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Gradient Formulae for Nonlinear Probabilistic Constraints with Non-convex Quadratic Forms

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Abstract

Probability functions appearing in chance constraints are an ingredient of many practical applications. Understanding differentiability, and providing explicit formulae for gradients, allow us to build nonlinear programming methods for solving these optimization problems from practice. Unfortunately, differentiability of probability functions cannot be taken for granted. In this paper, motivated by gas network applications, we investigate differentiability of probability functions acting on non-convex quadratic forms. We establish continuous differentiability for the broad class of elliptical random vectors under mild conditions.

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Acknowledgements

Both authors would like to acknowledge the contributions of René Henrion, through long deep discussions over many years, to this work. The authors would also like to acknowledge the anonymous reviewer for his extremely detailed lecture and valuable suggestions. Finally, the second author acknowledges partial support by CONICYT grants: Fondecyt Regular 1190110 and CONICYT grant: MATH-AmSud 20-MATH-08.

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Correspondence to Wim van Ackooij.

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van Ackooij, W., Pérez-Aros, P. Gradient Formulae for Nonlinear Probabilistic Constraints with Non-convex Quadratic Forms. J Optim Theory Appl (2020). https://doi.org/10.1007/s10957-020-01634-9

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Keywords

  • Stochastic optimization
  • Probabilistic constraints
  • Chance constraints
  • Gradients of probability functions

Mathematics Subject Classification

  • 90C15