Abstract
In this paper, we introduce a new class of nonsmooth convex functions called SOS-convex semialgebraic functions extending the recently proposed notion of SOS-convex polynomials. This class of nonsmooth convex functions covers many common nonsmooth functions arising in the applications such as the Euclidean norm, the maximum eigenvalue function and the least squares functions with ℓ 1-regularization or elastic net regularization used in statistics and compressed sensing. We show that, under commonly used strict feasibility conditions, the optimal value and an optimal solution of SOS-convex semialgebraic programs can be found by solving a single semidefinite programming problem (SDP). We achieve the results by using tools from semialgebraic geometry, convex-concave minimax theorem and a recently established Jensen inequality type result for SOS-convex polynomials. As an application, we show that robust SOS-convex optimization proble ms under restricted spectrahedron data uncertainty enjoy exact SDP relaxations. This extends the existing exact SDP relaxation result for restricted ellipsoidal data uncertainty and answers an open question in the literature on how to recover a robust solution of uncertain SOS-convex polynomial programs from its semidefinite programming relaxation in this broader setting.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Ahmadi, A.A., Parrilo, P.A.: A complete characterization of the gap between convexity and SOS-convexity. SIAM J. Optim. 23, 811–833 (2013)
Belousov, E.G., Klatte, D.: A Frank-Wolfe type theorem for convex polynomial programs. Comp. Optim. Appl. 22, 37–48 (2002)
Ben-Tal, A., Nemirovski, A.: Lectures on modern convex optimization. analysis, algorithms, and engineering applications. MPS/SIAM Series on Optimization. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA; Mathematical Programming Society (MPS), Philadelphia PA (2001)
Ben-Tal, A., El Ghaoui, L., Nemirovski, A.: Robust Optimization. Princeton U.P., Princeton (2009)
Bochnak, J., Coste, M., Roy, M.F.: Real Algebraic Geometry, vol. 36, p. x + 430. Springer, Berlin (1998)
Borwein, J.M., Vanderwerff, J.D.: Convex Functions: Constructions, Characterizations and Counterexamples Encyclopedia of Mathematics and Its Applications, vol. 109. Cambridge University Press, Cambridge (2010)
Boyd, S., Vandenberghe, L.: Convex Optimization. Cambridge University Press, Cambridge (2004)
Bertsimas, D., Sim, M.: The price of robustness. Oper. Res. 52, 35–53 (2004)
Bertsimas, D., Brown, D.B., Caramanis, C.: Theory and applications of robust optimization. SIAM Rev. 53(3), 464–501 (2011)
CVX Research, Inc.C.VX.: Matlab software for disciplined convex programming, version 2.0. http://cvxr.com/cvx (2011)
Dinh, D., Goberna, M.A., López, M.A., Volle, M.: A unifying approach to robust convex infinite optimization duality. to appear in J. Optim. Theory & Appl. https://doi.org/10.1007/s10957-017-1136-x (2017)
Goberna, M.A., Jeyakumar, V., Li, G., López, M.A.: Robust linear semi-infinite programming duality under uncertainty. Math. Program. 139, 185–203 (2013)
Goldfarb, D., Iyengar, G.: Robust convex quadratically constrained programs. Math. Program. 97, 495–515 (2003)
Grant, M., Boyd, S.: Graph implementations for nonsmooth convex programs, recent advances in learning and control. In: Blondel, V., Boyd, S., Kimura, H. (eds.) Lecture notes in control and information sciences, pp. 95–110. Springer (2008)
Helton, J.W., Nie, J.W.: Semidefinite representation of convex sets. Math. Program. 122, 21–64 (2010)
Jeyakumar, V., Li, G.: Strong duality in robust convex programming: complete characterizations. SIAM J. Optim. 20, 3384–3407 (2010)
Jeyakumar, V., Li, G.: A new class of alternative theorems for SOS-convex inequalities and robust optimization. Appl. Anal. 94, 56–74 (2015)
Jeyakumar, V., Li, G.: Exact SDP relaxations for classes of nonlinear semidefinite programming problems. Oper. Res. Letters 40, 529–536 (2012)
Jeyakumar, V., Li, G., Vicente-Pérez, J.: Robust SOS-convex polynomial optimization problems: exact SDP relaxations. Optim. Letters 9, 1–18 (2015)
Jeyakumar, V., Vicente-Pérez, J.: Dual semidefinite programs without duality gaps for a class of convex minimax programs. J. Optim. Theory Appl. 162, 735–753 (2014)
Lasserre, J.B.: Moments, Positive Polynomials and their Applications. Imperial College Press, London (2009)
Lasserre, J.B.: Convexity in semialgebraic geometry and polynomial optimization. SIAM J. Optim. 19, 1995–2014 (2008)
Jeyakumar, V., Li, G.Y., Lee, G.M.: A robust von Neumann minimax theorem for zero-sum games under bounded payoff uncertainty. Oper. Res. Letters 39(2), 109–114 (2011)
Lee, G.M., Pham, T.S.: Stability and genericity for semialgebraic compact programs. J. Optim. Theory Appl. 169, 473–495 (2016)
Li, G., Jeyakumar, V., Lee, G.M.: Robust conjugate duality for convex optimization under uncertainty with application to data classification. Nonlinear Anal. 74(6), 2327–2341 (2011)
Zou, H., Hastie, T.: Regularization and variable selection via the elastic net. J. R. Statist. Soc. B 67, 310–320 (2005)
Author information
Authors and Affiliations
Corresponding author
Additional information
Dedicated to the memory of Jon Borwein who was of great inspiration to us
Research was partially supported by aresearch grant from Australian Research Council. Research was also partially supported by the Vietnam National Foundation 465 for Science and Technology Development (NAFOSTED) under grant 101.01-2017.325.
Rights and permissions
About this article
Cite this article
Chieu, N.H., Feng, J.W., Gao, W. et al. SOS-Convex Semialgebraic Programs and its Applications to Robust Optimization: A Tractable Class of Nonsmooth Convex Optimization. Set-Valued Var. Anal 26, 305–326 (2018). https://doi.org/10.1007/s11228-017-0456-1
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11228-017-0456-1
Keywords
- Nonsmooth optimization
- Convex optimization
- SOS-convex polynomial
- Semidefinite program
- Robust optimization