An almost cyclic 2-coordinate descent method for singly linearly constrained problems

A block decomposition method is proposed for minimizing a (possibly non-convex) continuously differentiable function subject to one linear equality constraint and simple bounds on the variables. The proposed method iteratively selects a pair of coordinates according to an almost cyclic strategy that does not use first-order information, allowing us not to compute the whole gradient of the objective function during the algorithm. Using first-order search directions to update each pair of coordinates, global convergence to stationary points is established for different choices of the stepsize under an appropriate assumption on the level set. In particular, both inexact and exact line search strategies are analyzed. Further, linear convergence rate is proved under standard additional assumptions. Numerical results are finally provided to show the effectiveness of the proposed method.Comment: Computational Optimization and Application

Parallel decomposition methods for linearly constrained problems subject to simple bound with application to the SVMs training

We consider the convex quadratic linearly constrained problem with bounded variables and with huge and dense Hessian matrix that arises in many applications such as the training problem of bias support vector machines. We propose a decomposition algorithmic scheme suitable to parallel implementations and we prove global convergence under suitable conditions. Focusing on support vector machines training, we outline how these assumptions can be satisfied in practice and we suggest various specific implementations. Extensions of the theoretical results to general linearly constrained problem are provided. We included numerical results on support vector machines with the aim of showing the viability and the effectiveness of the proposed scheme

Contour Dynamics Methods

In an early paper on the stability of fluid layers with uniform vorticity Rayleigh (1880) remarks: "... In such cases, the velocity curve is composed of portions of straight lines which meet each other at finite angles. This state of things may be supposed to be slightly disturbed by bending the surfaces of transition, and the determination of the subsequent motion depends upon that of the form of these surfaces. For co retains its constant value throughout each layer unchanged in the absence of friction, and by a well-known theorem the whole motion depends upon [omega]." We can now recognize this as essentially the principal of contour dynamics (CD), where [omega] is the uniform vorticity. The theorem referred to is the Biot-Savart law. Nearly a century later Zabusky et al (1979) presented numerical CD calculations of nonlinear vortex patch evolution. Subsequently, owing to its compact form conferring a deceptive simplicity, CD has become a widely used method for the investigation of two-dimensional rotational flow of an incompressible inviscid fluid. The aim of this article is to survey the development, technical details, and vortex-dynamic applications of the CD method in an effort to assess its impact on our understanding of the mechanics of rotational flow in two dimensions at ultrahigh Reynolds numbers. The study of the dynamics of two- and three-dimensional vortex mechanics by computational methods has been an active research area for more than two decades. Quite apart from many practical applications in the aerodynamics of separated flows, the theoretical and numerical study of vortices in incompressible fluids has been stimulated by the idea that turbulent fluid motion may be viewed as comprising ensembles of more or less coherent laminar vortex structures that interact via relatively simple dynamics and by the appeal of the vorticity equation, which does not contain the fluid pressure. Two-dimensional vortex interactions have been perceived as supposedly relevant to the origins of coherent structures observed experimentally in mixing layers, jets, and wakes, and for models of large-scale atmospheric and oceanic turbulence. Interest has often focused on the limit of infinite Reynolds number, where in the absence of boundaries, the inviscid Euler equations are assumed to properly describe the flow dynamics. The numerous surveys of progress in the study of vorticity and the use of numerical methods applied to vortex mechanics include articles by Saffman & Baker (1979) and Saffman (1981) on inviscid vortex interactions and Aref (1983) on two-dimensional flows. Numerical methods have been surveyed by Chorin (1980), and Leonard (1980, 1985). Caflisch (1988) describes work on the mathematical aspects of the subject. Zabusky (1981), Aref (1983), and Melander et al (1987b) discuss various aspects of CD. The review of Dritschel (1989) gives emphasis to numerical issues in CD and to recent computations with contour surgery. This article is confined to a discussion of vortices on a two-dimensional surface. We generally follow Saffman & Baker (1979) in matters of definition. In two dimensions a vortex sheet is a line of discontinuity in velocity while a vortex jump is a line of discontinuity in vorticity. We shall, however, use filament to denote a two-dimensional ribbon of vorticity surrounded by fluid with vorticity of different magnitude (which may be zero), rather than the more usual three-dimensional idea of a vortex tube. The ambiguity is unfortunate but is already in the literature. Additionally, a vortex patch is a finite, singly connected area of uniform vorticity while a vortex strip is an infinite strip of uniform vorticity with finite thickness, or equivalently, an infinite filament. Contour Dynamics will refer to the numerical solution of initial value problems for piecewise constant vorticity distributions by the Lagrangian method of calculating the evolution of the vorticity jumps. Such flows are often related to corresponding solutions of the Euler equations that are steady in some translating or rotating frame of reference. These solutions will be called vortex equilibria, and the numerical technique for computing their shapes based on CD is often referred to as contour statics. The mathematical foundation for the study of vorticity was laid primarily by the well-known investigations of Helmholtz, Kelvin, J. J. Thomson, Love, and others. In our century, efforts to produce numerical simulations of flows governed by the Euler equations have utilized a variety of Eulerian, Lagrangian, and hybrid methods. Among the former are the class of spectral methods that now comprise the prevailing tool for large-scale two- and three-dimensional calculations (see Hussaini & Zang 1987). The Lagrangian methods for two-dimensional flows have been predominantly vortex tracking techniques based on the Helmholtz vorticity laws. The first initial value calculations were those of Rosenhead (193l) and Westwater (1935) who attempted to calculate vortex sheet evolution by the motion of O(10) point vortices. Subsequent efforts by Moore (1974) (see also Moore 1983, 1985) and others to produce more refined computations for vortex sheets have failed for reasons related to the tendency for initially smooth vortex sheet data to produce singularities (Moore 1979). Discrete vortex methods used to study the nonlinear dynamics of vortex patches and layers have included the evolution of assemblies of point vortices by direct summation (e.g. Acton 1976) and the cloud in cell method (Roberts & Christiansen 1972, Christiansen & Zabusky 1973, Aref & Siggia 1980, 1981). For reviews see Leonard (1980) and Aref (1983). These techniques have often been criticized for their lack of accuracy and numerical convergence and because they may be subject to grid scale dispersion. However, many qualitative vortex phenomena observed in nature and in experiments, such as amalgamation events and others still under active investigation (e.g. filamentation) were first simulated numerically with discrete vortices. The contour dynamics approach is attractive because it appears to allow direct access, at least for small times, to the inviscid dynamics for vorticity distributions smoother than those of either point vortices or vortex sheets, while at the same time enabling the mapping of the two-dimensional Euler equations to a one-dimensional Lagrangian form. In Section 2 we discuss the formulation and numerical implementation of contour dynamics for the Euler equations in two dimensions. Section 3 is concerned with applications to isolated and multiple vortex systems and to vortex layers. An attempt is made to relate this work to calculations of the relevant vortex equilibria and to results obtained with other methods. Axisymmetric contour dynamics and the treatment of the multi-layer model of quasigeostrophic flows are described in Section 4 while Section 5 is devoted to a discussion of the tendency shown by vorticity jumps to undergo the strange and subtle phenomenon of filamentation

