# Higher-order total variation bounds for expectations of periodic functions and simple integer recourse approximations

## Abstract

We derive bounds on the expectation of a class of periodic functions using the total variations of higher-order derivatives of the underlying probability density function. These bounds are a strict improvement over those of Romeijnders et al. (Math Program 157:3–46, 2016b), and we use them to derive error bounds for convex approximations of simple integer recourse models. In fact, we obtain a hierarchy of error bounds that become tighter if the total variations of additional higher-order derivatives are taken into account. Moreover, each error bound decreases if these total variations become smaller. The improved bounds may be used to derive tighter error bounds for convex approximations of more general recourse models involving integer decision variables.

## Keywords

Stochastic integer programming Convex approximations Error bounds## 1 Introduction

*Q*is defined for the tender variables \(z \in {\mathbb {R}}^m\) as

*v*is defined for \(s \in {\mathbb {R}}^m\) as

*x*has to be made while the realization of \(\omega \) is not yet available, whereas in the second stage, the realization of \(\omega \) is known and recourse actions

*y*can be taken to repair infeasibilities of the random constraints \(Tx \ge \omega \). The model is called an integer recourse model if \(Y = {\mathbb {Z}}^{n_2}_+\). If in addition \(W = I_m\), then the model is referred to as a simple integer recourse (SIR) model. More general formulations of (1) exist, with uncertainty in the technology matrix

*T*and cost parameters

*q*, see e.g. the textbooks (Birge and Louveaux 1997; Shapiro et al. 2009).

*Q*is finite everywhere.

- (i)
The recourse is complete: for all \(s \in {\mathbb {R}}^m\), there exists a \(y \in Y\) such that \(Wy \ge s\), so that \(v(s) < \infty \).

- (ii)
The recourse is sufficiently expensive: \(v(s) > -\infty \) for all \(s \in {\mathbb {R}}^m\).

- (iii)
\({\mathbb {E}}[|\omega _i|]\) is finite for all \(i=1,\ldots ,m\).

Unfortunately, solving integer recourse problems is generally time-consuming and practically impossible, because the recourse function *Q* is in general non-convex (Rinnooy Kan and Stougie 1988). Traditional solution methods typically combine ideas from deterministic mixed-integer programming and stochastic continuous programming, see e.g. Laporte and Louveaux (1993), Schultz et al. (1998), Carøe and Schultz (1999), Ahmed et al. (2004), Sen and Higle (2005), Ntaimo (2013), Zhang and Küçükyavuz (2014) and Gade et al. (2014), and the survey papers by Schultz (2003), Sen (2005) and Romeijnders et al. (2014).

However, in this paper we focus on an alternative solution methodology introduced by van der Vlerk (1995). His approach is to approximate the non-convex recourse function *Q* by a convex approximation \(\hat{Q}\), obtaining an approximating model for (1). The advantage is that the approximating model can be solved efficiently using known convex optimization techniques to obtain an approximate solution \(({\hat{x}}, {\hat{z}})\) for (1).

In the literature, convex approximations are typically derived by simultaneously modifying the underlying recourse structure (*Y*, *q*, *W*) and the distribution of the random vector \(\omega \). For example, Klein Haneveld et al. (2006) propose the so-called \(\alpha \)-approximations for SIR models, van der Vlerk (2004) studies a class of convex approximations for the general integer case, and Romeijnders et al. (2016b) propose a convex approximation, the shifted LP-relaxation, for integer recourse problems with a totally unimodular (TU) recourse matrix *W*. The latter approximation is generalized to the general mixed-integer recourse case by Romeijnders et al. (2016a).

*Q*and its convex approximation \(\hat{Q}\). In this way, bounds on \(\Vert \hat{Q}-Q\Vert _{\infty }\) are obtained by deriving total variation bounds on the expectation of periodic functions. Since we also exploit this relationship between expectations of periodic functions and the difference between \(\hat{Q}\) and

*Q*, we explain this relationship in more detail in Sect. 2.

Romeijnders et al. (2016b) use worst-case analysis to prove that their error bounds are tight for certain piecewise constant pdf. For other pdf, there may be a considerable difference between the error bound and the actual error as shown by numerical experiments on a fleet allocation and routing problem and on an investment problem in stochastic activity networks (Romeijnders et al. 2017). Motivated by these observations, we improve the error bounds of Romeijnders et al. (2016b) by using information on the higher-order derivatives of the underlying pdf of the random variables in the model. To be specific, we use that the total variation of the higher-order derivatives of the underlying pdf can be used to improve the error bound. The intuition behind our approach is that by imposing restrictions on the higher-order derivatives of the underlying pdf we are able to exclude the piecewise constant pdf, which have jump discontinuities.

