Generalized cutting plane method for solving nonlinear stochastic programming problems

ABSTRACT

A tiny subclass of minimum-type functions, called ${\mathcal{G}}_{n+1}$, is introduced. We show that abstract convex functions generated by ${\mathcal{G}}_{n+1}$ and those generated by the whole class of minimum-type functions coincide. Other concepts from abstract convex analysis such as support set, subdifferential and conjugate function with respect to ${\mathcal{G}}_{n+1}$ are investigated. We will use these results to establish a stochastic version of generalized cutting plane method (SGCPM) to solve two-stage nonconvex programming problems. Under mild conditions, we will show that every limit point of the sequence generated by SGCPM is an optimal solution.

Keywords:

• Generalized convexity
• degree-one calm functions
• minimum-type subgradients
• two-stage nonconvex programming problems
• cutting plane method

2010 AMS Mathematics Subject Classifications:

• 26B25
• 90C26
• 90C15

Acknowledgments

The authors are very grateful of the referees for their valuable remarks, which improved the presentation of the paper.

Disclosure statement

No potential conflict of interest was reported by the author.

Additional information

Funding

This work has been supported by the Center for International Scientific Studies and Collaboration (CISSC).