We consider statistical methods which invoke a min-max distributionally robust formulation to extract good out-of-sample performance in data-driven optimization and learning problems. Acknowledging the distributional uncertainty in learning from limited samples, the min-max formulations introduce an adversarial inner player to explore unseen covariate data. The resulting Distributionally Robust Optimization (DRO) formulations, which include Wasserstein DRO formulations (our main focus), are specified using optimal transportation phenomena. Upon describing how these infinite-dimensional min-max problems can be approached via a finite-dimensional dual reformulation, the tutorial moves into its main component, namely, explaining a generic recipe for optimally selecting the size of the adversary's budget. This is achieved by studying the limit behavior of an optimal transport projection formulation arising from an inquiry on the smallest confidence region that includes the unknown population risk minimizer. Incidentally, this systematic prescription coincides with those in specific examples in high-dimensional statistics and results in error bounds that are free from the curse of dimensions. Equipped with this prescription, we present a central limit theorem for the DRO estimator and provide a recipe for constructing compatible confidence regions that are useful for uncertainty quantification. The rest of the tutorial is devoted to insights into the nature of the optimizers selected by the min-max formulations and additional applications of optimal transport projections.

ISTRIBUTED heuristics are a promising field for current and future energy systems control and optimization tasks, In [12] we showed that different communication topologies and have been designed and evaluated in recent years on have an effect on the performance of the reflected algorithm agent-based systems [1] [2] [3]. While conventional control class: Highly meshed topologies converged into good solutions systems - centralized or hierarchical in their control paradigm - reliably and quickly, but increased communication overhead and perfectly fit to centralized generation and transmission systems, premature convergence. In contrast, results for sparsely meshed distributed renewable energy systems show properties that topologies were much less reliable. In the application domain promote the application of distributed optimization systems: of energy systems as critical infrastructures, this behavior is First, future energy systems can be regarded as complex highly unwanted. We presume that dynamically adjusting the systems of systems, sometimes framed as cyber-physical multienergy topology during runtime leads to a beneficial transition of systems, coupling communication systems, power, heat exploration and exploitation of the search space for distributed and gas systems.

In this paper, we consider the geometric landscape connection of the widely studied manifold and factorization formulations in low-rank positive semidefinite (PSD) and general matrix optimization. We establish an equivalence on the set of first-order stationary points (FOSPs) and second-order stationary points (SOSPs) between the manifold and the factorization formulations. We further give a sandwich inequality on the spectrum of Riemannian and Euclidean Hessians at FOSPs, which can be used to transfer more geometric properties from one formulation to another. Similarities and differences on the landscape connection under the PSD case and the general case are discussed. To the best of our knowledge, this is the first geometric landscape connection between the manifold and the factorization formulations for handling rank constraints. In the general low-rank matrix optimization, the landscape connection of two factorization formulations (unregularized and regularized ones) is also provided. By applying these geometric landscape connections, we are able to solve unanswered questions in literature and establish stronger results in the applications on geometric analysis of phase retrieval, well-conditioned low-rank matrix optimization, and the role of regularization in factorization arising from machine learning and signal processing.

The field of artificial intelligence (AI) is witnessing a recent upsurge in research, tools development, and deployment of applications. Multiple software companies are shifting their focus to developing intelligent systems; and many others are deploying AI paradigms to their existing processes. In parallel, the academic research community is injecting AI paradigms to provide solutions to traditional engineering problems. Similarly, AI has evidently been proved useful to software engineering (SE). When one observes the SE phases (requirements, design, development, testing, release, and maintenance), it becomes clear that multiple AI paradigms (such as neural networks, machine learning, knowledge-based systems, natural language processing) could be applied to improve the process and eliminate many of the major challenges that the SE field has been facing. This survey chapter is a review of the most commonplace methods of AI applied to SE. The review covers methods between years 1975-2017, for the requirements phase, 46 major AI-driven methods are found, 19 for design, 15 for development, 68 for testing, and 15 for release and maintenance. Furthermore, the purpose of this chapter is threefold; firstly, to answer the following questions: is there sufficient intelligence in the SE lifecycle? What does applying AI to SE entail? Secondly, to measure, formulize, and evaluate the overlap of SE phases and AI disciplines. Lastly, this chapter aims to provide serious questions to challenging the current conventional wisdom (i.e., status quo) of the state-of-the-art, craft a call for action, and to redefine the path forward.

Given a matrix $X\in \mathbb{R}^{m\times n}_+$ with non-negative entries, the cone factorization problem over a cone $\mathcal{K}\subseteq \mathbb{R}^k$ concerns computing $\{ a_1,\ldots, a_{m} \} \subseteq \mathcal{K}$ and $\{ b_1,\ldots, b_{n} \} \subseteq~\mathcal{K}^*$ belonging to its dual so that $X_{ij} = \langle a_i, b_j \rangle$ for all $i\in [m], j\in [n]$. Cone factorizations are fundamental to mathematical optimization as they allow us to express convex bodies as feasible regions of linear conic programs. In this paper, we introduce and analyze the symmetric-cone multiplicative update (SCMU) algorithm for computing cone factorizations when $\mathcal{K}$ is symmetric; i.e., it is self-dual and homogeneous. Symmetric cones are of central interest in mathematical optimization as they provide a common language for studying linear optimization over the nonnegative orthant (linear programs), over the second-order cone (second order cone programs), and over the cone of positive semidefinite matrices (semidefinite programs). The SCMU algorithm is multiplicative in the sense that the iterates are updated by applying a meticulously chosen automorphism of the cone computed using a generalization of the geometric mean to symmetric cones. Using an extension of Lieb's concavity theorem and von Neumann's trace inequality to symmetric cones, we show that the squared loss objective is non-decreasing along the trajectories of the SCMU algorithm. Specialized to the nonnegative orthant, the SCMU algorithm corresponds to the seminal algorithm by Lee and Seung for computing Nonnegative Matrix Factorizations.