To obtain these error bounds we improve the existing bounds on the expectation of periodic functions by Romeijnders et al. (2016b) using higher-order total variations, i.e. total variations of the higher-order derivatives of the underlying marginal pdfs. This is the main contribution of this paper since these bounds may be used to improve error bounds of convex approximations for mixed-integer recourse models in general. We illustrate their potential by improving error bounds for the shifted LP-relaxation approximation of SIR models by Romeijnders et al. (2016b). The improved error bounds decrease with the total variations of the underlying pdf and its higher-order derivatives.

The remainder of this paper is organized as follows. Section 2 describes in more detail the relationship between error bounds for the shifted LP-relaxation of integer recourse models and bounds on the expectation of periodic functions. In Sect. 3, we improve bounds on the expectation of a class of periodically monotone functions, and in Sect. 4 we illustrate how these results can be applied in the setting of SIR models. In Sect. 5 we conclude and summarize our results.

## 2 Total variation error bounds

In this section, we describe parts of the procedure employed by Romeijnders et al. (2016b) to derive an error bound for the so-called *shifted LP-relaxation approximation* \({\hat{Q}}\) of *Q* for TU integer recourse models. The idea behind this approximation is to simultaneously relax the integrality restrictions in the model and to perturb the random right-hand side vector \(\omega \).

### Definition 1

*shifted LP-relaxation*\({\hat{Q}}\) of the mixed-integer recourse function Q is defined as

*m*-dimensional all-one vector.

*Q*reduces to

The error bounds are based on a *worst-case approach* with respect to the total variation \(|{\varDelta }| f\) of the underlying pdf *f* of \(\omega \), since the expectation in (2) is intractable in general. The following definition of total variation is taken directly from Romeijnders et al. (2016b).

### Definition 2

*I*. Then, the

*total variation*of

*f*on

*I*, denoted \(|{\varDelta }|f(I)\), is defined as

*f*is said to be of bounded variation if \(|{\varDelta }|f\) is finite.

### Theorem 1

*Q*defined as

*f*of bounded variation. Then

### Proof

See Romeijnders et al. (2016b). \(\square \)

### Remark 1

Observe that \(\hat{f}\) can be interpreted as a pdf with \(|{\varDelta }|\hat{f}' = +\infty \).

In the next sections we will assume that *f* is continuously differentiable and that its derivative \(f'\) is of bounded variation. In this way, we exclude piecewise constant densities such as \(\hat{f}\). By using the higher-order total variations of *f* we are able to derive tighter bounds.

## 3 Improving bounds on the expectation of periodic functions

*half-periodic*functions \(\psi :{\mathbb {R}} \rightarrow {\mathbb {R}}\), which are of the form

The organization of this section is as follows. In Sect. 3.1, we consider packed densities, a concept introduced by Romeijnders et al. (2016b) which we generalize to higher-order derivatives. Next, in Sect. 3.2, we introduce *point-symmetric periodic functions*, the class of periodic functions for which we are able to derive results. In Sect. 3.3 we use the concept of bound propagation to derive a hierarchy of total variation bounds. In Sect. 3.4, we present bounds on the expectation of periodic and half-periodic functions.

### 3.1 Higher-order derivatives of packed densities

The main contribution of this paper is to generalize the results by Romeijnders et al. (2016b) to higher-order derivatives. The key insight here is that including information on the total variation and the maximum norm of higher-order derivatives of the underlying pdf leads to improved expectation bounds. We generalize several definitions by Romeijnders et al. (2016b) to allow for these elements to be included in our analysis. They consider the set \({\mathcal {F}}\) containing all one-dimensional pdf of bounded variation. In Definition 3, we introduce the sets \({\mathcal {F}}_n\), \(n \in {\mathbb {N}}\), which satisfy the relationship \({\mathcal {F}}\supset {\mathcal {F}}_0 \supset {\mathcal {F}}_1 \supset \cdots \).

### Definition 3

Let \({\mathcal {F}}_n\) denote the set of one-dimensional pdf *f*, such that the first *n* derivatives of *f* exist, are continuous, and are of bounded variation. Denote the *k*-th derivative of *f* by \(f^{(k)}\) and write \(f = f^{(0)}\).

