Abstract
In this paper, we deal with exact semidefinite programming (SDP) reformulations for a class of adjustable robust quadratic optimization problems with affine decision rules. By virtue of a special semidefinite representation of the non-negativity of separable non-convex quadratic functions on box uncertain sets, we establish an exact SDP reformulation for this adjustable robust quadratic optimization problem on spectrahedral uncertain sets. Note that the spectrahedral uncertain set contains commonly used uncertain sets, such as ellipsoids, polytopes, and boxes. As special cases, we also establish exact SDP reformulations for this adjustable robust quadratic optimization problems when the uncertain sets are ellipsoids, polytopes, and boxes, respectively. As applications, we establish the corresponding results for fractionally adjustable robust quadratic optimization problems.
Similar content being viewed by others
References
Akbay, M.A., Kalayci, C.B., Polat, O.: A parallel variable neighborhood search algorithm with quadratic programming for cardinality constrained portfolio optimization. Knowl. Based Syst. 198, 105944 (2020)
Al-Sultan, K.S., Murty, K.G.: Exterior point algorithms for nearest points and convex quadratic programs. Math. Program. 57, 145–161 (1992)
Ben-Tal, A., Nemirovski, A.: Robust solutions of uncertain linear programs. Oper. Res. 25, 1–13 (1999)
Ben-Tal, A., Nemirovski, A.: Robust optimization-methodology and applications. Math. Program. 92, 453–480 (2002)
Ben-Tal, A., Goryashko, A., Guslitzer, E., Nemirovski, A.: Adjustable robust solutions of uncertain linear programs. Math. Program. 99, 351–376 (2004)
Ben-Tal, A., Ghaoui, L.E., Nemirovski, A.: Robust Optimization. Princeton University Press, Princeton (2009)
Bertsimas, D., de Ruiter, F.J.: Duality in two-stage adaptive linear optimization: faster computation and stronger bounds. Inf. J. Comput. 28, 500–511 (2016)
Chen, X., Zhang, Y.: Uncertain linear programs: extended affinely adjustable robust counterparts. Oper. Res. 57, 1469–1482 (2009)
Chen, J.W., Köbis, E., Yao, J.C.: Optimality conditions and duality for robust nonsmooth multiobjective optimization problems with constraints. J. Optim. Theory Appl. 181, 411–436 (2019)
Chen, J.W., Li, J., Li, X.B., Lv, Y.B., Yao, J.C.: Radius of robust feasibility of system of convex inequalities with uncertain data. J. Optim. Theory Appl. 184, 384–399 (2020)
Chuong, T.D., Jeyakumar, V.: Generalized Farkas lemma with adjustable variables and two-stage robust linear programs. J. Optim. Theory Appl. 187, 488–519 (2020)
Chuong, T.D., Jeyakumar, V., Li, G., Woolnough, D.: Exact SDP reformulations of adjustable robust linear programs with box uncertainties under separable quadratic decision rules via SOS representations of non-negativity. J. Global Optim. 81, 1095–1117 (2021)
Chuong, T.D., Mak-Hau, V.H., Yearwood, J., Dazeley, R., Nguyen, M.T., Cao, T.: Robust Pareto solutions for convex quadratic multiobjective optimization problems under data uncertainty. Ann. Oper. Res. 319, 1533–1564 (2022)
Chuong, T.D., Jeyakumar, V., Li, G., Woolnough, D.: Exact dual semi-definite programs for affinely adjustable robust SOS-convex polynomial optimization problems. Optimization 71, 3539–3569 (2022)
Fang, S., Tsao, H.S.J.: An unconstrained convex programming approach to solving convex quadratic programming problems. Optimization 27, 235–243 (1993)
Friedlander, M.P., Orban, D.: A primal-dual regularized interior-point method for convex quadratic programming. Math. Program. Comput. 4, 71–107 (2012)
Gabrel, V., Murat, C., Thiele, A.: Recent advances in robust optimization: an overview. Eur. J. Oper. Res. 235, 471–483 (2014)
