Mathematics

E-commerce with major online retailers is changing the way people consume. The goal of increasing delivery speed while remaining cost-effective poses significant new challenges for supply chains as they race to satisfy the growing and fast-changing demand. In this paper, we consider a warehouse with a Robotic Mobile Fulfillment System (RMFS), in which a fleet of robots stores and retrieves shelves of items and brings them to human pickers. To adapt to changing demand, uncertainty, and differentiated service (e.g., prime vs. regular), one can dynamically modify the storage allocation of a shelf. The objective is to define a dynamic storage policy to minimise the average cycle time used by the robots to fulfil requests. We propose formulating this system as a Partially Observable Markov Decision Process, and using a Deep Q-learning agent from Reinforcement Learning, to learn an efficient real-time storage policy that leverages repeated experiences and insightful forecasts using simulations. Additionally, we develop a rollout strategy to enhance our method by leveraging more information available at a given time step. Using simulations to compare our method to traditional storage rules used in the industry showed preliminary results up to 14\% better in terms of travelling times.

Obtaining solutions to Optimal Transportation (OT) problems is typically intractable when the marginal spaces are continuous. Recent research has focused on approximating continuous solutions with discretization methods based on i.i.d. sampling, and has proven convergence as the sample size increases. However, obtaining OT solutions with large sample sizes requires intensive computation effort, that can be prohibitive in practice. In this paper, we propose an algorithm for calculating discretizations with a given number of points for marginal distributions, by minimizing the (entropy-regularized) Wasserstein distance, and result in plans that are comparable to those obtained with much larger numbers of i.i.d. samples. Moreover, a local version of such discretizations which is parallelizable for large scale applications is proposed. We prove bounds for our approximation and demonstrate performance on a wide range of problems.

We propose a principled method for projecting an arbitrary square matrix to the non-convex set of asymptotically stable matrices. Leveraging ideas from large deviations theory, we show that this projection is optimal in an information-theoretic sense and that it simply amounts to shifting the initial matrix by an optimal linear quadratic feedback gain, which can be computed exactly and highly efficiently by solving a standard linear quadratic regulator problem. The proposed approach allows us to learn the system matrix of a stable linear dynamical system from a single trajectory of correlated state observations. The resulting estimator is guaranteed to be stable and offers explicit statistical bounds on the estimation error.

We investigate the problem of parameter selection for the scaled trust-region Newton (STRN) algorithm in solving bound-constrained nonlinear equations. Numerical experiments were performed on a large number of test problems to find the best value range of parameters that give the least algorithm iterations and function evaluations. Our experiments demonstrate that, in general, there is no best parameter to be chosen and each specific value shows an efficient performance on some problems and weak performance on other ones. In this research, we report the performance of STRN for various choices of parameters and then suggest the most effective one.

A novel approach to efficiently treat pure-state equality constraints in optimal control problems (OCPs) using a Riccati recursion algorithm is proposed. The proposed method transforms a pure-state equality constraint into a mixed state-control constraint such that the constraint is expressed by variables at a certain previous time stage. It is showed that if the solution satisfies the second-order sufficient conditions of the OCP with the transformed mixed state-control constraints, it is a local minimum of the OCP with the original pure-state constraints. A Riccati recursion algorithm is derived to solve the OCP using the transformed constraints with linear time complexity in the grid number of the horizon, in contrast to a previous approach that scales cubically with respect to the total dimension of the pure-state equality constraints. Numerical experiments on the whole-body optimal control of quadrupedal gaits that involve pure-state equality constraints owing to contact switches demonstrate the effectiveness of the proposed method over existing approaches.

