Computational Optimization and Applications

, Volume 58, Issue 2, pp 347–379 | Cite as

Robust least square semidefinite programming with applications

Article

Abstract

In this paper, we consider a least square semidefinite programming problem under ellipsoidal data uncertainty. We show that the robustification of this uncertain problem can be reformulated as a semidefinite linear programming problem with an additional second-order cone constraint. We then provide an explicit quantitative sensitivity analysis on how the solution under the robustification depends on the size/shape of the ellipsoidal data uncertainty set. Next, we prove that, under suitable constraint qualifications, the reformulation has zero duality gap with its dual problem, even when the primal problem itself is infeasible. The dual problem is equivalent to minimizing a smooth objective function over the Cartesian product of second-order cones and the Euclidean space, which is easy to project onto. Thus, we propose a simple variant of the spectral projected gradient method (Birgin et al. in SIAM J. Optim. 10:1196–1211, 2000) to solve the dual problem. While it is well-known that any accumulation point of the sequence generated from the algorithm is a dual optimal solution, we show in addition that the dual objective value along the sequence generated converges to a finite value if and only if the primal problem is feasible, again under suitable constraint qualifications. This latter fact leads to a simple certificate for primal infeasibility in situations when the primal feasible set lies in a known compact set. As an application, we consider robust correlation stress testing where data uncertainty arises due to untimely recording of portfolio holdings. In our computational experiments on this particular application, our algorithm performs reasonably well on medium-sized problems for real data when finding the optimal solution (if exists) or identifying primal infeasibility, and usually outperforms the standard interior-point solver SDPT3 in terms of CPU time.

Keywords

Robust optimization Uncertain least square SDP Strong duality Infeasibility Spectral gradient projection method Robust correlation stress testing 