Grant, M., Boyd, S.: CVX: Matlab software for disciplined convex programming, version 2.1. http://cvxr.com/cvx (2014)
Jeyakumar, V., Li, G.: Exact second-order cone programming relaxations for some nonconvex minimax quadratic optimization problems. SIAM J. Optim. 28, 760–787 (2018)
Jiao, L., Lee, J.H.: Fractional optimization problems with support functions: exact SDP relaxations. Linear Nonlinear Anal. 5, 255–268 (2019)
Köbis, E.: On robust optimization: relations between scalar robust optimization and unconstrained multicriteria optimization. J. Optim. Theory Appl. 167, 969–984 (2015)
Lee, J.H., Jiao, L.: Solving fractional multicriteria optimization problems with sum of squares convex polynomial data. J. Optim. Theory Appl. 176, 428–455 (2018)
Liu, P., Fattahi, S., Gómez, A., Küçükyavuz, S.: A graph-based decomposition method for convex quadratic optimization with indicators. Math. Program. 200, 669–701 (2023)
Mittal, A., Gokalp, C., Hanasusanto, G.A.: Robust quadratic programming with mixed-integer uncertainty. Inf. J. Comput. 32, 201–218 (2020)
Ramana, M., Goldman, A.J.: Some geometric results in semidefinite programming. J. Global Optim. 7, 33–50 (1995)
Robust convex quadratically constrained programs: Goldfarb, D., Iyengar. G. Math. Program. 97, 495–515 (2003)
Sun, X.K., Teo, K.L., Zeng, J., Guo, X.L.: On approximate solutions and saddle point theorems for robust convex optimization. Optim. Lett. 14, 1711–1730 (2020)
Sun, X.K., Teo, K.L., Long, X.J.: Some characterizations of approximate solutions for robust semiinfinite optimization problems. J. Optim. Theory Appl. 191, 281–310 (2021)
Sun, X.K., Tan, W., Teo, K.L.: Characterizing a class of robust vector polynomial optimization via sum of squares conditions. J. Optim. Theory Appl. 197, 737–764 (2023)
Vinzant, C.: What is a spectrahedron? Notices Am. Math. Soc. 61, 492–494 (2014)
Wei, H.Z., Chen, C.R., Li, S.J.: Characterizations for optimality conditions of general robust optimization problems. J. Optim. Theory Appl. 177, 835–856 (2018)
Woolnough, D., Jeyakumar, V., Li, G.: Exact conic programming reformulations of two-stage adjustable robust linear programs with new quadratic decision rules. Optim. Lett. 15, 25–44 (2021)
Xia, Y.S., Feng, G.: An improved neural network for convex quadratic optimization with application to real-time beamforming. Neurocomputing 64, 359–374 (2005)
Yang, Y.: A polynomial arc-search interior-point algorithm for convex quadratic programming. Eur. J. Oper. Res. 215, 25–38 (2011)
Yanikoglu, I., Gorissen, B.L., den Hertog, D.: A survey of adjustable robust optimization. Eur. J. Oper. Res. 277, 799–813 (2019)
Zhang, S., Huang, Y.: Complex quadratic optimization and semidefinite programming. SIAM J. Optim. 16, 871–890 (2006)
Zhang, H., Sun, X.K., Li, G.H.: On second-order conic programming duals for robust convex quadratic optimization problems. J. Ind. Manag. Optim. 19, 8114–8128 (2023)
Funding
This research is supported by the Science and Technology Research Program of Chongqing Municipal Education Commission (KJZDK202100803), the Team Building Project for Graduate Tutors in Chongqing (yds223010) and the Innovation Project of CTBU (yjscxx2023-211-72).
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Jen-Chih Yao.
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Zhang, H., Sun, X. & Teo, K.L. Exact SDP Reformulations for Adjustable Robust Quadratic Optimization with Affine Decision Rules. J Optim Theory Appl 203, 2206–2232 (2024). https://doi.org/10.1007/s10957-023-02371-5
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10957-023-02371-5
Keywords
- Adjustable robust optimization
- Quadratic optimization
- Semidefinite programming reformulation
- Spectrahedral uncertain sets