Mathematics

We study strategic interactions between firms with heterogeneous beliefs about future climate impacts. To that end, we propose a Cournot-type equilibrium model where firms choose mitigation efforts and production quantities such as to maximize the expected profits under their subjective beliefs. It is shown that optimal mitigation efforts are increased by the presence of uncertainty and act as substitutes; i.e., one firm's lack of mitigation incentivizes others to act more decidedly, and vice versa.

This work is devoted to an analysis of exact penalty functions and optimality conditions for nonsmooth two-stage stochastic programming problems. To this end, we first study the co-/quasi-differentiability of the expectation of nonsmooth random integrands and obtain explicit formulae for its co- and quasidifferential under some natural assumptions on the integrand. Then we analyse exact penalty functions for a variational reformulation of two-stage stochastic programming problems and obtain sufficient conditions for the global exactness of these functions with two different penalty terms. In the end of the paper, we combine our results on the co-/quasi-differentiability of the expectation of nonsmooth random integrands and exact penalty functions to derive optimality conditions for nonsmooth two-stage stochastic programming problems in terms of codifferentials.

A method to compute optimal collision avoidance maneuvers for short-term encounters is presented. The maneuvers are modeled as multiple-impulses to handle impulsive cases and to approximate finite burn arcs associated either with short alert times or the use of low-thrust propulsion. The maneuver design is formulated as a sequence of convex optimization problems solved in polynomial time by state-of-the-art primal-dual interior-point algorithms. The proposed approach calculates optimal solutions without assumptions about the thrust arc structure and thrust direction. The execution time is fraction of a second for an optimization problem with hundreds of variables and constraints, making it suitable for autonomous calculations.

A decentralized feedback controller for multi-agent systems, inspired by vehicle platooning, is proposed. The closed-loop resulting from the decentralized control action has three distinctive features: the generation of collision-free trajectories, flocking of the system towards a consensus state in velocity, and asymptotic convergence to a prescribed pattern of distances between agents. For each feature, a rigorous dynamical analysis is provided, yielding a characterization of the set of parameters and initial configurations where collision avoidance, flocking, and pattern formation is guaranteed. Numerical tests assess the theoretical results presented.

The control of bilinear systems has attracted considerable attention in the field of systems and control for decades, owing to their prevalence in diverse applications across science and engineering disciplines. Although much work has been conducted on analyzing controllability properties, the mostly used tool remains the Lie algebra rank condition. In this paper, we develop alternative approaches based on theory and techniques in combinatorics to study controllability of bilinear systems. The core idea of our methodology is to represent vector fields of a bilinear system by permutations or graphs, so that Lie brackets are represented by permutation multiplications or graph operations, respectively. Following these representations, we derive combinatorial characterization of controllability for bilinear systems, which consequently provides novel applications of symmetric group and graph theory to control theory. Moreover, the developed combinatorial approaches are compatible with Lie algebra decompositions, including the Cartan and non-intertwining decomposition. This compatibility enables the exploitation of representation theory for analyzing controllability, which allows us to characterize controllability properties of bilinear systems governed by semisimple and reductive Lie algebras.

We investigate fast and communication-efficient algorithms for the classic problem of minimizing a sum of strongly convex and smooth functions that are distributed among n different nodes, which can communicate using a limited number of bits. Most previous communication-efficient approaches for this problem are limited to first-order optimization, and therefore have \emph{linear} dependence on the condition number in their communication complexity. We show that this dependence is not inherent: communication-efficient methods can in fact have sublinear dependence on the condition number. For this, we design and analyze the first communication-efficient distributed variants of preconditioned gradient descent for Generalized Linear Models, and for Newton's method. Our results rely on a new technique for quantizing both the preconditioner and the descent direction at each step of the algorithms, while controlling their convergence rate. We also validate our findings experimentally, showing fast convergence and reduced communication.

Semidefinite programming is an important tool to tackle several problems in data science and signal processing, including clustering and community detection. However, semidefinite programs are often slow in practice, so speed up techniques such as sketching are often considered. In the context of community detection in the stochastic block model, Mixon and Xie [9] have recently proposed a sketching framework in which a semidefinite program is solved only on a subsampled subgraph of the network, giving rise to significant computational savings. In this short paper, we provide a positive answer to a conjecture of Mixon and Xie about the statistical limits of this technique for the stochastic block model with two balanced communities.

This paper studies the benefits of autonomous vehicles in ride-sharing platforms dedicated to serving commuting needs. It considers the Commute Trip Sharing Problem with Autonomous Vehicles (CTSPAV), the optimization problem faced by a reservation-based platform that receives daily commute-trip requests and serves them with a fleet of autonomous vehicles. The CTSPAV can be viewed as a special case of the Dial- A-Ride Problem (DARP). However, this paper recognizes that commuting trips exhibit special spatial and temporal properties that can be exploited in a branch and cut algorithm that leverages a redundant modeling approach. In particular, the branch and cut algorithm relies on a MIP formulation that schedules mini routes representing inbound or outbound trips. This formulation is effective in finding high-quality solutions quickly but its relaxation is relatively weak. To remedy this limitation, the mini-route MIP is complemented by a DARP formulation which is not as effective in obtaining primal solutions but has a stronger relaxation. The benefits of the proposed approach are demonstrated by comparing it with another, more traditional, exact branch and cut procedure and a heuristic method based on mini routes. The methodological contribution is complemented by a comprehensive analysis of a CTSPAV platform for reducing vehicle counts, travel distances, and congestion. In particular, the case study for a medium-sized city reveals that a CTSPAV platform can reduce daily vehicle counts by a staggering 92% and decrease vehicles miles by 30%. The platform also significantly reduces congestion, measured as the number of vehicles on the road per unit time, by 60% during peak times. These benefits, however, come at the expense of introducing empty miles. Hence the paper also highlights the tradeoffs between future ride-sharing and car-pooling platforms.

Inverse optimization (IO) aims to determine optimization model parameters from observed decisions. However, IO is not part of a data scientist's toolkit in practice, especially as many general-purpose machine learning packages are widely available as an alternative. When encountering IO, practitioners face the question of when, or even whether, investing in developing IO methods is worthwhile. Our paper provides a starting point toward answering these questions, focusing on the problem of imputing the objective function of a parametric convex optimization problem. We compare the predictive performance of three standard supervised machine learning (ML) algorithms (random forest, support vector regression and Gaussian process regression) to the performance of the IO model of Keshavarz, Wang, and Boyd (2011). While the IO literature focuses on the development of methods tailored to particular problem classes, our goal is to evaluate general "out-of-the-box" approaches. Our experiments demonstrate that determining whether to use an ML or IO approach requires considering (i) the training set size, (ii) the dependence of the optimization problem on external parameters, (iii) the level of confidence with regards to the correctness of the optimization prior, and (iv) the number of critical regions in the solution space.

We provide a dual subgradient method and a primal-dual subgradient method for standard convex optimization problems with complexity O( ε ?? ) and O( ε ??r ) , for all r>1 , respectively. They are based on recent Metel-Takeda's work in [arXiv:2009.12769, 2020, pp. 1-12] and Boyd's method in [Lecture notes of EE364b, Stanford University, Spring 2013-14, pp. 1-39]. The efficiency of our methods is numerically illustrated in a comparison to the others.