Smoothing sample average approximation method for solving stochastic second-order-cone complementarity problems.

Area of Science:

• Optimization Theory
• Mathematical Programming
• Numerical Analysis

Background:

• Stochastic second-order-cone complementarity problems (SSOCCP) present significant computational challenges.
• Existing methods may struggle with the inherent uncertainty and complexity of SSOCCP.

Purpose of the Study:

• To develop a robust and efficient method for solving SSOCCP.
• To introduce an expected residual minimization (ERM) model for SSOCCP.
• To analyze the convergence properties of the proposed numerical methods.

Main Methods:

• Formulation of an expected residual minimization (ERM) model using a second-order-cone complementarity function.
• Introduction of a smoothing technique to create an approximate ERM model.
• Application of the sample average approximate (SAA) method to address the expectation in the ERM model.

Main Results:

• Demonstrated convergence of the smoothing approximate ERM model's solutions to the original ERM model's solutions.
• Proved that the global optimal solution of the smoothing SAA problem converges to the ERM problem's global optimal solution with probability one.
• Established convergence results for weak stationary points of the smoothing SAA problem.

Conclusions:

• The proposed smoothing and sample average approximate methods provide a feasible and effective approach for solving SSOCCP.
• The theoretical convergence analyses support the reliability of the developed algorithms.
• Numerical examples confirm the practical applicability of the methods.