Parallel computing in network optimization

Caption title.Includes bibliographical references (p. 82-95).Supported by the NSF. CCR-9103804Dimitri Bertsekas ... [et al.]

Optimising arrival management in air traffic control

Efficient landing is a key component of improving air transport, both for passengers' experience and fuel consumption. With just two runways, Heathrow airport was running at 98% capacity before the pandemic. The existing queuing system allows buffer time for the aircraft to land on one runway, but this can add delays to journeys and be fuel inefficient. With the recovery of the travel industry after the 2020 pandemic, improving landing procedures remains a pertinent problem to NATS (who manages all the air traffic in UK airspace). In this thesis, we develop alternative methods to sequence aircraft as they approach for landing at Heathrow. In the first part of this thesis, we cast the arrival management problem in a reinforcement learning framework. We design a basic air traffic model and apply both table representation methods and nonlinear approximation with a neural network. Specifically, we compare the performance of Q-learning, SARSA, and DDPG on this environment. Further we explore dimension reduction/feature representation through path signatures. Finally we design multi-grid inspired neural network structures and see that these lead to faster training but ultimately comparable performance. For the second part, we look at the problem from a different perspective. We take inspiration from the theory of optimal transport and formulate an entropy-regularised optimisation problem. We design an algorithm with block gradient descent-like steps and note that the conflicts in the set-up of our problem introduces non-convexity even when working in the (convex) space of distributions over arrival times. By adding an additional 'considerate' cost, akin to a Pigouvian tax, the performance of the algorithm is enhanced. Finally, the last part of this thesis shows the flexibility of our approach. We adapt our work to apply to a new air traffic design concept being researched at NATS