Robust least square semidefinite programming with applications

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.

Appendix: Proof of Lemma 2.1

Proof

By a translation, we may assume without loss of generality that X ₀=0. For a closed convex set Ω, let σ _Ω denote the usual support function of Ω (that is, σ _Ω (X)=sup _Y∈Ω tr(YX) for all $X \in {\mathcal{S}}^{n}$) and let epiσ _Ω be the epigraph of the support function. Now, notice that if $(Y_{1},\alpha_{1})\in {\mathrm{epi}}\,\sigma_{\varOmega_{1}}$ and $(Y_{2},\alpha_{2})\in {\mathrm{epi}}\,\sigma_{\varOmega_{2}}$ are such that $\alpha_{1}+\alpha_{2}=\sigma_{\varOmega_{1}\cap \varOmega_{2}}(Y_{1}+Y_{2})$ and ∥Y ₁+Y ₂∥ _F ≤1, then we have $0\le \sigma_{\varOmega_{1}}(Y_{1})\le \alpha_{1}$ and $\alpha_{2}\le \sigma_{\varOmega_{1}\cap \varOmega_{2}}(Y_{1}+Y_{2})\le R$. From these we obtain further that $\delta\|Y_{2}\|_{F}\le \sigma_{\varOmega_{2}}(Y_{2})\le \alpha_{2}\le R$ and consequently

$$\|Y_1\|_F\le \|Y_1+Y_2 \|_F+\|Y_2\|_F\le 1+\frac{R}{\delta}. $$

The proof of this lemma now follows similarly as in [23, Lemma 4.10], making use of the bounds derived above. □