Quantile Formulation for Optimization Under a Qualitative Risk Constraint

This paper focuses on the pathologies of common gradient-based algorithms for solving optimization problems under probability constraints. These problems are both important and difficult to handle. We provide a thorough analysis of what we believe is the main pathology of the distributional formulation of the problem, using both examples and rigorous proofs. With this knowledge, we then present an alternative formulation that we call the quantile formulation of the problem and prove that this formulation is exempt of the numerical pathologies of the original formulation. Our theoretical framework is developed for univariate distributions. To complete the work, we present generalizations to a class of multivariate problems motivated by the design of a robotic medical device to guide micro-surgeries, which must satisfy a specific structural reliability.