### Definition 4

### Remark 2

In Definition 4, we could have suppressed the constants \(C_0,\ldots ,C_n\), since an upper bound \(B_k\) on \(|{\varDelta }|f^{(k)}\) directly implies an upper bound \(\frac{B_k}{2}\) on \(||f^{(k)}||_{\infty }\). However, these bounds may be larger than \(C_k\) so that by including these constants we may obtain tighter bounds.

*p*] such that for either periodic of half-periodic functions \(\varphi \) with period

*p*,

*half-packed density*. This packed density does not include the values of \(f(x + pk)\) for \(k < 0\), since \(\varphi \) is only half-periodic and thus \(\varphi (x + pk) = 0\) for \(k < 0\). Definition 5 below contains the definitions of both the original and the new packed density.

### Definition 5

*classical packed density*\(f_p :[0,p] \rightarrow {\mathbb {R}}\) as

*half-packed density*\(\hat{f}_p:[0,p] \rightarrow {\mathbb {R}}\) as

### Remark 3

Note that the half-packed density does not integrate to one in general, as opposed to the classical packed density. As a result, the half-packed density cannot be interpreted as a pdf.

### Remark 4

We define the (higher-order) derivatives of the classical packed density and the half-packed density at the endpoints of the closed interval [0, *p*] by their one-sided derivatives, provided that they exist.

Romeijnders et al. (2016b) show that the total variation of *f* can be used to bound the total variation of \(f_p\) on [0, *p*]. Interestingly, similar bounds can be derived for the total variation of half-packed densities \(\hat{f}_p\) on [0, *p*]. Moreover, similar bounds also hold for higher-order derivatives \(f^{(k)}, k=1,\ldots ,n\), of *f*.

### Lemma 1

- (i)
\(f_p^{(k)}\) and \({\hat{f}}_p^{(k)}\) exist and are continuous on [0,

*p*], - (ii)
\(f_p^{(k)}(0)= f_p^{(k)}(p)\) and \({\hat{f}}_p^{(k)}(0)= {\hat{f}}_p^{(k)}(p) + f^{(k)}(0)\), and

- (iii)\(|{\varDelta }| f_p^{(k)}([0,p]) \le |{\varDelta }| f^{(k)}\) and \(|{\varDelta }|{\hat{f}}_p^{(k)}([0,p]) \le |{\varDelta }|f^{(k)} - |f^{(k)}(0)|\). Moreover, for all bounded integrable periodic functions \(\varphi \) with period
*p*, and half-periodic functions \(\psi \) of the form$$\begin{aligned} \psi (x) = \left\{ \begin{array}{ll}0 ,&{} x < 0 ,\\ \varphi (x),&{} x \ge 0,\end{array}\right. \end{aligned}$$ - (iv)
\({\mathbb {E}}_{f}[\varphi (\omega )] = \int _0^p\varphi (x) f_p(x)dx\) and \({\mathbb {E}}_{f}[\psi (\omega )] = \int _0^p\varphi (x){\hat{f}}_p(x)dx\).

### Proof

We refer to the proof of corresponding properties involving packed densities in Romeijnders et al. (2016b). \(\square \)

Observe that the properties involving the classical packed density generalize readily to higher-order derivatives. With respect to the half-packed density, notice the additional terms \(f^{(k)}(0)\) and \(-|f^{(k)}(0)|\) in properties (ii) and (iii), respectively. These terms result from the fact that the half-packed density only sums over the non-negative integers.

In our analysis in the next sections, we will initially not make a distinction between the classical packed density and the half-packed density. This is possible because they belong to the broader class of packed densities, which we introduce in Definition 6.

### Definition 6

For a given \(n \in {\mathbb {N}}\) and \(p \in {\mathbb {R}}\) with \(p >0\), let \(g:[0,p]\rightarrow {\mathbb {R}}\) be an *n* times continuously differentiable function such that its first *n* derivatives are of bounded variation on the interval [0, *p*]. Assume \(\gamma _k := g^{(k)}(p) - g^{(k)}(0)\) is bounded for \(k = 0,\ldots ,n\). Then *g* is referred to as a packed density with discontinuities \( \gamma = (\gamma _0,\ldots ,\gamma _n)\).

