A new model for solving stochastic second-order cone complementarity problem and its convergence analysis.

Area of Science:

• Optimization Theory
• Mathematical Programming
• Operations Research

Background:

• The stochastic second-order cone complementarity problem (SSOCCP) can lack solutions due to inherent stochastic variables.
• Existing methods may struggle with the non-smooth and expectation-containing objective functions of direct SSOCCP models.

Purpose of the Study:

• To develop a robust deterministic model for solving the SSOCCP.
• To address the challenges of non-smoothness and expectation calculation in SSOCCP modeling.
• To propose approximation techniques for practical solvability.

Main Methods:

• Formulating a conditional value-at-risk (CVaR) model using the merit function as a loss function.
• Employing a smoothing method to handle the non-smooth objective function.
• Utilizing the sample average approximation (SAA) technique to approximate the expectation.

Main Results:

• The proposed CVaR model transforms the SSOCCP into a solvable deterministic problem.
• Convergence results for global optimal solutions of the approximation problems are established.
• Convergence results for stationary points of the approximation problems are also provided.

Conclusions:

• The developed approximation approach effectively addresses the challenges of solving the SSOCCP.
• The study provides theoretical convergence guarantees for the proposed approximation methods.
• This work offers a practical framework for tackling stochastic optimization problems in second-order cone settings.