We propose a Riemannian conjugate gradient (CG) optimization method for finding low rank approximations of incomplete tensors. Our main contribution consists of an explicit expression of the geodesics on the Segre manifold. These are exploited in our algorithm to perform the retractions. We apply our method to movie rating predictions in a recommender system for the MovieLens dataset, and identification of pure fluorophores via fluorescent spectroscopy with missing data. In this last application, we recover the tensor decomposition from less than $10\%$ of the data.

We consider a two-stage electricity market comprising a forward and a real-time settlement. The former pre-dispatches the power system following a least-cost merit order and facing an uncertain net demand, while the latter copes with the plausible deviations with respect to the forward schedule by making use of power regulation during the actual operation of the system. Standard industry practice deals with the uncertain net demand in the forward stage by replacing it with a good estimate of its conditional expectation (usually referred to as a point forecast), so as to minimize the need for power regulation in real time. However, it is well known that the cost structure of a power system is highly asymmetric and dependent on its operating point, with the result that minimizing the amount of power imbalances is not necessarily aligned with minimizing operating costs. In this paper, we propose a mixed-integer program to construct, from the available historical data, an alternative estimate of the net demand that accounts for the power system's cost asymmetry. Furthermore, to accommodate the strong dependence of this cost on the power system's operating point, we use clustering to tailor the proposed estimate to the foreseen net-demand regime. By way of an illustrative example and a more realistic case study based on the European power system, we show that our approach leads to substantial cost savings compared to the customary way of doing.

This thesis explores the benefits machine learning algorithms can bring to online planning and scheduling for autonomous vehicles in off-road situations. Mainly, we focus on typical problems of interest which include computing itineraries that meet certain objectives, as well as computing scheduling strategies to execute synchronized maneuvers with other vehicles. We present a range of learning-based heuristics to assist different itinerary planners. We show that these heuristics allow a significant increase in performance for optimal planners. Furthermore, in the case of approximate planning, we show that not only does the running time decrease, the quality of the itinerary found also becomes almost always better. Finally, in order to synthesize strategies to execute synchronized maneuvers, we propose a novel type of scheduling controllability and a learning-assisted algorithm. The proposed framework achieves significant improvement on known benchmarks in this controllability type over the performance of state-of-the-art works in a related controllability type. Moreover, it is able to find strategies on complex scheduling problems for which previous works fail to do so.

Omni-directional mobile robot (OMR) systems have been very popular in academia and industry for their superb maneuverability and flexibility. Yet their potential has not been fully exploited, where the extra degree of freedom in OMR can potentially enable the robot to carry out extra tasks. For instance, gimbals or sensors on robots may suffer from a limited field of view or be constrained by the inherent mechanical design, which will require the chassis to be orientation-aware and respond in time. To solve this problem and further develop the OMR systems, in this paper, we categorize the tasks related to OMR chassis into orientation transition tasks and position transition tasks, where the two tasks can be carried out at the same time. By integrating the parallel task goals in a single planning problem, we proposed an orientation-aware planning architecture for OMR systems to execute the orientation transition and position transition in a unified and efficient way. A modified trajectory optimization method called orientation-aware timed-elastic-band (OATEB) is introduced to generate the trajectory that satisfies the requirements of both tasks. Experiments in both 2D simulated environments and real scenes are carried out. A four-wheeled OMR is deployed to conduct the real scene experiment and the results demonstrate that the proposed method is capable of simultaneously executing parallel tasks and is applicable to real-life scenarios.

Networks are landmarks of many complex phenomena where interweaving interactions between different agents transform simple local rule-sets into nonlinear emergent behaviors. While some recent studies unveil associations between the network structure and the underlying dynamical process, identifying stochastic nonlinear dynamical processes continues to be an outstanding problem. Here we develop a simple data-driven framework based on operator-theoretic techniques to identify and control stochastic nonlinear dynamics taking place over large-scale networks. The proposed approach requires no prior knowledge of the network structure and identifies the underlying dynamics solely using a collection of two-step snapshots of the states. This data-driven system identification is achieved by using the Koopman operator to find a low dimensional representation of the dynamical patterns that evolve linearly. Further, we use the global linear Koopman model to solve critical control problems by applying to model predictive control (MPC)--typically, a challenging proposition when applied to large networks. We show that our proposed approach tackles this by converting the original nonlinear programming into a more tractable optimization problem that is both convex and with far fewer variables.