*g*of Definition 6. It equals the expectation of a periodic function if

*g*is a classical packed density, in which case the discontinuities corresponding to

*g*are \( \gamma = (0,\ldots ,0)\). Furthermore, for \(\varphi \) defined as in (3), the quantity \(D(\varphi ,g)\) is equal to the approximation error of the shifted-LP relaxation of a simple integer recourse model if

*g*is a half-packed density, \(\hat{f}_p\) of Definition 5, with \( \gamma = (f^{(0)}(0),\ldots , f^{(n)}(0))\).

### 3.2 Point-symmetric periodic functions

We restrict our attention to *point-symmetric periodic functions*, a concept we introduce in Definition 7, since the underlying periodic function that arises when studying the shifted LP-relaxation for TU integer recourse models is point-symmetric periodic with period \(p=1\). However, our results apply to all point-symmetric periodic functions \(\varphi \) with period \(p >0\).

### Definition 7

*p*satisfying

*point-symmetric periodic*, or a PSP(

*p*) function.

For a packed density *g* of Definition 6 and a PSP(*p*) function \(\varphi \), we are able to derive an upper bound on \(D(\varphi ,g)\). The first step we take in obtaining such a bound is to prove in Lemma 2 that there exists a function \(g_p\) that is symmetric in the same sense as \(\varphi \) and carries all relevant information of *g* for computing \(D(\varphi ,g)\). Moreover, the total variation of \(g_p\) on [0, *p*] does not exceed that of *g* on [0, *p*]. Next, in Lemma 3, we obtain bounds on higher-order derivatives of \(g_p\) by combining the symmetry of \(g_p\) with the mean-value theorem. In Sect. 3.3, we translate these bounds into a single bound on \(g_p\), a process we refer to as *bound propagation*.

### Lemma 2

*p*) function. For a packed density

*g*of Definition 6, define \(g_p:[0,p] \rightarrow {\mathbb {R}}\) as

*k*satisfying \(0\le k \le n\),

### Proof

*p*) function. Using (iv) in Lemma 1 we have

*f*of bounded variation, and where the inequality follows from \(|{\varDelta }|(f + g)(I) \le |{\varDelta }|f(I) + |{\varDelta }|g(I)\) for

*f*and

*g*of bounded variation. \(\square \)

We now state and prove a number of properties of \(g_p\) that we will use to prove the results in Sects. 3.3 and 3.4. We make a distinction between odd and even-order derivatives of \(g_p\), because they share different properties.

### Lemma 3

*k*, \(0 \le k \le n\),

*k*, \(1 \le k \le n\),

### Proof

*k*and assume for contradiction that \(\left| g_p^{(k)}(t^*)\right| > w_k\) for some \(t^* \in \left[ 0,p/2\right) \). This implies that

Property (v) follows in a similar fashion, since by combining (iv) with (8) it is possible to arrive at a contradiction with property (ii) in Lemma 2 if (v) does not hold. \(\square \)

### 3.3 Bound propagation

In this section, we use the bounds on \(g_p^{(k)}, k =0,\ldots ,n\), in Lemma 3 to derive a single tighter bound on \(g_p\) as defined in (7) for a given packed density *g* with known discontinuities \((\gamma _0,\ldots ,\gamma _n)\). We use information on the total variation of *g* and its higher-order derivatives to bound \(g_p\). Clearly, we obtain tighter bounds if we include information on more higher-order derivatives of *g*.

For illustration, suppose that information is available on *g* and \(g^{(1)}\). Given a bound on \(|{\varDelta }|g\), property (iii) in Lemma 3 directly yields a uniform bound on \(|g_p(x)|\) for \(x \in [0,p]\), denoted \(w_0\). An additional bound on \(|{\varDelta }|g^{(1)}\) yields a bound on \(|g_p^{(1)}(x)|\), denoted \(w_1\), using property (v) in Lemma 3. The latter bound can be used to improve the bound on \(|g_p(x)|\) itself, using the concept of bound propagation, which we will now demonstrate.

*q*(

*x*) be such that \(\left| g_p^{(1)}(x)\right| \le q(x)\) for \(x \in \left[ 0,p/2\right] \). Using that \(g_p\left( p/2\right) = 0\) according to property (i) in Lemma 3, and that

*q*(

*x*) on \(|g_p^{(1)}(x)|\) for all \(x \in [0,p/2]\) into an upper bound on \(|g_p(x)|\) for every \(x \in [0,p/2]\). We formalize this transformation by defining an operator

*T*that maps the upper bound

*q*on \(|g_p^{(1)}|\) into an upper bound

*Tq*on \(|g_p|\). This operator is defined as,

*Tq*)(

*x*) provides a tighter bound than the bound based on \(|{\varDelta }|g\) alone, which is given by \(w_0\).