References

  1. 1.
    Ben-Tal, A., Nemirovski, A.: Robust solutions of linear programming problems contaminated with uncertain data. Math. Program. 88, 411–424 (2000) CrossRefMATHMathSciNetGoogle Scholar
  2. 2.
    Ben-Tal, A., Nemirovski, A.: Selected topics in robust convex optimization. Math. Program. 112, 125–158 (2008) CrossRefMATHMathSciNetGoogle Scholar
  3. 3.
    Ben-Tal, A., El Ghaoui, L., Nemirovski, A.: Robust Optimization. Princeton Series in Applied Mathematics. Princeton University Press, Princeton (2009) MATHGoogle Scholar
  4. 4.
    Berkowitz, J.: A coherent framework for stress testing. J. Risk 2(2), 5–15 (1999) Google Scholar
  5. 5.
    Bertsimas, D., Brown, D.B.: Constructing uncertainty sets for robust linear optimization. Oper. Res. 57, 1483–1495 (2009) CrossRefMATHMathSciNetGoogle Scholar
  6. 6.
    Bertsimas, D., Brown, D.B., Caramanis, C.: Theory and applications of robust optimization. SIAM Rev. 53, 464–501 (2011) CrossRefMATHMathSciNetGoogle Scholar
  7. 7.
    Birge, J.R., Louveaux, F.: Introduction to Stochastic Programming. Springer, Berlin (1997) MATHGoogle Scholar
  8. 8.
    Birgin, E.G., Martínez, J.M., Raydan, M.: Nonmonotone spectral projected gradient methods on convex sets. SIAM J. Optim. 10, 1196–1211 (2000) CrossRefMATHMathSciNetGoogle Scholar
  9. 9.
    Chou, C.-C., Ng, K.-F., Pang, J.-S.: Minimizing and stationary sequences of constrained optimization problems. SIAM J. Control Optim. 36, 1908–1936 (1998) CrossRefMATHMathSciNetGoogle Scholar
  10. 10.
    Eckstein, J., Bertsekas, D.P.: On the Douglas-Rachford splitting method and the proximal point algorithm for maximal monotone operators. Math. Prog. 55, 293–318 (1992) CrossRefMATHMathSciNetGoogle Scholar
  11. 11.
    El Ghaoui, L., Lebret, H.: Robust solutions to least-squares problems with uncertain data. SIAM J. Matrix Anal. Appl. 18, 1035–1064 (1997) CrossRefMATHMathSciNetGoogle Scholar
  12. 12.
    Fortin, M., Glowinski, R.: On decomposition-coordination methods using an augmented Lagrangian. In: Fortin, M., Glowinski, R. (eds.) Augmented Lagrangion Methods: Applications to the Solution of Boundary Problems. North-Holland, Amsterdam (1983) Google Scholar
  13. 13.
    Fukushima, M., Luo, Z.-Q., Tseng, P.: Smoothing functions for second-order-cone complementarity problems. SIAM J. Optim. 12, 436–460 (2001) CrossRefMATHMathSciNetGoogle Scholar
  14. 14.
    Gabay, D.: Applications of the method of multipliers to variational inequalities. In: Fortin, M., Glowinski, R. (eds.) Augmented Lagrangion Methods: Applications to the Solution of Boundary Problems. North-Holland, Amsterdam (1983) Google Scholar
  15. 15.
    Gabay, D., Mercier, B.: A dual algorithm for the solution of nonlinear variational problems via finite element approximations. Comput. Math. Appl. 2, 17–40 (1976) CrossRefMATHGoogle Scholar
  16. 16.
    Gafni, E.M., Bertsekas, D.P.: Two metric projection methods for constrained optimization. SIAM J. Control Optim. 22, 936–964 (1984) CrossRefMATHMathSciNetGoogle Scholar
  17. 17.
    Gao, Y., Sun, D.: Calibrating least squares semidefinite programming with equality and inequality constraints. SIAM J. Matrix Anal. Appl. 31, 1432–1457 (2009) CrossRefMathSciNetGoogle Scholar
  18. 18.
    Glowinski, R., Marroco, A.: Sur l’approximation, par elements finis d’ordre un, et la resolution, par penalisation-dualit’e, d’une classe de problemes de Dirichlet non lineares. Revue Francaise d’Automatique Informatique et Recherche Op’erationelle 9(r–2), 41–76 (1975) MATHGoogle Scholar
  19. 19.
    He, B., Liao, L., Han, D., Yang, H.: A new inexact alternating directions method for monotone variational inequalities. Math. Program. 92, 103–118 (2002) CrossRefMATHMathSciNetGoogle Scholar
  20. 20.
    Hu, H.: Geometric condition measures and smoothness condition measures for closed convex sets and linear regularity of infinitely many closed convex sets. J. Optim. Theory Appl. 126, 287–308 (2005) CrossRefMATHMathSciNetGoogle Scholar
  21. 21.
    Kim, J., Finger, C.C.: A stress test to incorporate correlation breakdown. J. Risk 2(3), 5–19 (2000) Google Scholar
  22. 22.
    Kupiec, P.H.: Stress testing in a value at risk framework. J. Deriv. 6, 7–24 (1998) CrossRefGoogle Scholar
  23. 23.
    Li, C., Ng, K.F., Pong, T.K.: The SECQ, linear regularity, and the strong CHIP for an infinite system of closed convex sets in normed linear spaces. SIAM J. Optim. 18, 643–665 (2007) CrossRefMATHMathSciNetGoogle Scholar
  24. 24.
    Lu, Z., Zhang, Y.: An augmented Lagrangian approach for sparse principal component analysis. Math. Program. 135, 149–193 (2012) CrossRefMATHMathSciNetGoogle Scholar
  25. 25.
    Nesterov, Y.: A method for solving a convex programming problem with convergence rate O(1/k 2). Sov. Math. Dokl. 27(2), 372–376 (1983) MATHGoogle Scholar
  26. 26.
    Nesterov, Y.: Introductory Lectures on Convex Optimization: A Basic Course. Kluwer Academic, Amsterdam (2003) Google Scholar
  27. 27.
    Nesterov, Y.: Excessive gap technique in nonsmooth convex minimization. SIAM J. Optim. 16, 235–249 (2005) CrossRefMATHMathSciNetGoogle Scholar
  28. 28.
    Nesterov, Y.: Smooth minimization of non-smooth functions. Math. Program. 103, 127–152 (2005) CrossRefMATHMathSciNetGoogle Scholar
  29. 29.
    Qi, H., Sun, D.: Correlation stress testing for value-at-risk: an unconstrained convex optimization approach. Comput. Optim. Appl. 45, 427–462 (2010) CrossRefMATHMathSciNetGoogle Scholar
  30. 30.
    Rebonato, R., Jäckel, P.: The most general methodology for creating a valid correlation matrix for risk management and option pricing purposes. J. Risk 2(2), 17–27 (1999) Google Scholar
  31. 31.
    Robinson, S.M.: An application of error bounds for convex programming in a linear space. SIAM J. Control 13, 271–273 (1975) CrossRefMATHMathSciNetGoogle Scholar
  32. 32.
    Rockafellar, R.T.: Convex Analysis. Princeton University Press, Princeton (1970) MATHGoogle Scholar
  33. 33.
    Shapiro, A., Dentcheva, D., Ruszczynski, A.: Lectures on Stochastic Programming: Modeling and Theory. SIAM, Philadelphia (2009) CrossRefGoogle Scholar
  34. 34.
    Soyster, A.L.: Convex programming with set-inclusive constraints and applications to inexact linear programming. Oper. Res. 21, 1154–1157 (1973) CrossRefMATHMathSciNetGoogle Scholar
  35. 35.
    Toh, K.C., Todd, M.J., Tütüncü, R.H.: SDPT3—a Matlab software package for semidefinite programming. Optim. Method Softw. 11, 545–581 (1999) CrossRefGoogle Scholar
  36. 36.
    Toint, Ph.L.: Global convergence of a class of trust-region methods for nonconvex minimization in Hilbert space. IMA J. Numer. Anal. 8, 231–252 (1988) CrossRefMATHMathSciNetGoogle Scholar
  37. 37.
    Tseng, P.: Approximation accuracy, gradient methods, and error bound for structured convex optimization. Math. Program. 125, 263–295 (2010) CrossRefMATHMathSciNetGoogle Scholar
  38. 38.
    Tseng, P., Yun, S.: A coordinate gradient descent method for nonsmooth separable minimization. Math. Program. 117, 387–423 (2007) CrossRefMathSciNetGoogle Scholar
  39. 39.
    Turkay, S., Epperlein, E., Christofides, N.: Correlation stress testing for value-at-risk. J. Risk 5(4), 75–89 (2003) Google Scholar
  40. 40.
    Tütüncü, R.H., Toh, K.C., Todd, M.J.: Solving semidefinite-quadratic-linear programs using SDPT3. Math. Program. Series B 95, 189–217 (2003) CrossRefMATHGoogle Scholar
  41. 41.
    Yang, J., Zhang, Y.: Alternating direction algorithms for 1-problems in compressive sensing. SIAM J. Sci. Comput. 33, 250–278 (2011) CrossRefMATHMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  • Guoyin Li
    • 1
  • Alfred Ka Chun Ma
    • 2
  • Ting Kei Pong
    • 3
  1. 1.Department of Applied MathematicsUniversity of New South WalesSydneyAustralia
  2. 2.CASH Financial Services Group Limited and Quant-Finance Lab.Kowloon BayHong Kong
  3. 3.Department of Combinatorics and OptimizationUniversity of WaterlooWaterlooCanada

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