Derivatives of Probability Functions: Unions of Polyhedra and Elliptical Distributions

Abstract

In many practical applications models exhibiting chance constraints play a role. Since, in practice one is also typically interesting in numerically solving the underlying optimization problems, an interest naturally arises in understanding analytical properties, such as differentiability, of probability functions. However in order to build nonlinear programming methods, not only knowledge of differentiability, but also explicit formulæ for gradients are important. Unfortunately, differentiability of probability functions cannot be taken for granted. In this paper, motivated by applications from energy management, wherein we face a variety of non-linear transforms of underlying Elliptical distributions, we investigate probability functions acting on decision dependent union of polyhedra. Union of polyhedra naturally occur as soon as one approaches the components of “difference-of-convex” (DC) functions with their respective cutting plane models. In this work, we will establish that the probability functions are locally Lipschitzian and exhibit explicit formulæ for “the” Clarke sub-gradients, under very mild conditions. We also highlight, on a numerical example, that the formulæ can be put to use “in practice”.

Appendices

Appendix

A An approximate projected gradient algorithm

We present in this Appendix a generic algorithm, similar to the one used in [67, Section 6] for solving the following problem:

$$ \begin{array}{@{}rcl@{}} \min\limits_{x \in X} \quad & f(x) \\ \text{s.t.} \quad & \phi(x) \ge p, \end{array} $$

where f is assumed to be affine, and ϕ is as defined in (1). We also assume that a point $x_{0} \in X^{\prime } := \{x\in X : \mathbb {P}[ g(x,\xi ) \leq 0] \ge p\}$ is known (and therefore this set is not empty). Let us define $\tau _{X^{\prime }}^{x_{0}} : X \mapsto [0,1]$ as follows:

$$ \tau_{X^{\prime}}^{x_{0}}(x) = \left \{ \begin{array}{cc} \min_{t\in [0,1]} & t \\ \text{s.t.} & \mathbb{P}[ g((1-t)x + tx_{0},\xi) \le 0] \ge p \end{array} \right. $$

and let us define $P_{X^{\prime }}^{x_{0}} : X \rightarrow X^{\prime }$ as $P_{X^{\prime }}^{x_{0}}(x) = (1-\tau _{X^{\prime }}^{x_{0}}(x))x + \tau _{X^{\prime }}^{x_{0}}(x) x_{0}$. Since $X^{\prime }$ is a closed set, $\tau _{X^{\prime }}^{x_{0}}$ and thus $P_{X^{\prime }}^{x_{0}}$ are well defined. Since, it is not a proper projection, this algorithm is sometimes referred to as an approximate projected gradient algorithm. Convergence of such an algorithm is studied for example in [73]. Obtaining a numerical value for $\tau _{X^{\prime }}^{x_{0}}$ can be done using a bisection method, having at each iteration k to compute $\mathbb {P}[g((1-t_{k})x + t_{k} x_{0},\xi ) \le 0]$ and stopping when it reaches p up to a given tolerance.

The formulæ (33) can be exploited to compute ∇ϕ(x_k) at a given trial point upon verifying condition (40).