We generalize this idea to higher-order derivatives by introducing appropriate operators in Definition 8. The idea is that a bound *q* on \(|g_p^n|\) may be propogated in dynamic programming fashion to obtain a bound \(T \cdots Tq\) on \(|g_p|\). In our case, however, we have to define two operators \(T^1\) and \(T^2\) since the process of bound propagation is different for odd and even higher-order derivatives. In Definition 9, we use these operators to define functions \(q_0^n\), \(n \in {\mathbb {N}}\), which represent a bound on \(\left| g_p(x)\right| \) based on information on the first *n* derivatives of *g*. Intuitively, including more derivatives should lead to sharper bounds. This intuition is confirmed in Corollary 1.

### Definition 8

### Definition 9

Before we are ready to prove that \(q_0^n\) yields a bound on \(|g_p|\), we need some elementary properties of the operators introduced in Definition 8. In Lemma 4, we prove that \(T^j\), \(j = 1,2\), are non-negative and monotone operators.

### Lemma 4

- (i)
\(T^j_{w,\gamma }q\) is a non-negative function, \(j = 1,2\), and

- (ii)
\(T^j_{w,\gamma }{\bar{q}} \ge T^i_{w,\gamma }q\), \(j = 1,2\).

### Proof

Property (i) follows directly from the non-negativity of *q*. Property (ii) is a direct consequence of \({\bar{q}} \ge q\). \(\square \)

### Proposition 1

### Proof

*m*separately. For even

*m*, we have to show that

*m*then follows by combining (10), (11), and (12).

*m*.

*m*, we have to show that

*m*, and the proof of (9). \(\square \)

We now present a corollary of Lemma 4, which states the intuitive result that the bounds on \(g_p\) become sharper if more higher-order derivatives are included.

### Corollary 1

### Proof

Fix an \(l \in \{1,\ldots ,n\}\). We will use backward induction to prove that \(q_k^{l-1} \ge q_k^{l}\) for \(k = 0,\ldots ,l-1\). The claim then follows by setting \(k= 0\).

It follows directly from the definition of \(q_k^n\) that \(q_{l-1}^{l-1} \ge q_{l-1}^{l}\). For the induction step, suppose that \(q_k^{l-1} \ge q_k^{l}\) for some *k*, \(0<k\le l - 1\). It follows from the monotonicity of \(T^j\), \(j = 1,2\), that \(q_{k-1}^{l-1} \ge q_{k-1}^{l}\), completing the proof. \(\square \)

### 3.4 Error bounds

We are now ready to state the main results of this section. In Theorem 2, we formulate an improved bound on the expectation of PSP(*p*) functions and Theorem 3 states the bound on the expectation of half-periodic functions where the underlying periodic function is PSP(*p*).

### Theorem 2

*p*) function and let \( B = (B_0,\ldots ,B_n)\) and \( C = (C_0,\ldots ,C_n)\) be positive constants. Then, for \(M_n\) defined as

### Proof

*p*) function \(\varphi \) and pdf \(f \in {\mathcal {F}}_n\), we know from Lemma 1 that

*f*. The function \(f_p\) is a packed density with discontinuities \(\gamma = (0,\ldots ,0)\). Lemma 2 informs us that

### Remark 5

Note that the bound presented in Theorem 2 is independent of *C*. This is because the discontinuities of the classical packed density are zero.

### Theorem 3

*p*) function. Let \(B = (B_0,\ldots ,B_n)\) and \( C = (C_0,\ldots ,C_n)\) be positive constants. Then, for \(M_n\) defined as

### Proof

*p*) function \(\varphi \) and pdf \(f \in {\mathcal {F}}_n\), we know from Lemma 1 that

*f*. The function \({\hat{f}}_p\) is a packed density with discontinuities \(\gamma = (f^{(0)}(0),\ldots ,f^{(n)}(0))\). Lemma 2 informs us that

## 4 Applications and examples

In this section, we apply the results in Sect. 3 to specific functions. As in Sect. 3, we make a distinction between periodic and half-periodic functions. First, we apply Theorem 2 to a particular PSP(*p*) function. Second, we derive an improved error bound for the shifted LP-relaxation approximation of one-dimensional SIR models. Since the underlying difference function is half-periodic, we can use Theorem 3 to derive such a bound. This error bound can be generalized to higher-dimensional SIR models. Finally, we conduct numerical experiments to compare the performance of the error bound by Romeijnders et al. (2016b) and the improved bound derived in this section.