Full waveform inversion (FWI) requires an accurate estimation of source signatures. Due to the coupling between the source signatures and the subsurface model, small errors in the former can translate into large errors in the latter. When direct methods are used to solve the forward problem, classical frequency-domain FWI efficiently processes multiple sources for source signature and wavefield estimations once a single Lower-Upper (LU) decomposition of the wave-equation operator has been performed. However, this efficient FWI formulation is based on the exact solution of the wave equation and hence is highly sensitive to the inaccuracy of the velocity model due to the cycle skipping pathology. Recent extended-space FWI variants tackle this sensitivity issue through a relaxation of the wave equation combined with data assimilation, allowing the wavefields to closely match the data from the first inversion iteration. Then, the subsurface parameters are updated by minimizing the wave-equation violations. When the wavefields and the source signatures are jointly estimated with this approach, the extended wave equation operator becomes source dependent, hence making direct methods ineffective. In this paper, we propose a simple method to bypass this issue and estimate source signatures efficiently during extended FWI. The proposed method replaces each source with a blended source during each data-assimilated wavefield reconstruction to make the extended wave equation operator source independent. Besides computational efficiency, the additional degrees of freedom introduced by spatially distributing the sources allows for a better signature estimation at the physical location when the velocity model is rough. Numerical tests on the Marmousi II and 2004 BP salt synthetic models confirm the efficiency and the robustness of the proposed method.

The influence maximization problem (IMP) is a hot topic, which asks for identifying a limited number of key individuals to spread influence in a social network such that the expected number of influenced individuals is maximized. The stochastic maximal covering location problem (SMCLP) formulation is a mixed integer programming formulation that approximates the IMP by the Monte-Carlo sampling. However, the SMCLP formulation cannot be solved efficiently using existing exact algorithms due to its large problem size. In this paper, we concentrate on deriving presolving methods to reduce the problem size and hence enhance the capability of employing exact algorithms in solving the IMP. In particular, we propose two effective presolving methods, called the strongly connected nodes aggregation (SCNA) and the isomorphic nodes aggregation (INA). The SCNA enables us to build a new SMCLP formulation that is much more compact than the existing one. For the INA, an analysis is given on the one-way bipartite social network. Specifically, we show that under certain conditions, the problem size of the reduced SMCLP formulation depends only on the size of the given network but not on the number of samplings and this reduced formulation is strongly polynomialtime solvable. Finally, we integrate the proposed presolving methods SCNA and INA into the Benders decomposition algorithm, which is recognized as one of the state-of-the-art exact algorithms for solving the IMP. Numerical results demonstrate that with the SCNA and INA, the Benders decomposition algorithm is much more effective in solving the IMP in terms of solution time.

We propose a method to non-rigidly align a three-dimensional (3D) volumetric image with a two-dimensional (2D) planar image representing a projection of the deformed volume. The application in mind comes from biological studies in which 2D intravital microscopy videos of living tissue are recorded, after which the tissue is excised and a more detailed 3D volume microscopy is performed. Coregistration of both data sets allows to combine the temporal (but 2D) information with more detailed spatial 3D information. Our approach is variational and uses a hyperelastic deformation regularization, as is appropriate for biological material. As a particular feature, the out of plane deformation is estimated based on the out of focus blur inside the 2D microscopy image. The approach becomes computationally feasible through the use of a coarse-to-fine optimization strategy and higher order optimization methods.

We consider a manufacturer who controls the inventory of spare parts in the end-of-life phase and takes one of three actions at each period: (1) place an order, (2) use existing inventory, (3) stop holding inventory and use an outside/alternative source. Two examples of this source are discounts for a new generation product and delegating operations. Demand is described by a non-homogeneous Poisson process, and the decision to stop holding inventory is described by a stopping time. After formulating this problem as an optimal stopping problem with additional decisions and presenting its dynamic programming algorithm, we use martingale theory to facilitate the calculation of the value function. Moreover, we show analytical results to compute several metrics of interest including the expected number of orders placed throughout the end-of-life phase. Furthermore, we devise an expandable taxonomy and relate the benchmark models to the literature. Analytical insights from the models as well as an extensive numerical analysis show the value of our approach. The results indicate that the loss can be high in case the manufacturer does not exploit flexibility in placing orders or use an outside source. Also, an outside source can be a significant alternative for the end-of-life inventory management problem. Finally, we show that some counter-intuitive strategies can be valuable for future analysis.

We study the convergence of entropically regularized optimal transport to optimal transport. The main result is concerned with the convergence of the associated optimizers and takes the form of a large deviations principle quantifying the local exponential convergence rate as the regularization parameter vanishes. The exact rate function is determined in a general setting and linked to the Kantorovich potential of optimal transport. Our arguments are based on the geometry of the optimizers and inspired by the use of c -cyclical monotonicity in classical transport theory. The results can also be phrased in terms of Schrödinger bridges.