Analytic approximation and differentiability of joint chance constraints

ABSTRACT

An inner–outer approximation approach was recently developed to solve single chance constrained optimization (SCCOPT) problems. In this paper, we extend this approach to address joint chance constrained optimization (JCCOPT) problems. Using an inner–outer approximation, two smooth parametric optimization problems are defined whose feasible sets converge to the feasible set of JCCOPT from inside and outside, respectively. Any optimal solution of the inner approximation problem is a priori feasible to the JCCOPT. As the approximation parameter tends to zero, a subsequence of the solutions of the inner and outer problems, respectively, converge asymptotically to an optimal solution of the JCCOPT. As a main result, the continuous differentiability of the probability function of a joint chance constraint is obtained by examining the uniform convergence of the gradients of the parametric approximations.

KEYWORDS:

• Joint chance constraints
• stochastic optimization
• smoothing
• inner–outer approximation
• differentiability

AMS CLASSIFICATIONS:

• 90C15
• 90C31
• 90C30
• 26A42
• 57R12
• 58B10

Disclosure statement

No potential conflict of interest was reported by the authors.

ORCID

P. Li http://orcid.org/0000-0001-6481-9961

Notes

1. If (LICQ) is violated and only the gradients of g w.r.t $\hat{\xi }$ are non-zero then Lebesgue's Theorem is not applicable and the proof becomes more complicated and will be considered in a future work.

Additional information

Funding

The authors would like to express their indebtedness to the Deutsche Forschungsgemeinschaft (DFG) for the financial support with grants Nr. LI 806/13-4.