Optimization algorithms for the solution of the frictionless normal contact between rough surfaces

This paper revisits the fundamental equations for the solution of the frictionless unilateral normal contact problem between a rough rigid surface and a linear elastic half-plane using the boundary element method (BEM). After recasting the resulting Linear Complementarity Problem (LCP) as a convex quadratic program (QP) with nonnegative constraints, different optimization algorithms are compared for its solution: (i) a Greedy method, based on different solvers for the unconstrained linear system (Conjugate Gradient CG, Gauss-Seidel, Cholesky factorization), (ii) a constrained CG algorithm, (iii) the Alternating Direction Method of Multipliers (ADMM), and ($iv$) the Non-Negative Least Squares (NNLS) algorithm, possibly warm-started by accelerated gradient projection steps or taking advantage of a loading history. The latter method is two orders of magnitude faster than the Greedy CG method and one order of magnitude faster than the constrained CG algorithm. Finally, we propose another type of warm start based on a refined criterion for the identification of the initial trial contact domain that can be used in conjunction with all the previous optimization algorithms. This method, called Cascade Multi-Resolution (CMR), takes advantage of physical considerations regarding the scaling of the contact predictions by changing the surface resolution. The method is very efficient and accurate when applied to real or numerically generated rough surfaces, provided that their power spectral density function is of power-law type, as in case of self-similar fractal surfaces.Comment: 38 pages, 11 figure

A posteriori multi-stage optimal trading under transaction costs and a diversification constraint

This paper presents a simple method for a posteriori (historical) multi-variate multi-stage optimal trading under transaction costs and a diversification constraint. Starting from a given amount of money in some currency, we analyze the stage-wise optimal allocation over a time horizon with potential investments in multiple currencies and various assets. Three variants are discussed, including unconstrained trading frequency, a fixed number of total admissable trades, and the waiting of a specific time-period after every executed trade until the next trade. The developed methods are based on efficient graph generation and consequent graph search, and are evaluated quantitatively on real-world data. The fundamental motivation of this work is preparatory labeling of financial time-series data for supervised machine learning.Comment: 25 pages, 4 figures, 6 table

Forward-backward truncated Newton methods for convex composite optimization

This paper proposes two proximal Newton-CG methods for convex nonsmooth optimization problems in composite form. The algorithms are based on a a reformulation of the original nonsmooth problem as the unconstrained minimization of a continuously differentiable function, namely the forward-backward envelope (FBE). The first algorithm is based on a standard line search strategy, whereas the second one combines the global efficiency estimates of the corresponding first-order methods, while achieving fast asymptotic convergence rates. Furthermore, they are computationally attractive since each Newton iteration requires the approximate solution of a linear system of usually small dimension

A Convex Feasibility Approach to Anytime Model Predictive Control

This paper proposes to decouple performance optimization and enforcement of asymptotic convergence in Model Predictive Control (MPC) so that convergence to a given terminal set is achieved independently of how much performance is optimized at each sampling step. By embedding an explicit decreasing condition in the MPC constraints and thanks to a novel and very easy-to-implement convex feasibility solver proposed in the paper, it is possible to run an outer performance optimization algorithm on top of the feasibility solver and optimize for an amount of time that depends on the available CPU resources within the current sampling step (possibly going open-loop at a given sampling step in the extreme case no resources are available) and still guarantee convergence to the terminal set. While the MPC setup and the solver proposed in the paper can deal with quite general classes of functions, we highlight the synthesis method and show numerical results in case of linear MPC and ellipsoidal and polyhedral terminal sets.Comment: 8 page

Direct data-driven control of constrained linear parameter-varying systems: A hierarchical approach

In many nonlinear control problems, the plant can be accurately described by a linear model whose operating point depends on some measurable variables, called scheduling signals. When such a linear parameter-varying (LPV) model of the open-loop plant needs to be derived from a set of data, several issues arise in terms of parameterization, estimation, and validation of the model before designing the controller. Moreover, the way modeling errors affect the closed-loop performance is still largely unknown in the LPV context. In this paper, a direct data-driven control method is proposed to design LPV controllers directly from data without deriving a model of the plant. The main idea of the approach is to use a hierarchical control architecture, where the inner controller is designed to match a simple and a-priori specified closed-loop behavior. Then, an outer model predictive controller is synthesized to handle input/output constraints and to enhance the performance of the inner loop. The effectiveness of the approach is illustrated by means of a simulation and an experimental example. Practical implementation issues are also discussed.Comment: Preliminary version of the paper "Direct data-driven control of constrained systems" published in the IEEE Transactions on Control Systems Technolog

A solar cooling plant: a benchmark for hybrid systems control

This paper describes the hybrid model of a solar cooling plant. This modelconsiders all possible operating modes of the process, which are modelled as a nitestate machine whose transition conditions are given by the discrete variables. Thediscrete variables are the electrovalves and pumps. The model has been written as amixed logical dynamical system and is simulated using State ow/Simulink Matlab.The model has been validated using real data from the plant. This plant is being usedas a benchmark for hybrid control experiences by many European researchers in theframework of the HYCON Network of ExcellenceUnión Europea HYCON(FP6-511368

Uncertainties in polarimetric 3D reconstructions of coronal mass ejections

This work is aimed at quantifying the uncertainties in the 3D reconstruction of the location of coronal mass ejections (CMEs) obtained with the polarization ratio technique. The method takes advantage of the different distributions along the line of sight (LOS) of total (tB) and polarized (pB) brightnesses to estimate the average location of the emitting plasma. To this end, we assumed two simple electron density distributions along the LOS (a constant density and Gaussian density profiles) for a plasma blob and synthesized the expected tB and pB for different distances $z$ of the blob from the plane of the sky (POS) and different projected altitudes $\rho$. Reconstructed locations of the blob along the LOS were thus compared with the real ones, allowing a precise determination of uncertainties in the method. Independently of the analytical density profile, when the blob is centered at a small distance from the POS (i.e. for limb CMEs) the distance from the POS starts to be significantly overestimated. Polarization ratio technique provides the LOS position of the center of mass of what we call folded density distribution, given by reflecting and summing in front of the POS the fraction of density profile located behind that plane. On the other hand, when the blob is far from the POS, but with very small projected altitudes (i.e. for halo CMEs, $\rho < 1.4$ R$_\odot$), the inferred distance from that plane is significantly underestimated. Better determination of the real blob position along the LOS is given for intermediate locations, and in particular when the blob is centered at an angle of $20^\circ$ from the POS. These result have important consequences not only for future 3D reconstruction of CMEs with polarization ratio technique, but also for the design of future coronagraphs aimed at providing a continuous monitoring of halo-CMEs for space weather prediction purposes