Robust Solutions to a System of Stochastic Vertical Linear Complementarity Problems

We propose a stochastic minimization model to find a robust solution of a system of stochastic vertical linear complementarity problems. This model aims to minimize a risk function under stochastic vertical linear complementarity constraints. We reformulate the model with a finite support set as a linearly constrained piecewise smooth minimization problem by a penalty method. We prove the existence of exact penalty parameters regarding global and local minimizers. We define a smoothing function of the piecewise smooth objective function and show that the smoothing function satisfies the Kurdyka–Łojasiewicz (KL) property. Moreover, we propose a smoothing block coordinate descent algorithm and prove that the sequence generated by the algorithm globally converges to an $ϵ$-Clarke stationary point of the penalty problem by the KL property for any $ϵ>0$. Finally, we apply our model and algorithm to portfolio selection problems with real data. Numerical results demonstrate the robustness of our model.

Funding: X. Chen was supported in part by the Hong Kong Research Grant Council [Grants PolyU15300123, and PolyU15300322] and the CAS-Croucher Funding Scheme for the AMSS-PolyU Joint Laboratory. Z. Allen-Zhao was supported in part by the National Natural Science Foundation of China [Grant 12301405], the Shaanxi Fundamental Science Research Project for Mathematics and Physics [Grant 23JSZ010], and the Fundamental Research Fund for Central Universities of China [Grant ZYTS25201].