Mathematics

How to dynamically measure the local-to-global spatio-temporal coherence between demand and supply networks is a fundamental task for ride-sourcing platforms, such as DiDi. Such coherence measurement is critically important for the quantification of the market efficiency and the comparison of different platform policies, such as dispatching. The aim of this paper is to introduce a graph-based equilibrium metric (GEM) to quantify the distance between demand and supply networks based on a weighted graph structure. We formulate GEM as the optimal objective value of an unbalanced transport problem, which can be efficiently solved by optimizing an equivalent linear programming. We examine how the GEM can help solve three operational tasks of ride-sourcing platforms. The first one is that GEM achieves up to 70.6% reduction in root-mean-square error over the second-best distance measurement for the prediction accuracy. The second one is that the use of GEM for designing order dispatching policy increases answer rate and drivers' revenue for more than 1%, representing a huge improvement in number. The third one is that GEM is to serve as an endpoint for comparing different platform policies in AB test.

We explore generalizations of some integrated learning and optimization frameworks for data-driven contextual stochastic optimization that can adapt to heteroscedasticity. We identify conditions on the stochastic program, data generation process, and the prediction setup under which these generalizations possess asymptotic and finite sample guarantees for a class of stochastic programs, including two-stage stochastic mixed-integer programs with continuous recourse. We verify that our assumptions hold for popular parametric and nonparametric regression methods.

This paper presents a static generation expansion planning formulation in which operational and investment decisions for a wide range of technologies are co-optimized from a centralized perspective. The location, type and quantity of new generation and storage capacities are provided such that system demand and flexibility requirements are satisfied. Depending on investments in new intermittent renewables (wind, PV), the flexibility requirements are adapted in order to fully capture RES integration costs and ensure normal system operating conditions. To position candidate units, we incorporate DC constraints, nodal demand and production of existing generators as well as imports and exports from other interconnected zones. To show the capabilities of the formulation, high-temporal resolution simulations are conducted on a 162-bus system consisting of the full Swiss transmission grid and aggregated neighboring countries.

This paper builds Wasserstein ambiguity sets for the unknown probability distribution of dynamic random variables leveraging noisy partial-state observations. The constructed ambiguity sets contain the true distribution of the data with quantifiable probability and can be exploited to formulate robust stochastic optimization problems with out-of-sample guarantees. We assume the random variable evolves in discrete time under uncertain initial conditions and dynamics, and that noisy partial measurements are available. All random elements have unknown probability distributions and we make inferences about the distribution of the state vector using several output samples from multiple realizations of the process. To this end, we leverage an observer to estimate the state of each independent realization and exploit the outcome to construct the ambiguity sets. We illustrate our results in an economic dispatch problem involving distributed energy resources over which the scheduler has no direct control.

We describe homogeneous sub-Riemannian geodesics for the standard sub-Riemannian structure on the group of proper motions of the plane SE(2). We show that this structure is not geodesically orbital, although the cut time is invariant w.r.t. shift of the initial point along geodesic.

In this article, we propose a mathematical model -- based on a cellular automaton -- for the redistribution of patients within a network of hospitals with limited available resources, in order to reduce the risks of a local/global collapse of the healthcare system. We attempt at developing a conceptual tool to support making rational decisions relevant to the optimisation of the allocation of patients into accessible medical facilities. The strategy is based on a version of the Abelian Sandpile model for the Self-Organised Criticality, with the idea of testing the paradigm for the management of patients among the COVID-19 hospitals in Italian regions. In particular, we compare the novel proposal to the standard management of connections between hospitals, showing a number of advantages at a local and global level, by means of a reliable indicator function introduced for measuring the effectiveness of the allocation strategies.

In this paper, we consider an online stochastic resource allocation problem which takes a linear program as its underlying form. We analyze an adaptive allocation algorithm and derives a constant regret bound that is not dependent on the number of time periods (number of decision variables) under the condition that the objective coefficient of the linear program is linear in the corresponding constraint coefficients. Furthermore, the constant regret bound does not assume the knowledge of underlying distribution.

Amenability is a notion of facial exposedness for convex cones that is stronger than being facially dual complete (or "nice") which is, in turn, stronger than merely being facially exposed. Hyperbolicity cones are a family of algebraically structured closed convex cones that contain all spectrahedra (linear sections of positive semidefinite cones) as special cases. It is known that all spectrahedra are amenable. We establish that all hyperbolicity cones are amenable. As part of the argument, we show that any face of a hyperbolicity cone is a hyperbolicity cone. As a corollary, we show that the intersection of two hyperbolicity cones, not necessarily sharing a common relative interior point, is a hyperbolicity cone.

Statistical preconditioning can be used to design fast methods for distributed large-scale empirical risk minimization problems, for strongly convex and smooth loss functions, allowing fewer communication rounds. Multiple worker nodes compute gradients in parallel, which are then used by the central node to update the parameter by solving an auxiliary (preconditioned) smaller-scale optimization problem. However, previous works require an exact solution of an auxiliary optimization problem by the central node at every iteration, which may be impractical. This paper proposes a method that allows the inexact solution of the auxiliary problem, reducing the total computation time. Moreover, for loss functions with high-order smoothness, we exploit the structure of the auxiliary problem and propose a hyperfast second-order method with complexity O ~ ( κ 1/5 ) , where κ is the local condition number. Combining these two building blocks (inexactness and hyperfast methods), we show complexity estimates for the proposed algorithm, which is provably better than classical variance reduction methods and has the same convergence rate as statistical preconditioning with exact solutions. Finally, we illustrate the proposed method's practical efficiency by performing large-scale numerical experiments on logistic regression models.

A renewal system divides the slotted timeline into back to back time periods called renewal frames. At the beginning of each frame, it chooses a policy from a set of options for that frame. The policy determines the duration of the frame, the penalty incurred during the frame (such as energy expenditure), and a vector of performance metrics (such as instantaneous number of jobs served). The starting points of this line of research are Chapter 7 of the book [Nee10a], the seminal work [Nee13a], and Chapter 5 of the PhD thesis of Chih-ping Li [Li11]. These works consider stochastic optimization over a single renewal system. By way of contrast, this thesis considers optimization over multiple parallel renewal systems, which is computationally more challenging and yields much more applications. The goal is to minimize the time average overall penalty subject to time average overall constraints on the corresponding performance metrics. The main difficulty, which is not present in earlier works, is that these systems act asynchronously due to the fact that the renewal frames of different renewal systems are not aligned. The goal of the thesis is to resolve this difficulty head-on via a new asynchronous algorithm and a novel supermartingale stopping time analysis which shows our algorithms not only converge to the optimal solution but also enjoy fast convergence rates. Based on this general theory, we further develop novel algorithms for data center server provision problems with performance guarantees as well as new heuristics for the multi-user file downloading problems.