Computational Optimization and Applications

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

Robust least square semidefinite programming with applications

  • Guoyin Li
  • Alfred Ka Chun Ma
  • Ting Kei Pong


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.


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


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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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