Convergence analysis of sample average approximation for a class of stochastic nonlinear complementarity problems:...

Area of Science:

• Optimization
• Mathematical Programming
• Operations Research

Background:

• Stochastic Nonlinear Complementarity Problems (SNCPs) are complex optimization models with inherent uncertainty.
• The Sample Average Approximation (SAA) method is a common approach for solving such problems by approximating expected values with sample averages.
• Understanding the convergence properties of SAA is crucial for ensuring the reliability of solutions for SNCPs.

Purpose of the Study:

• To analyze the convergence properties of the Sample Average Approximation (SAA) approach for a class of stochastic nonlinear complementarity problems (SNCPs).
• To investigate the convergence of SAA for two-stage SNCPs with specific differentiability and continuity conditions.
• To extend these convergence results to a broader class of multistage SNCPs.

Main Methods:

• The study employs the Sample Average Approximation (SAA) technique.
• Convergence analysis is performed for two-stage SNCPs where the first stage is continuously differentiable and the second stage is locally Lipschitz continuous.
• The methodology is extended to multistage SNCPs with adjacent-stage variable dependencies.

Main Results:

• The convergence of the SAA counterparts for two-stage SNCPs is established under the specified conditions.
• Convergence results are successfully extended to a class of multistage SNCPs.
• Preliminary numerical tests provide empirical support for the theoretical convergence findings.

Conclusions:

• The SAA approach is a viable method for solving the considered classes of SNCPs.
• The established convergence properties enhance confidence in using SAA for these complex problems.
• The findings contribute to the theoretical understanding and practical application of SAA in stochastic optimization.