A Note on Stability for Risk-Averse Stochastic Complementarity Problems
- 176 Downloads
In this paper, we propose a new approach to stochastic complementarity problems which allows to take into account various notions of risk aversion. Our model enhances the expected residual minimization formulation of Chen and Fukushima (Math Oper Res 30:1022–1038, 2005) by replacing the expectation with a more general convex, nondecreasing and law-invariant risk functional. Relevant examples of such risk functionals include the Expected Excess and the Conditional Value-at-Risk. We examine qualitative stability of the resulting one-stage stochastic optimization problem with respect to perturbations of the underlying probability measure. Considering the topology of weak convergence, we prove joint continuity of the objective function with respect to both the decision vector and the entering probability measure. By a classical result from parametric optimization, this implies upper semicontinuity of the optimal value function. Throughout the analysis, we assume for building the model that a nonlinear complementarity function fulfills a certain polynomial growth condition. We conclude the paper demonstrating that this assumption holds in the vast majority of all practically relevant cases.
KeywordsStochastic complementarity problem Nonlinear complementarity function Expected residual minimization Risk aversion Stability
Mathematics Subject Classification90C15 90C31 90C33
The authors thank Prof. R. Schultz for helpful discussions.
- 10.Claus, M., Krätschmer, V., Schultz, R.: Weak continuity of risk functionals with applications to stochastic programming. https://www.uni-due.de/imperia/md/content/mathematik/2015_preprint_790 (2015)
- 12.Luo, M.-J., Lu, Y.: Properties of expected residual minimization model for a class of stochastic complementarity problems. J. Appl. Math., Article ID 497586 (2013). doi: 10.1155/2013/497586
- 18.Krätschmer, V., Schied, A., Zähle, H.: Domains of weak continuity of statistical functionals with a view on robust statistics. http://arxiv.org/pdf/1511.08677v1 (2015)
- 21.Ulbrich, M.: Semismooth Newton methods for operator equations in function spaces. Technical Rep. 00-11, Department of Computational and Applied Mathematics, Rice University (2000)Google Scholar