### 4.1 Point symmetric periodic functions

### Example 1

*f*denote the pdf of a normally distributed random variable \(\omega \) with variance \(\sigma ^2\). Then,

Note that our results only apply to continuously differentiable pdf, nevertheless, for continuous pdf *f* that are not continuously differentiable we can still apply our results by considering the right derivative of *f*, denoted \(f'_+\), instead of the derivative of *f*. The reason for this lies in the fact that there exists a continuously differentiable approximation \({\tilde{f}}\) of *f* such that \(|{\varDelta }|f = |{\varDelta }|{\tilde{f}}\), \(|{\varDelta }|f'_+ = |{\varDelta }|\tilde{f}'\), and \({\mathbb {E}}_f[\varphi (\omega - z)]\) is arbitrarily close to \({\mathbb {E}}_{{\tilde{f}}}[\varphi (\omega - z)]\). We illustrate this in the next example.

### Example 2

*a*,

*b*] and mode

*m*, \(a< m < b\). See Fig. 3 for illustration.

*f*. Write \(d := b -a\) and note that \(f(m) = \frac{2}{d}\), so that,

*m*, however, changing

*m*does affect \(|{\varDelta }|f'_+\). In fact, for given

*a*and

*b*, \(|{\varDelta }| f'_+\) is minimized by \(m = \frac{1}{2}(a + b)\). Hence, as the mode is closer to the midpoint of the support of \(\omega \), \(|{\varDelta }|f'_+\) is smaller and we obtain tighter bounds. \(\lozenge \)

### 4.2 Error bound shifted-LP relaxation

In this section, we consider the underlying difference function \(\psi \), defined in (4), for the shifted LP-relaxation for SIR models. Recall that bounds on the expectation of \(\psi \) can be used directly to derive error bounds for the shifted LP-relaxation for SIR models.

*f*,

*f*is unimodal. For this reason, we take \(C_0 = \frac{B_0}{2}\), so that the resulting bound holds for all pdf \(f \in {\mathcal {F}}_1\). For unimodal pdf’s the resulting bound is the tightest bound we can provide, whereas tighter bounds can be derived for non-unimodal pdf. Simple computations yield

*x*and

*y*. We thus have that for any \(f \in {\mathcal {F}}_1\),

### Example 3

We numerically evaluate \(||{\hat{Q}} - Q||_{\infty }\) in the one-dimensional SIR case, where \({\hat{Q}}\) denotes the shifted-LP relaxation by Romeijnders et al. (2016b) as in Definition 1 and where \(\omega \) follows a normal distribution with arbitrary mean and variance \(\sigma ^2\), for \(\sigma \in [0.2,4]\). We compare the actual error to the error bound based on Romeijnders et al. (2016b) and the improved error bounds based on \(M_1\) and \(M_2\). We omit the analytical expression for the bound based on \(M_2\), which is tractable but cumbersome.

*f*. To compute the (improved) error bounds, we make use of the expressions for \(|{\varDelta }|f\) and \( |{\varDelta }|f'\) found in Example 1. It follows from these expressions that

## 5 Conclusion

We consider existing convex approximations for two-stage mixed-integer recourse models. We construct a hierarchy of bounds on the expectation of periodic and half-periodic functions using total variations of higher-order derivatives of the underlying probability density function. We use these results to derive improved error bounds for the shifted LP-relaxation of simple integer recourse models. Moreover, the results presented here may be used to improve error bounds of convex approximations for general mixed-integer recourse model approximations.

There are multiple directions for future research. One extension is to generalize our results to a to a higher-dimensional setting. A first step in this direction may be to consider totally unimodular integer recourse models. Another avenue is to apply the results to a particular application of integer recourse models. Finally, our results may be extended to a larger class of periodic functions, which may be useful for other types of convex approximations.

## Notes

### Acknowledgements

We are grateful to Wim Klein Haneveld for his constructive comments on a preliminary version of our manuscript.

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