Convex Optimization Procedure Research Articles

Motion planning with sequential convex optimization and convex collision checking

We present a new optimization-based approach for robotic motion planning among obstacles. Like CHOMP (Covariant Hamiltonian Optimization for Motion Planning), our algorithm can be used to find collision-free trajectories from naïve, straight-line initializations that might be in collision. At the core of our approach are (a) a sequential convex optimization procedure, which penalizes collisions with a hinge loss and increases the penalty coefficients in an outer loop as necessary, and (b) an efficient formulation of the no-collisions constraint that directly considers continuous-time safety Our algorithm is implemented in a software package called TrajOpt. We report results from a series of experiments comparing TrajOpt with CHOMP and randomized planners from OMPL, with regard to planning time and path quality. We consider motion planning for 7 DOF robot arms, 18 DOF full-body robots, statically stable walking motion for the 34 DOF Atlas humanoid robot, and physical experiments with the 18 DOF PR2. We also apply TrajOpt to plan curvature-constrained steerable needle trajectories in the SE(3) configuration space and multiple non-intersecting curved channels within 3D-printed implants for intracavitary brachytherapy. Details, videos, and source code are freely available at: http://rll.berkeley.edu/trajopt/ijrr.

Quadratic MPC with ℓ 0 -input constraint

Abstract In this paper we propose a novel quadratic model predictive control technique that constrains the number of active inputs at each control horizon instant. This problem is known as sparse control. We use an iterative convex optimization procedure to solve the corresponding optimization problem subject to sparsity constraints defined by means of the l 0 -norm. We also derive a sufficient condition on the minimum number of active of inputs that guarantees the exponential stability of the closed-loop system. A simulation example illustrates the benefits of the control design method proposed in the paper.

A Rank-Constrained Optimization approach: Application to Factor Analysis

In this paper, we present a general method for rank-constrained optimization. We use an iterative convex optimization procedure where it is possible to include any extra convex constraints. The proposed approach has potential application in several areas. We focus on the problem of Factor Analysis. In this case, our approach provides sufficient flexibility to handle correlated errors. The benefits of the method is demonstrated via a simulation study.

Lower bound analysis ofH ∞ performance achievable via decentralized LTI controllers

SUMMARY This paper is concerned with the decentralized H ∞ controller synthesis problem for discrete-time LTI systems. Despite of intensive research efforts over the last several decades, this problem is believed to be nonconvex and still outstanding in general. Therefore, most of existing approaches resort to heuristic optimization algorithms that do not allow us to draw any definite conclusion on the quality of the designed controllers. To get around this difficulty, in this paper, we propose convex optimization procedures for computing lower bounds of the H ∞ performance that is achievable via decentralized LTI controllers of any order. In particular, we will show that sharpened lower bounds can be obtained by making good use of structures of the LTI plant typically observed in the decentralized control setting. We illustrate via numerical examples that these lower bounds are indeed useful to ensure the good quality of decentralized controllers designed by a heuristic optimization. Copyright © 2013 John Wiley & Sons, Ltd.

Solving a two-colour problem by applying probabilistic approach to a full-colour multi-frame image super-resolution

Reconstruction based algorithms play an important role in the multi-frame super-resolution problem. A group of images of the same scene are fused together to produce an image with higher spatial resolution, or with more visible details in the high spatial frequency features. Demosaicing algorithms interpolate missing pixels in a raw image taken from one Charged Coupled Device (CCD) array, upsampling the number of the pixels present in the image. Since super-resolution (SR) and demosaicing are the two faces of the same problem it is natural to address them together. In this paper it is: (i) shown that correct modelling of the Bayer pattern in the generative process improves the super-resolution performance for colour images, and (ii) an algorithm that incorporates the two colour prior into the probabilistic model is designed. The algorithm presented in this paper focuses on the classes of images that have two dominant colours, i.e. most of the areas in the image are uniformly coloured. A convex optimization procedure for joint super-resolution and demosaicing is developed which outperforms state-of-the-art algorithms.

