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We consider two-stage risk-averse mixed-integer recourse models with law invariant coherent risk measures. As in the risk-neutral case, these models are generally non-convex as a result of the integer restrictions on the second-stage decision variables and hence, hard to solve. To overcome this issue, we propose a convex approximation approach. We derive a performance guarantee for this approximation in the form of an asymptotic error bound, which depends on the choice of risk measure. This error bound, which extends an existing error bound for the conditional value at risk, shows that our approximation method works particularly well if the distribution of the random parameters in the model is highly dispersed. For special cases we derive tighter, non-asymptotic error bounds. Whereas our error bounds are valid only for a continuously distributed second-stage right-hand side vector, practical optimization methods often require discrete distributions. In this context, we show that our error bounds provide statistical error bounds for the corresponding (discretized) sample average approximation (SAA) model. In addition, we construct a Benders' decomposition algorithm that uses our convex approximations in an SAA-framework and we provide a performance guarantee for the resulting algorithm solution. Finally, we perform numerical experiments which show that for certain risk measures our approach works even better than our theoretical performance guarantees suggest. [ABSTRACT FROM AUTHOR]

In the regulation of network tariffs, the compensation for the opportunity costs of capital through the Weighted Average Costs of Capital (WACC) plays a crucial role. Determining the appropriate level for the WACC is, though, problematic because of the uncertainty about the future conditions in capital markets. When the WACC is set above the future opportunity costs of capital, consumers will pay too much, while when the WACC is below that level, network operators may be unable to finance investments affecting quality of network services. In this paper, we explicitly take this uncertainty into account when we determine the optimal WACC for the tariff regulation of an electricity network. By trading off consumer surplus and expected disruption costs in the electricity grid, we conclude that from a social-welfare perspective in most cases the optimal WACC in tariff regulation is above the historical mean costs of capital. Only in case of high uncertainty about the true costs of capital while network operators are able to quickly increase investment levels, the optimal WACC is below the historical mean because then it is less likely that the WACC is constantly insufficient to cover actual costs of capital. However, when network operators cannot quickly increase investment levels the optimal WACC is always above the historical mean cost of capital. [ABSTRACT FROM AUTHOR]

We propose a new class of convex approximations for two-stage mixed-integer recourse models, the so-called generalized alpha-approximations. The advantage of these convex approximations over existing ones is that they are more suitable for efficient computations. Indeed, we construct a loose Benders decomposition algorithm that solves large problem instances in reasonable time. To guarantee the performance of the resulting solution, we derive corresponding error bounds that depend on the total variations of the probability density functions of the random variables in the model. The error bounds converge to zero if these total variations converge to zero. We empirically assess our solution method on several test instances, including the SIZES and SSLP instances from SIPLIB. We show that our method finds near-optimal solutions if the variability of the random parameters in the model is large. Moreover, our method outperforms existing methods in terms of computation time, especially for large problem instances. [ABSTRACT FROM AUTHOR]

In traditional two-stage mixed-integer recourse models, the expected value of the total costs is minimized. In order to address risk-averse attitudes of decision makers, we consider a weighted mean-risk objective instead. Conditional value-at-risk is used as our risk measure. Integrality conditions on decision variables make the model non-convex and hence, hard to solve. To tackle this problem, we derive convex approximation models and corresponding error bounds, that depend on the total variations of the density functions of the random right-hand side variables in the model. We show that the error bounds converge to zero if these total variations go to zero. In addition, for the special cases of totally unimodular and simple integer recourse models we derive sharper error bounds. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]