A Discussion of Probability Functions and Constraints from a Variational Perspective

Abstract

Probability constraints are a popular modelling mechanism in applications. They help to model feasible decisions when the latter are taken prior to observing uncertainty and both decisions and uncertainty are involved in a constraint structure of an optimization problem. The popularity of this paradigm is attested by a vast literature using probability constraints. In this work we try to provide, with variational analysis in mind, an introduction to the topic. We wish to highlight questions regarding the understanding of theoretical properties, such as continuity, (generalized) differentiability, convexity, but also regarding algorithms. We try to highlight open research avenues whenever possible.

Appendix: Supplementary material

For the convenience of the reader we will provide directly the definition of some basic notions. The first notion discusses continuity properties of set-valued map**s (see, e.g., Def. 5.4 [67]):

Definition A.1

Let $M : \Re ^{n} \rightrightarrows \Re ^{m}$ be a set-valued map**. The map** M is said to be outer semicontinuous at $\bar x \in \Re ^{n}$ if

$$ \limsup_{x \rightarrow \bar x} M(x) \subseteq M(\bar x), $$

which means that any (possible) cluster point z of $\left \{z_{n}\right \}_{n \geq 0}$ must belong to $M(\bar x)$, where z_n ∈ M(x_n), and x_n is a sequence converging to $\bar x$.

The map** M is said to be inner semicontinuous at $\bar x$ if

$$ M(\bar x) \subseteq \liminf_{x \rightarrow \bar x} M(x), $$

which means that any $\bar z \in M(\bar x)$, can be seen as the limit of a sequence z_n ∈ M(x_n), $x_{n} \rightarrow \bar x$.

We have also employed generalized concavity. Once more for the convenience of the reader, we formally introduce this concept. This is best done through the use of an auxiliary map**:

Definition A.2

Let $m_{\alpha } : \mathbb {R}_{+} \times \mathbb {R}_{+} \times [0,1] \rightarrow \mathbb {R}$ (for a given $\alpha \in [-\infty ,\infty ]$) be defined as follows:

$$ \text{if} ab = 0 \text{and} \alpha \leq 0, \qquad m_{\alpha}(a,b,\lambda) = 0 $$

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else, for λ ∈ [0, 1], we let:

$$ m_{\alpha}(a,b,\lambda) = \left \{ \begin{array}{ccc} a^{\lambda}b^{1-\lambda} & \text{if} & \alpha=0 \\ \min\left\{a,b\right\} & \text{if} & \alpha=-\infty \\ \max\left\{a,b\right\} & \text{if} & \alpha=\infty \\ (\lambda a^{\alpha} + (1-\lambda)b^{\alpha})^{\frac{1}{\alpha}} & \text{else} & \end{array} \right. $$

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Definition A.3

Let C be a convex subset of Rⁿ. We say that a function $f : C \rightarrow \mathbb {R}_{+}$ is α-concave if

$$ f(\lambda x + (1-\lambda) y) \geq m_{\alpha}(f(x),f(y),\lambda), $$

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for all x,y ∈ C and λ ∈ [0, 1].

The map** is said to be α-convex if

$$ f(\lambda x + (1-\lambda) y) \leq m_{\alpha}(f(x),f(y),\lambda), $$

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for all x,y ∈ C and λ ∈ [0, 1].