Time-frequency mixed-norm estimates: Sparse M/EEG imaging with non-stationary source activations

Magnetoencephalography (MEG) and electroencephalography (EEG) allow functional brain imaging with high temporal resolution. While solving the inverse problem independently at every time point can give an image of the active brain at every millisecond, such a procedure does not capitalize on the temporal dynamics of the signal. Linear inverse methods (minimum-norm, dSPM, sLORETA, beamformers) typically assume that the signal is stationary: regularization parameter and data covariance are independent of time and the time varying signal-to-noise ratio (SNR). Other recently proposed non-linear inverse solvers promoting focal activations estimate the sources in both space and time while also assuming stationary sources during a time interval. However such a hypothesis holds only for short time intervals. To overcome this limitation, we propose time-frequency mixed-norm estimates (TF-MxNE), which use time-frequency analysis to regularize the ill-posed inverse problem. This method makes use of structured sparse priors defined in the time-frequency domain, offering more accurate estimates by capturing the non-stationary and transient nature of brain signals. State-of-the-art convex optimization procedures based on proximal operators are employed, allowing the derivation of a fast estimation algorithm. The accuracy of the TF-MxNE is compared with recently proposed inverse solvers with help of simulations and by analyzing publicly available MEG datasets.

A nonlinear coupling between the firing threshold and the membrane potential enhances coding of rapid signals

Event Abstract Back to Event A nonlinear coupling between the firing threshold and the membrane potential enhances coding of rapid signals Christian Pozzorini1*, Skander Mensi1, Olivier Hagens2 and Wulfram Gerstner1 1 EPFL, Brain Mind Institute, Switzerland 2 EPFL, Brain Mind Institute, Switzerland In the field of computational neuroscience it is of crucial importance to dispose of simplified spiking models that carefully capture the spiking activity of real neurons. Generalized Integrate-and-Fire (GIF) models have recently been shown to predict the occurrence of individual spikes of both inhibitory and excitatory neurons with millisecond precision. However, while the f-I curves of inhibitory fast-spiking neurons are in good agreement with the ones predicted by GIF models, the same is not true for excitatory pyramidal neurons. In particular, standard GIF models do not capture saturation at relatively low rates. Moreover, in contrast to what has been observed in pyramidal neurons, the firing threshold of standard GIF models does not depend on the speed at which the membrane potential approaches this threshold. To solve this problem, we propose a new model in which a stochastic GIF model equipped with both a spike-triggered current and a spike-triggered movement of the firing threshold is extended with a subthreshold adaptation mechanism consisting of a nonlinear coupling between the firing threshold and the membrane potential. This additional mechanism can be interpreted as a phenomenological model of sodium channel inactivation. Importantly, all the model parameters, including the timescale and the functional shape of the nonlinear coupling, are not assumed a priori but are extracted from in vitro recordings using a convex optimization procedure. Our results demonstrate that, in pyramidal neurons, the firing threshold and the subthreshold membrane potential are indeed nonlinearly coupled. Consistent with the dynamics of sodium channel inactivation, we found that this mechanism operates on a relatively short timescale (5 ms) and makes the firing threshold dependent on the speed at which the threshold is approached. The precise shape of the nonlinear coupling extracted from the experimental data accounts for both the saturation at low rates and the noise sensitivity observed in pyramidal neurons. Moreover, accounting for sodium channel inactivation significantly improved the ability to predict individual spikes with millisecond precision. Our results suggest that the firing threshold dynamics enhances sensitivity to rapid fluctuations of the input and makes pyramidal neurons respond as differentiate-and-fire rather than integrate-and-fire. Figure 1 Keywords: spiking neuron models, Single Neuron Dynamics, Neural coding, Single Neuron Adaptation, spike-frequency adaptation Conference: Neuroinformatics 2013, Stockholm, Sweden, 27 Aug - 29 Aug, 2013. Presentation Type: Poster Topic: Computational neuroscience Citation: Pozzorini C, Mensi S, Hagens O and Gerstner W (2013). A nonlinear coupling between the firing threshold and the membrane potential enhances coding of rapid signals. Front. Neuroinform. Conference Abstract: Neuroinformatics 2013. doi: 10.3389/conf.fninf.2013.09.00119 Copyright: The abstracts in this collection have not been subject to any Frontiers peer review or checks, and are not endorsed by Frontiers. They are made available through the Frontiers publishing platform as a service to conference organizers and presenters. The copyright in the individual abstracts is owned by the author of each abstract or his/her employer unless otherwise stated. Each abstract, as well as the collection of abstracts, are published under a Creative Commons CC-BY 4.0 (attribution) licence (https://creativecommons.org/licenses/by/4.0/) and may thus be reproduced, translated, adapted and be the subject of derivative works provided the authors and Frontiers are attributed. For Frontiers’ terms and conditions please see https://www.frontiersin.org/legal/terms-and-conditions. Received: 29 Apr 2013; Published Online: 11 Jul 2013. * Correspondence: Mr. Christian Pozzorini, EPFL, Brain Mind Institute, Lausanne, Vaud, CH-1015, Switzerland, christian.pozzorini@epfl.ch Login Required This action requires you to be registered with Frontiers and logged in. To register or login click here. Abstract Info Abstract The Authors in Frontiers Christian Pozzorini Skander Mensi Olivier Hagens Wulfram Gerstner Google Christian Pozzorini Skander Mensi Olivier Hagens Wulfram Gerstner Google Scholar Christian Pozzorini Skander Mensi Olivier Hagens Wulfram Gerstner PubMed Christian Pozzorini Skander Mensi Olivier Hagens Wulfram Gerstner Related Article in Frontiers Google Scholar PubMed Abstract Close Back to top Javascript is disabled. Please enable Javascript in your browser settings in order to see all the content on this page.

