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Characterizing robust solution sets of convex programs under data uncertainty. (English) Zbl 1307.90136
Summary: This paper deals with convex optimization problems in the face of data uncertainty within the framework of robust optimization. It provides various properties and characterizations of the set of all robust optimal solutions of the problems. In particular, it provides generalizations of the constant subdifferential property as well as the constant Lagrangian property for solution sets of convex programming to robust solution sets of uncertain convex programs. The paper shows also that the robust solution sets of uncertain convex quadratic programs and sum-of-squares convex polynomial programs under some commonly used uncertainty sets of robust optimization can be expressed as conic representable sets. As applications, it derives robust optimal solution set characterizations for uncertain fractional programs. The paper presents several numerical examples illustrating the results.

90C25 Convex programming
90C20 Quadratic programming
90C46 Optimality conditions and duality in mathematical programming
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[1] Burke, JV; Ferris, M, Characterization of solution sets of convex programs, Oper. Res. Lett., 10, 57-60, (1991) · Zbl 0719.90055
[2] Castellani, M; Giuli, M, A characterization of the solution set of pseudoconvex extremum problems, J. Convex Anal., 19, 113-123, (2012) · Zbl 1232.47040
[3] Jeyakumar, V; Lee, GM; Dinh, N, Lagrange multiplier conditions characterizing optimal solution sets of cone-constrained convex programs, J. Optim. Theory Appl., 123, 83-103, (2004) · Zbl 1114.90091
[4] Jeyakumar, V; Lee, GM; Dinh, N, Characterizations of solution sets of convex vector minimization problems, Eur. J. Oper. Res., 174, 1380-1395, (2006) · Zbl 1103.90090
[5] Jeyakumar, V; Yang, XQ, On characterizing the solution sets of pseudolinear programs, J. Optim. Theory Appl., 87, 747-755, (1995) · Zbl 0840.90118
[6] Lalitha, CS; Mehta, M, Characterizations of solution sets of mathematical programs in terms of Lagrange multipliers, Optimization, 58, 995-1007, (2009) · Zbl 1175.90373
[7] Mangasarian, OL, A simple characterization of solution sets of convex programs, Oper. Res. Lett., 7, 21-26, (1988) · Zbl 0653.90055
[8] Penot, JP, Characterization of solution sets of quasiconvex programs, J. Optim. Theory Appl., 117, 627-636, (2003) · Zbl 1043.90071
[9] Son, TQ; Dinh, N, Characterizations of optimal solution sets of convex infinite programs, TOP, 16, 147-163, (2008) · Zbl 1201.90158
[10] Zhao, KQ; Yang, XM, Characterizations of the solution set for a class of nonsmooth optimization problems, Optim. Lett., 7, 685-694, (2013) · Zbl 1269.90112
[11] Wu, ZL; Wu, SY, Characterizations of the solution sets of convex programs and variational inequality problems, J. Optim. Theory Appl., 130, 339-358, (2006) · Zbl 1152.90565
[12] Ben-Tal, A., Ghaoui, L.E., Nemirovski, A.: Robust Optimization. Princeton University Press, Princeton and Oxford (2009) · Zbl 1221.90001
[13] Bertsimas, D; Brown, D; Caramanis, C, Theory and applications of robust optimization, SIAM Rev., 53, 464-501, (2011) · Zbl 1233.90259
[14] Beck, A; Ben-Tal, A, Duality in robust optimization: primal worst equals dual best, Oper. Res. Lett., 37, 1-6, (2009) · Zbl 1154.90614
[15] Goberna, MA; Jeyakumar, V; Li, G; Lopez, M, Robust linear semi-infinite programming duality, Math. Program Ser. B, 139, 185-203, (2013) · Zbl 1282.90204
[16] Jeyakumar, V; Li, G, Characterizing robust set containments and solutions of uncertain linear programs without qualifications, Oper. Res. Lett., 38, 188-194, (2010) · Zbl 1220.90067
[17] Jeyakumar, V; Li, G, Strong duality in robust convex programming: complete characterizations, SIAM J. Optim., 20, 3384-3407, (2010) · Zbl 1228.90075
[18] Jeyakumar, V; Li, G; Wang, JH, Some robust convex programs without a duality gap, J. Convex Anal., 20, 377-394, (2013) · Zbl 1277.90086
[19] Jeyakumar, V; Wang, JH; Li, G, Lagrange multiplier characterizations of robust best approximations under constraint data uncertainty, J. Math. Anal. Appl., 393, 285-297, (2012) · Zbl 1308.41028
[20] Ben-Tal, A., Nemirovski, A.: Lectures on Modern Convex Optimization: Analysis, Algorithms and Engineering Applications. MPS-SIAM Series on Optimization. SIAM, Philadelphia (2001) · Zbl 0986.90032
[21] Jeyakumar, V; Li, G, Exact SDP relaxations for classes of nonlinear semidefinite programming problems, Oper. Res. Lett., 40, 529-536, (2012) · Zbl 1287.90047
[22] Jeyakumar, V., Vicente-Perez, J.: Dual semidefinite programming duals without duality gaps for a class of convex minimax programs. J. Optim. Theory Appl. doi:10.1007/s10957-013-0496-0 · Zbl 1312.90087
[23] Jeyakumar, V., Li, G., Vicente-Perez, J.: Robust SOS-convex polynomial programs: exact SDP relaxations. Optim. Lett. (2014). doi:10.1007/s11590-014-0732-z · Zbl 1338.90452
[24] Boyd, S., Vandenberghe, L.: Convex Optimization. Cambridge University Press, Cambridge (2004) · Zbl 1058.90049
[25] Rockafellar, R.T.: Convex Analysis. Princeton University Press, Princeton (1970) · Zbl 0193.18401
[26] Zalinescu, C.: Convex Analysis in General Vector Spaces. World Scientific, Singapore (2002) · Zbl 1023.46003
[27] Ahmadi, AA; Parrilo, PA, A convex polynomial that is not SOS-convex, Math. Program, 135, 275-292, (2012) · Zbl 1254.90159
[28] Helton, JW; Nie, J, Semidefinite representation of convex sets, Math. Program Ser. A, 112, 21-64, (2010) · Zbl 1192.90143
[29] Goldfarb, D; Iyengar, G, Robust convex quadratically constrained programs, Math. Program Ser. B, 515, 97-495, (2003) · Zbl 1106.90365
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