Mathematics

Tukey's halfspace depth can be seen as a stochastic program and as such it is not guarded against optimizer's curse, so that a limited training sample may easily result in a poor out-of-sample performance. We propose a generalized halfspace depth concept relying on the recent advances in distributionally robust optimization, where every halfspace is examined using the respective worst-case distribution in the Wasserstein ball of radius δ?? centered at the empirical law. This new depth can be seen as a smoothed and regularized classical halfspace depth which is retrieved as δ?? . It inherits most of the main properties of the latter and, additionally, enjoys various new attractive features such as continuity and strict positivity beyond the convex hull of the support. We provide numerical illustrations of the new depth and its advantages, and develop some fundamental theory. In particular, we study the upper level sets and the median region including their breakdown properties.

We consider a distributionally robust second-order stochastic dominance constrained optimization problem, where the true distribution of the uncertain parameters is ambiguous. The ambiguity set contains all probability distributions close to the empirical distribution under the Wasserstein distance. We adopt the sample approximation technique to develop a linear programming formulation that provides a lower bound. We propose a novel split-and-dual decomposition framework which provides an upper bound. We prove that both lower and upper bound approximations are asymptotically tight when there are enough samples or pieces. We present quantitative error estimation for the upper bound under a specific constraint qualification condition. To efficiently solve the non-convex upper bound problem, we use a sequential convex approximation algorithm. Numerical evidences on a portfolio selection problem valid the efficiency and asymptotically tightness of the proposed two approximation methods.

The work is devoted to ways of modeling street traffic on a street layout without traffic lights of an established topology. The behavior of traffic participants takes into account the individual inclinations of drivers to creatively interpret traffic rules. Participant interactions describe game theory models that provide information for simulation algorithms based on cellular automata. Driver diversification comes down to two types often considered in such research: DE(fective)-agent and CO(operative)-agent. Various ways of using the description of traffic participants to examine the impact of behavior on street traffic dynamics were shown. Directions for the further detailed analysis were indicated, which requires basic research in the field of game theory models.

In this note we observe that for constrained convex minimization problems min x?�P f(x) over a polytope P , dual prices for the linear program min z?�P ?�f(x)z obtained from linearization at approximately optimal solutions x have a similar interpretation of rate of change in optimal value as for linear programming, providing a convex form of sensitivity analysis. This is of particular interest for Frank--Wolfe algorithms (also called conditional gradients), forming an important class of first-order methods, where a basic building block is linear minimization of gradients of f over P , which in most implementations already compute the dual prices as a by-product.

The purpose of this short note is to show that the Christoffel-Darboux polynomial, useful in approximation theory and data science, arises naturally when deriving the dual to the problem of semi-algebraic D-optimal experimental design in statistics. It uses only elementary notions of convex analysis. Geometric interpretations and algorithmic consequences are mentioned.

We propose a method for solving non-smooth optimization problems on manifolds. In order to obtain superlinear convergence, we apply a Riemannian Semi-smooth Newton method to a non-smooth non-linear primal-dual optimality system based on a recent extension of Fenchel duality theory to Riemannian manifolds. We also propose an inexact version of the Riemannian Semi-smooth Newton method and prove conditions for local linear and superlinear convergence. Numerical experiments on l2-TV-like problems confirm superlinear convergence on manifolds with positive and negative curvature.

In this paper we propose a novel consensus protocol for discrete-time multi-agent systems (MAS), which solves the dynamic consensus problem on the max value, i.e., the dynamic max-consensus problem. In the dynamic max-consensus problem to each agent is fed a an exogenous reference signal, the objective of each agent is to estimate the instantaneous and time-varying value of the maximum among the signals fed to the network, by exploiting only local and anonymous interactions among the agents. The absolute and relative tracking error of the proposed distributed control protocol is theoretically characterized and is shown to be bounded and by tuning its parameters it is possible to trade-off convergence time for steady-state error. The dynamic Max-consensus algorithm is then applied to solve the distributed size estimation problem in a dynamic setting where the size of the network is time-varying during the execution of the estimation algorithm. Numerical simulations are provided to corroborate the theoretical analysis.

This manuscript provides a theoretical foundation for the Dynamic Mode Decomposition (DMD) of control affine dynamical systems through vector valued reproducing kernel Hilbert spaces (RKHSs). Specifically, control Liouville operators and control occupation kernels are introduced to separate the drift dynamics from the control effectiveness components. Given a known feedback controller that is represented through a multiplication operator, a DMD analysis may be performed on the composition of these operators to make predictions concerning the system controlled by the feedback controller.

In this study, we analyse the convergence and stability of dynamic system optimal (DSO) traffic assignment with fixed departure times. We first formulate the DSO traffic assignment problem as a strategic game wherein atomic users select routes that minimise their marginal social costs, called a 'DSO game'. By utilising the fact that the DSO game is a potential game, we prove that a globally optimal state is a stochastically stable state under the logit response dynamics, and the better/best response dynamics converges to a locally optimal state. Furthermore, as an application of DSO assignment, we examine characteristics of the evolutionary implementation scheme of marginal cost pricing. Through theoretical comparison with a fixed pricing scheme, we found the following properties of the evolutionary implementation scheme: (i) the total travel time decreases smoother to an efficient traffic state as congestion externalities are perfectly internalised; (ii) a traffic state would reach a more efficient state as the globally optimal state is stabilised. Numerical experiments also suggest that these properties make the evolutionary scheme robust in the sense that they prevent a traffic state from going to worse traffic states with high total travel times.

This study investigates dynamic system-optimal (DSO) and dynamic user equilibrium (DUE) traffic assignment of departure/arrival-time choices in a corridor network. The morning commute problems with a many-to-one pattern of origin-destination demand and the evening commute problems with a one-to-many pattern are considered. In this study, a novel approach is developed to derive an analytical solution for the DSO problem. By utilizing the analytical solution, we prove that the queuing delay at a bottleneck in a DUE solution is equal to an optimal toll that eliminates the queue in a DSO solution under certain conditions of a schedule delay function. This enables us to derive a closed-form DUE solution by using the DSO solution. Numerical examples are provided to illustrate and verify analytical results.