We present an example illustrating that our main assumptions in the optimality theorem can be weakened.
Abstract
In this paper, we prove a necessary optimality theorem for a nonsmooth optimization problem in the face of data uncertainty, which is called a robust optimization problem. Recently, the robust optimization problems have been intensively studied by many authors. Moreover, we give examples showing that the convexity of the uncertain sets and the concavity of the constraint functions are essential in the optimality theorem. We present an example illustrating that our main assumptions in the optimality theorem can be weakened.
참고문헌 (20)
- J. P. Aubin, "Applied Nonlinear Analysis" (1984), John Wiley & Sons, Inc
- A. Beck, "Duality in robust optimization: primal worst equals dual best", Oper. Res. Lett, 37(1), pp.1~6 (2009)
- A. Ben-Tal, "Robust optimization methodology and applications", Math. Program, 92(3, Ser B), pp.453~480 (2002)
- A. Ben-Tal, "Robust Optimization" (2009), Princeton Series in Applied Mathematics
- A. Ben-Tal, "Selected topics in robust convex optimization", Math. Program, 112(1, Ser. B), pp.125~158 (2008)
- D. Bertsimas, "Constructing uncertainty sets for robust linear optimiza-tion", Oper. Res, 57(6), pp.1483~1495 (2009)
- D. Bertsimas, "Robust linear optimization under general norms", Oper. Res. Lett, 32(6), pp.510~516 (2004)
- F. H. Clarke, "Generelized gradients and applications", Trans. Amer. Math. Soc, 205, pp.247~262 (1975)
- F. H. Clarke, "A new approach to Lagrange multipliers", Math. Oper. Res, 1(2), pp.165~174 (1976)
- F. H. Clarke, "Generalized gradients of Lipschitz functionals", Adv. in Math, 40(1), pp.52~67 (1981)
- F. H. Clarke, "Optimization and Nonsmooth Analysis, Classics in Applied Mathematics" (1990), So-ciety for Industrial and Applied Mathematics
- J. M. Danskin, "The Theory of Max-min and Its Application to Weapons Allocation Problems" (1967), Springer-Verlag
- J. B. Hiriart-Urruty, "Tangent cones, generalized gradients and mathematical program-ming in Banach spaces", Math. Oper. Res, 4(1), pp.79~97 (1979)
- V. Jeyakumar, "Characterizing robust set containments and solutions of uncertain linear programs without qualifications", Oper. Res. Lett, 38(3), pp.188~194 (2010)
- V. Jeyakumar, "Robust Farkas’ lemma for uncertain linear systems with applications", Positivity, 15(2), pp.331~342 (2011)
- V. Jeyakumar, "Strong duality in robust convex programming: complete characterizations", SIAM J. Optim, 20(6), pp.3384~3407 (2010)
- V. Jeyakumar, "Robust duality for generalized convex program-ming problems under data uncertainty", Nonlinear Anal, 75(3), pp.1362~1373 (2012)
- G. Lebourg, "Generic differentiability of Lipschitzian functions", Trans. Amer. Math. Soc, 256, pp.125~144 (1979)
- G. Y. Li, "Robust conjugate duality for convex optimization under uncertainty with application to data classification", Nonlinear Anal, 74(6), pp.2327~2341 (2012)
- Olvi L. Mangasarian, "Classics in Applied Mathematics, 10" (1994), Society for Industrial and Applied Mathematics