Chance constrained programs with Gaussian mixture models

Abstract

In this article, we discuss input modeling and solution techniques for several classes of Chance constrained programs (CCPs). We propose to use a Gaussian Mixture Model (GMM) to fit the data available and to model the randomness. We demonstrate the merits of using a GMM. We consider several scenarios that arise from practical applications and analyze how the problem structures could embrace alternative optimization techniques. More specifically, for several scenarios, we study how to assess the gradient of the chance constraint and incorporate the results into gradient-based nonlinear optimization algorithms, and for a class of CCPs, we propose a spatial branch-and-bound procedure and solve the problems to global optimality. We also conduct numerical experiments to test the efficiency of our approach and propose an example of hedge fund portfolio to illustrate the practical application of the method.

Keywords:

• Stochastic programming
• chance constrained programs
• Gaussian mixture model
• spatial branch-and-bound
• value-at-risk

Acknowledgments

We thank Andrew Schaefer, the associate editor and two anonymous referees for their effort and valuable comments/suggestions.

Notes on contributors

Zhaolin Hu is a Professor in School of Economics and Management at Tongji University. He obtained his PhD degree in Industrial Engineering and Logistics Management from the Hong Kong University of Science and Technology, and his BSc degree in Mathematics and Applied Mathematics from Zhejiang University. His current research interests include stochastic optimization, stochastic simulation, machine learning and risk management.

Wenjie Sun is a PhD candidate affiliated with the Department of Analytics and Operations in the NUS Business School at National University of Singapore. His current research interests lie in the revenue management and inventory control.

Shushang Zhu is a Professor in School of Business at Sun Yat-Sen University. He obtained his PhD degree in Management Science and Engineering from Academy of Mathematics and Systems Science, Chinese Academy of Sciences, and his MSc degree in Applied Mathematics and BSc degree in Mathematical Statistics from Xiangtan University. His current research interests include portfolio optimization, systemic risk measurement and quantitative risk management.

Additional information

Funding

This research was partially supported by the Natural Science Foundation of China (Grants 71722009, 72071148, 71890973, 71721001).