Robust Solutions to a System of Stochastic Vertical Linear Complementarity Problems
提出一个随机最小化模型来求解随机垂直线性互补问题系统的鲁棒解，通过惩罚方法转化为分段光滑最小化问题，设计平滑块坐标下降算法并证明收敛性，应用于投资组合选择问题。
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 [Formula: see text]-Clarke stationary point of the penalty problem by the KL property for any [Formula: see text]. 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].