Bounds for probabilistic programming with application to a blend planning problem

In this paper, we derive deterministic inner approximations for single and joint independent or dependent probabilistic constraints based on classical inequalities from probability theory such as the onesided Chebyshev inequality, Bernstein inequality, Chernoff inequality and Hoeffding inequality (see Pinter, 1989). The dependent case has been modelled via copulas. New assumptions under which the bounds based approximations are convex allowing to solve the problem efficiently are derived. When the convexity condition can not hold, an efficient sequential convex approximation approach is further proposed to solve the approximated problem. Piecewise linear and tangent approximations are also provided for Chernoff and Hoeffding inequalities allowing to reduce the computational complexity of the associated optimization problem. Extensive numerical results on a blend planning problem under uncertainty are finally provided allowing to compare the proposed bounds with the Second Order Cone (SOCP) formulation and Sample Average Approximation (SAA). (c) 2021 Elsevier B.V. All rights reserved.

Automated summary

This paper presents deterministic inner approximations for single and joint independent or dependent probabilistic constraints based on classical inequalities from probability theory, modeled through copulas in the dependent case. New assumptions for convex bounds-based approximations are derived, allowing for efficient problem-solving. When convexity conditions cannot be met, an efficient sequential convex approximation approach is proposed, along with piecewise linear and tangent approximations for reducing computational complexity. Extensive numerical results on a blend planning problem under uncertainty are provided for comparison with the Second Order Cone (SOCP) formulation and Sample Average Approximation (SAA).