The Circulant Rational Covariance Extension Problem: The Complete Solution

The rational covariance extension problem to determine a rational spectral density given a finite number of covariance lags can be seen as a matrix completion problem to construct an infinite-dimensional positive-definite Toeplitz matrix the northwest corner of which is given. The circulant rational covariance extension problem considered in this paper is a modification of this problem to partial stochastic realization of periodic stationary process, which are better represented on the discrete unit circle rather than on the discrete real line . The corresponding matrix completion problem then amounts to completing a finite-dimensional Toeplitz matrix that is circulant. Another important motivation for this problem is that it provides a natural approximation, involving only computations based on the fast Fourier transform, for the ordinary rational covariance extension problem, potentially leading to an efficient numerical procedure for the latter. The circulant rational covariance extension problem is an inverse problem with infinitely many solutions in general, each corresponding to a bilateral ARMA representation of the underlying periodic process. In this paper, we present a complete smooth parameterization of all solutions and convex optimization procedures for determining them. A procedure to determine which solution that best matches additional data in the form of logarithmic moments is also presented.

Pathwise Optimization for Optimal Stopping Problems

We introduce the pathwise optimization (PO) method, a new convex optimization procedure to produce upper and lower bounds on the optimal value (the “price”) of a high-dimensional optimal stopping problem. The PO method builds on a dual characterization of optimal stopping problems as optimization problems over the space of martingales, which we dub the martingale duality approach. We demonstrate via numerical experiments that the PO method produces upper bounds of a quality comparable with state-of-the-art approaches, but in a fraction of the time required for those approaches. As a by-product, it yields lower bounds (and suboptimal exercise policies) that are substantially superior to those produced by state-of-the-art methods. The PO method thus constitutes a practical and desirable approach to high-dimensional pricing problems. Furthermore, we develop an approximation theory relevant to martingale duality approaches in general and the PO method in particular. Our analysis provides a guarantee on the quality of upper bounds resulting from these approaches and identifies three key determinants of their performance: the quality of an input value function approximation, the square root of the effective time horizon of the problem, and a certain spectral measure of “predictability” of the underlying Markov chain. As a corollary to this analysis we develop approximation guarantees specific to the PO method. Finally, we view the PO method and several approximate dynamic programming methods for high-dimensional pricing problems through a common lens and in doing so show that the PO method dominates those alternatives. This paper was accepted by Wei Xiong, stochastic models and simulation.

Receding Horizon Control Strategies for Constrained LPV Systems Based on a Class of Nonlinearly Parameterized Lyapunov Functions

In this technical note, we present a Receding Horizon Control (RHC) design method for linear parameter varying (LPV) systems subject to input and/or state constraints based on a class of nonlinearly parameterized Lyapunov functions recently introduced by Guerra and Vermeiren. As it will be made clear, their use gives rise to less conservative stabilizability conditions w.r.t. those arising from quadratic Lyapunov functions. A workable convex optimization procedure is first presented for control design purposes which allows the synthesis of stabilizing scheduling state-feedback control laws complying with the prescribed constraints. This control design method is then arranged into a receding horizon framework and its feasibility and stability properties are carefully analyzed. Numerical comparisons with existing RHC methods for LPV systems are reported in the final example.

