Total variation bounds on the expectation of periodic functions with applications to recourse approximations

Abstract

We derive a lower and upper bound for the expectation of periodic functions, depending on the total variation of the probability density function of the underlying random variable. Using worst-case analysis we derive tighter bounds for functions that are periodically monotone. These bounds can be used to evaluate the performance of approximations for both continuous and integer recourse models. In this paper, we introduce a new convex approximation for totally unimodular recourse models, and we show that this convex approximation has the best worst-case error bound possible, improving previous bounds with a factor 2. Moreover, we use similar analysis to derive error bounds for two types of discrete approximations of continuous recourse models with continuous random variables. Furthermore, we derive a tractable Lipschitz constant for pure integer recourse models.

Appendix

In this appendix we give the proofs of Lemmas 1 and 2 of Sect. 1.1.

Proof of Lemma 1

Properties (i) and (ii) follow trivially from the definition of $M$ and $N$.

To show (iii) let $r > 0$ be given, and consider $\bar{\varphi }_r(x) := \varphi (x/r), \, x \in \mathbb {R}^{}$. We will show that for every $f \in \mathcal{{F}}$ with $|\varDelta |f \le B$, there exists $g \in \mathcal{{F}}$ with $|\varDelta |g \le rB$ such that $\mathbb {E}_{f}[\bar{\varphi }_r(\omega )] = \mathbb {E}_g[\varphi (\omega )]$, and vice versa, that for every $f \in \mathcal{{F}}$ with $|\varDelta |f \le rB$ there exists $g \in \mathcal{{F}}$ with $|\varDelta |g \le B$ such that $\mathbb {E}_{f}[\varphi (\omega )] = \mathbb {E}_g[\bar{\varphi }_r(\omega )]$. Together these results imply that (iii) holds. Observe that for $f \in \mathcal{{F}}$ with $|\varDelta |f \le B$ the pdf $g$ defined as $g(x) := rf(rx), \, x \in \mathbb {R}^{}$ satisfies the first conditions and for $f \in \mathcal{{F}}$ with $|\varDelta |f \le rB$ the pdf $g$ defined as $g(x) := r^{-1}f(x/r)$ satisfies the latter.

Similarly, for $\hat{\varphi }(x) := \varphi (-x), \, x \in \mathbb {R}^{}$, let $f \in \mathcal{{F}}$ with $|\varDelta |f \le B$ be given. Then, $g(x) := f(-x), \, x \in \mathbb {R}^{},$ satisfies $g \in \mathcal{{F}}$ with $|\varDelta |g = |\varDelta |f \le B$ and $\mathbb {E}_{f}[\hat{\varphi }(\omega )] = \mathbb {E}_{g}[\varphi (\omega )]$, implying

$$\begin{aligned} M(\hat{\varphi },B) \ge M(\varphi ,B) \quad \text{ and } \quad N(\hat{\varphi },B) \le N(\varphi ,B). \end{aligned}$$

Since $\varphi (x) = \hat{\varphi }(-x), \, x \in \mathbb {R}^{}$, the reverse inequalities hold as well, proving (iv).

Finally, property (v) can be proven in a similar way, using that for every $f \in \mathcal{{F}}$ with $|\varDelta |f \le B$, and $\beta \in \mathbb {R}^{}$, the pdf $g(x) {:=} f(x+\beta ), \, x \in \mathbb {R}^{},$ satisfies $|\varDelta |g = |\varDelta |f \le B$.$\square$

Proof of Lemma 2

Since for every $f^0,f^1 \in \mathcal{{F}}$ with $|\varDelta |f^0 \le B$ and $|\varDelta |f^1 \le B$, and $0 \le t \le 1$, the pdf $f := (1-t)f^0 + tf^1$ satisfies

$$\begin{aligned} |\varDelta |f \le (1-t)|\varDelta |f^0 + t|\varDelta |f^1 \le B, \end{aligned}$$

it follows that the constraint $|\varDelta |f \le B$ in $M(\varphi ,B)$ and $N(\varphi ,B)$ is convex. Since, in addition, the objective $\mathbb {E}_{f}[\varphi (\omega )]$ is linear in $f$, we conclude that both $M(\varphi ,B)$ and $N(\varphi ,B)$ are convex optimization problems with a linear objective. Since $M(\varphi ,B)$ is a maximization problem and $N(\varphi ,B)$ is a minimization problem, it follows that $M(\varphi ,B)$ is concave in $B$, and $N(\varphi ,B)$ is convex in $B$, respectively.$\square$