A DECOMPOSITION ALGORITHM FOR TWO-STAGE STOCHASTIC PROGRAMS WITH NONCONVEX RECOURSE FUNCTIONS.

Area of Science:

• Optimization
• Mathematical Programming
• Computational Science

Background:

• Classical decomposition methods fail for nonconvex recourse functions in two-stage stochastic programs.
• Nonlinear parameterization by first-stage variables complicates second-stage objective and constraints.
• Clarke regularity failure prevents direct generalization of Benders or Lagrangian algorithms.

Purpose of the Study:

• To develop a novel decomposition framework for nonconvex two-stage stochastic programs.
• To address the limitations of existing methods when Clarke regularity fails.
• To provide an effective algorithm for problems with nonlinearly parameterized second-stage recourse functions.

Main Methods:

• Exploration of an implicitly convex-concave structure of the recourse function.
• Introduction of a decomposition framework utilizing the partial Moreau envelope.
• Successive generation of strongly convex quadratic approximations of the recourse function.
• Integration of approximations into the first-stage master problem via solutions of second-stage convex subproblems.

Main Results:

• Convergence established for both fixed and sequentially sampled scenarios.
• Demonstrated effectiveness of the proposed algorithm through numerical experiments.
• Successfully addressed the challenge of nonconvexity in two-stage stochastic programming.

Conclusions:

• The novel partial Moreau envelope-based decomposition framework is effective for nonconvex two-stage stochastic programs.
• The method overcomes limitations of classical decomposition algorithms.
• The approach offers a viable solution for complex optimization problems with nonlinearly parameterized recourse functions.