Gradient-Coil Design: A Multi-Objective Problem

In this work, the design of gradient coils for magnetic resonance imaging (MRI) is studied as a multi-objective optimization (MOP) problem, which is successfully solved by using Pareto optimality formalism. The proposed approach is illustrated using a stream function inverse boundary element method (IBEM), as the coil design paradigm that is capable of including numerous design requirements or objectives. These are frequently in conflict, which stresses the need to deal efficiently with the tradeoff between different coil properties. It is shown that the inclusion of many of the most commonly used coil design requirements (such as field homogeneity, uniformity, magnetic stored energy, power dissipated, torque balanced ... ) reduces the problem to a convex MOP, where Pareto optimal solutions can be efficiently found by using suitable convex optimization procedures. Pertinent examples are studied to illustrate the versatility of the proposed MOP approach, which can be used to obtain a comprehensive understanding of the coil design problem, as well as to handle the different coil requirements efficiently and how they should be combined to yield the best solution for a given problem.

Energy-Efficient Spectrum Sensing and Access for Cognitive Radio Networks

We consider a cognitive radio system with one secondary user (SU) accessing multiple channels via periodic sensing and spectrum handoff. We propose an optimal spectrum sensing and access mechanism such that the average energy cost of the SU, which includes the energy consumed by spectrum sensing, channel switching, and data transmission, is minimized, whereas multiple constraints on the reliability of sensing, the throughput, and the delay of the secondary transmission are satisfied. Optimality is achieved by jointly considering two fundamental tradeoffs involved in energy minimization, i.e., the sensing/transmission tradeoff and the wait/switch tradeoff. An efficient convex optimization procedure is developed to solve for the optimal values of the sensing slot duration and the channel switching probability. The advantages of the proposed spectrum sensing and access mechanism are shown through simulations.

Robust reliable guaranteed cost piecewise fuzzy control for discrete-time nonlinear systems with time-varying delay and actuator failures

In this paper, we investigate the delay-dependent robust reliable guaranteed cost (RRGC) fuzzy control problem for discrete-time nonlinear systems with time-varying delays. The delays may simultaneously appear in the state and in the control input. Also, both parametric uncertainties and control component failure may exist. Through Takagi–Sugeno fuzzy modelling of nonlinear delayed-systems and based on an appropriate piecewise Lyapunov-Krasovskii functional, a piecewise fuzzy controller is designed. Sufficient conditions for the existence of a RRGC controller are derived in terms of linear matrix inequalities (LMIs). Furthermore, a suboptimal RRGC fuzzy controller is given by means of a convex optimization procedure with LMI constraints which can not only guarantee the stability of the closed-loop fuzzy system, but also provides an optimized upper bound of the given cost performance despite possible actuator faults. Two numerical examples are presented in this paper to illustrate the feasibility of the theoretical developments.

A fast nonlinear control method for linear systems with input saturation

We present a novel, fast saturating nonlinear feedback law for single input systems with linear dynamics and input saturation. It is fast in the sense that it yields a better performance than a saturating linear control law. The control law is based on implicit soft variable-structure control. A convex optimization procedure for the controller synthesis based on linear matrix inequalities (LMIs) is derived at the price of some conservatism. As an example, we consider the control of a submarine.

Diffusion Estimation from Multiscale Data by Operator Eigenpairs

In this paper we present a new procedure for the estimation of diffusion processes from discretely sampled data. It is based on the close relation between eigenpairs of the diffusion operator and those of the conditional expectation operator , a relation stemming from the semigroup structure for . It allows for estimation without making time discretization errors, an aspect that is particularly advantageous in the case of data with low sampling frequency. After estimating eigenpairs of via eigenpairs of , we infer the drift and diffusion functions that determine by fitting to the estimated eigenpairs using a convex optimization procedure. We present numerical examples in which we apply the procedure to one- and two-dimensional diffusions, reversible as well as nonreversible. In the second part of the paper, we consider estimation of coarse-grained (homogenized) diffusion processes from multiscale data. We show that eigenpairs of the homogenized diffusion operator are asymptotically close to eigenpairs of the underlying multiscale diffusion operator. This implies that we can infer the correct homogenized process from data of the multiscale process, using the estimation procedure discussed in the first part of the paper. This is illustrated with numerical examples.

