Abstract Two-stage mean-risk stochastic integer programming (MR-SIP) with endogenous uncertainty provides a powerful modeling tool for real-life decision-making problems, as it allows to capture here-and-now decisions that influence future outcomes. However, such problems are difficult to solve due to their nonconvexity and large-scale nature. We derive a decomposition method for this class of MR-SIP and apply it to a critical problem in wildfire management: optimal fuel treatment planning (FTP) under uncertainty. The uncertainty stems from fuels (vegetation), fire occurrence, and weather conditions. Fuel treatment methods such as prescribed burning, mechanical thinning, mowing, grazing, and chemical treatments are aimed at reducing hazardous fuels and thus, influence the uncertainty. In this work, we formulate a novel MR-SIP FTP model that integrates fuel treatment and firefighting resource deployment planning before fires happen, which are typically addressed in isolation rather than in an integrated manner. The new model uses the expected excess risk measure, which given a target level of wildfire damage cost, minimizes the mean excess above the target level. We parameterize the FTP model through standard wildfire behavior simulation software for generating fire scenarios and apply it to a real study area in West Texas, U.S.A. The results provide several practical insights for FTP decision-making. For example, the results reveal that when considering fuel treatment alone, treatment coverage is spread across the study area to high-risk subareas. However, integrating fuel treatment with resource deployment reduces coverage near operation bases, prioritizing high-risk subareas and reducing fire damage cost by 80% on average.

This article presents a new approach for constructing convex enclosures of reachable sets of parametric ordinary differential equations (ODEs), for use in deterministic methods for global dynamic optimization. In our new approach, we modify an established ODE relaxation framework by Scott and Barton (2013), using kinematic intuition to replace certain discontinuous transitions between discrete modes with tighter, smoother transitions, and ultimately producing tighter, smoother relaxations of the original ODE solution that are more amenable to integration by off-the-shelf numerical ODE solvers. We refer to our new relaxation approach as “kinematic smoothing”. Our new ODE relaxations are straightforward to construct automatically based on established tools, and we present several numerical examples based on a proof-of-concept implementation in Julia.

Abstract Existing algorithms for generalized semi-infinite programs can only handle lower-level constraints containing equality constraints depending on upper-level variables (so-called coupling equality constraints) under limiting assumptions. More specifically, discretization-based algorithms require that the coupling equality constraints result in some lower-level variables being determined uniquely as implicit functions of the other lower-level and upper-level variables. We propose an adaptation of the discretization-based algorithm of Blankenship & Falk and demonstrate it can handle coupling equality constraints under the weaker assumption of stability of the solution set for these constraints in the sense of Lipschitz lower semi-continuity. The key idea is to allow a perturbation of the lower-level variable values from discretization points in connection with changes in the upper-level variables in the discretized upper-level problem. We enforce that these perturbed values satisfy the coupling equality constraints while remaining close to the discretization point, provided we can guarantee the stability of the solution in the sense that a nearby solution exists for small changes of the upper-level variables. We provide concrete realizations of the algorithm for three different situations: i) when knowledge about a certain Lipschitz constant is available, ii) when the coupling equality constraints are assumed to have full rank, and iii) when the coupling equality constraints are additionally linear in the lower-level variables. Numerical experiments on small test problems and a physically motivated problem related to power flow illustrate that the approach can be successfully applied to solve the challenging problems, but is currently limited in terms of scalability.