This paper presents an invariant Rauch-Tung-Striebel (IRTS) smoother applicable to systems with states that are an element of a matrix Lie group. In particular, the extended Rauch-Tung-Striebel (RTS) smoother is adapted to work within a matrix Lie group framework. The main advantage of the invariant RTS (IRTS) smoother is that the linearization of the process and measurement models is independent of the state estimate resulting in state-estimate-independent Jacobians when certain technical requirements are met. A sample problem is considered that involves estimation of the three dimensional pose of a rigid body on SE(3), along with sensor biases. The multiplicative RTS (MRTS) smoother is also reviewed and is used as a direct comparison to the proposed IRTS smoother using experimental data. Both smoothing methods are also compared to invariant and multiplicative versions of the Gauss-Newton approach to solving the batch state estimation problem.

Cutting planes need not be valid in stochastic integer optimization. Many practical problems under uncertainty, for example, in energy, logistics, and healthcare, can be modeled as mixed-integer stochastic programs (MISPs). However, such problems are notoriously difficult to solve. In “Pseudo-Valid Cutting Planes for Two-Stage Mixed-Integer Stochastic Programs with Right-Hand-Side Uncertainty,” Romeijnders and van der Laan introduce a novel approach to solve two-stage MISPs. Instead of using exact cuts that are always valid, they propose to use pseudo-valid cutting planes for the second-stage feasible regions that may cut away feasible integer second-stage solutions for some scenarios and may be overly conservative for others. The advantage of using such cutting planes is that the approximating problem remains convex in the first-stage decision variables and thus can be solved efficiently. Moreover, the performance of these cutting planes is good if the variability of the random parameters in the model is large enough.

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.

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Novel Optimality Cuts for Two-Stage Stochastic Mixed-Integer Programs The applicability and use of two-stage stochastic mixed-integer programs is well-established, thus calling for efficient decomposition algorithms to solve them. Such algorithms typically rely on optimality cuts to approximate the expected second stage cost function from below. In “A Converging Benders’ Decomposition Algorithm for Mixed-Integer Recourse Models,” van der Laan and Romeijnders derive a new family of optimality cuts that is sufficiently rich to identify the optimal solution of two-stage stochastic mixed-integer programs in general. That is, general mixed-integer decision variables are allowed in both stages, and all data elements are allowed to be random. Moreover, these new optimality cuts require computations that decompose by scenario, and thus, they can be computed efficiently. Van der Laan and Romeijnders demonstrate the potential of their approach on a range of problem instances, including the DCAP instances from SIPLIB.