FIELD SYNTHESIS IN INHOMOGENEOUS MEDIA: JOINT CONTROL OF POLARIZATION, UNIFORMITY AND SAR IN MRI B<sub>1</sub>-FIELD

The homogeneity of the amplitude of one of the polarizations of the RF fleld B 1 is a crucial issue in Magnetic Resonance Imaging (MRI), and several methods have been proposed for enhancing this uniformity (\Shimming). The existing approaches aim at controlling the homogeneity of B + 1 and limiting the Speciflc Absorption Rate (SAR) of the RF fleld by independently controlling magnitude and phase of individual excitation currents in MRI scanners, either birdcage or TEM coil system. A novel approach is presented here which allows a joint control of B + 1 uniformity, SAR, and purity of polarization of the total RF B 1 fleld. We propose a convex optimization procedure with convex constraints, and special attention has been devoted to the issue of convexity of the proposed functional. The method is applied to MRI brain imaging; numerical tests have been performed on a realistic head model at low, medium, and high RF fleld in order to assess the efiectiveness of the proposed method. We found that maintaining a speciflc polarization plays an important role also in maintaining the homogeneity of B + 1 amplitude.

Robust PID design for second-order processes with time-delay and structured uncertainties.

AbstractThis paper deals with the problem of PID design for continuous-time systems with time delays. The system is assumed to be free of parametric disturbances and affected by a time-invariant discrete delay of known magnitude. The robustness of the PID control with respect to structured uncertainties is investigated with the small-gain theorem and better performance is sought through the minimization of an upper bound to the closed-loop system ℋ∞ norm. A Lyapunov-Krasovskii type functional is used yielding delay-dependent design conditions. The controller design is accomplished by means of a convex optimization procedure formulated using linear matrix inequalities (LMIs). Numerical experiments are provided to illustrate the main characteristics of the proposed design method. The particular case of a recycle process controller is addressed.

Robust [formula omitted] and [formula omitted] filter design for uncertain linear systems via LMIs and polynomial matrices

This paper presents new convex optimization procedures for full order robust H2 and H∞ filter design for continuous and discrete-time uncertain linear systems. The time-invariant uncertain parameters are supposed to belong to a polytope with known vertices. Thanks to the use of a larger number of slack variables, linear matrix inequalities for the design of robust filters can be derived from the proposed conditions, outperforming the existing methods. The superiority and efficiency of the proposed method for filter design are illustrated by means of numerical comparisons in benchmark examples from the literature.

CORE-Net: exploiting prior knowledge and preferential attachment to infer biological interaction networks

The problem of reverse engineering in the topology of functional interaction networks from time-course experimental data has received considerable attention in literature, due to the potential applications in the most diverse fields, comprising engineering, biology, economics and social sciences. The present work introduces a novel technique, CORE-Net, which addresses this problem focusing on the case of biological interaction networks. The method is based on the representation of the network in the form of a dynamical system and on an iterative convex optimisation procedure. A first advantage of the proposed approach is that it allows to exploit qualitative prior knowledge about the network interactions, of the same kind as typically available from biological literature and databases. A second novel contribution consists of exploiting the growth and preferential attachment mechanisms to improve the inference performances when dealing with networks which exhibit a scale-free topology. The technique is first assessed through numerical tests on in silico random networks, subsequently it is applied to reverse engineering a cell cycle regulatory subnetwork in Saccharomyces cerevisiae from experimental microarray data. These tests show that the combined exploitation of prior knowledge and preferential attachment significantly improves the predictions with respect to other approaches.

Message-passing algorithms for compressed sensing

Compressed sensing aims to undersample certain high-dimensional signals yet accurately reconstruct them by exploiting signal characteristics. Accurate reconstruction is possible when the object to be recovered is sufficiently sparse in a known basis. Currently, the best known sparsity-undersampling tradeoff is achieved when reconstructing by convex optimization, which is expensive in important large-scale applications. Fast iterative thresholding algorithms have been intensively studied as alternatives to convex optimization for large-scale problems. Unfortunately known fast algorithms offer substantially worse sparsity-undersampling tradeoffs than convex optimization. We introduce a simple costless modification to iterative thresholding making the sparsity-undersampling tradeoff of the new algorithms equivalent to that of the corresponding convex optimization procedures. The new iterative-thresholding algorithms are inspired by belief propagation in graphical models. Our empirical measurements of the sparsity-undersampling tradeoff for the new algorithms agree with theoretical calculations. We show that a state evolution formalism correctly derives the true sparsity-undersampling tradeoff. There is a surprising agreement between earlier calculations based on random convex polytopes and this apparently very different theoretical formalism.