Logarithmic Barrier Research Articles

An efficient logarithmic barrier method without line search for convex quadratic programming

An efficient logarithmic barrier method without line search for convex quadratic programming

A weighted logarithmic barrier interior-point method for linearly constrained optimization

In this paper, a weighted logarithmic barrier interior-point method for solving the linearly convex constrained optimization problems is presented. Unlike the classical central-path, the barrier parameter associated with the per- turbed barrier problems, is not a scalar but is a weighted positive vector. This modi cation gives a theoretical exibility on its convergence and its numerical performance. In addition, this method is of a Newton descent direction and the computation of the step-size along this direction is based on a new e cient tech- nique called the tangent method. The practical e ciency of our approach is shown by giving some numerical results.

Bregman primal\u2013dual first-order method and application to sparse semidefinite programming

We present a new variant of the Chambolle–Pock primal–dual algorithm with Bregman distances, analyze its convergence, and apply it to the centering problem in sparse semidefinite programming. The novelty in the method is a line search procedure for selecting suitable step sizes. The line search obviates the need for estimating the norm of the constraint matrix and the strong convexity constant of the Bregman kernel. As an application, we discuss the centering problem in large-scale semidefinite programming with sparse coefficient matrices. The logarithmic barrier function for the cone of positive semidefinite completable sparse matrices is used as the distance-generating kernel. For this distance, the complexity of evaluating the Bregman proximal operator is shown to be roughly proportional to the cost of a sparse Cholesky factorization. This is much cheaper than the standard proximal operator with Euclidean distances, which requires an eigenvalue decomposition.

An efficient logarithmic barrier method for linear programming

In this paper, we are interested in the performance of the logarithmic barrier type of interior point method to solve a linear programming problem. We present a new alternative allowing to easily compute the step of displacement and without line search. This purpose is confirmed by numerical tests showing the efficiency of this approach versus classical line search methods.

Adaptive Neural Control and Modeling for Continuous Stirred Tank Reactor with Delays and Full State Constraints

In this paper, an adaptive neural network control method is described to stabilize a continuous stirred tank reactor (CSTR) subject to unknown time-varying delays and full state constraints. The unknown time delay and state constraints problem of the concentration in the reactor seriously affect the input-output ratio and stability of the entire system. Therefore, the design difficulty of this control scheme is how to debar the effect of time delay in CSTR systems. To deal with time-varying delays, Lyapunov–Krasovskii functionals (LKFs) are utilized in the adaptive controller design. The convergence of the tracking error to a small compact set without violating the constraints can be identified by the time-varying logarithm barrier Lyapunov function (LBLF). Finally, the simulation results on CSTR are shown to reveal the validity of the developed control strategy.

Adaptive CPG-Based Gait Planning With Learning-Based Torque Estimation and Control for Exoskeletons

In this letter, a new adaptable gait trajectory shaping method is proposed for lower-limb exoskeletons by defining central pattern generators (CPGs). These CPGs are synchronized across different joints and updated online in response to the human users' physical behavior to enhance their safety and comfort. In this CPG structure of a high-level control scheme, an overall locomotion frequency is defined for all joint motions that can be modulated as a function of the human-robot interaction (HRI) energy. The amplitude and equilibrium position of oscillation for each joint can be adjusted in real-time based on the HRI torque. Logarithmic barrier functions are also formulated for these connected CPGs to avoid exceeding safe bounds of the joints' motion. A supervised learning algorithm is employed to identify the exoskeleton-limb dynamics and estimate the active HRI torque on different joints based on an autoregressive network with exogenous inputs (NARX) model. In order to track the reference trajectories generated by CPGs, a proportional derivative (PD) controller with torque compensation is designed. In the experimental evaluation of this intelligent control strategy, an able-bodied person wearing the Indego exoskeleton could amend and personalize the gait features considerably over a short period of time by applying active torques on different joints. In these experiments, the user increased the motion amplitude of the hip and knee joints up to 14% and made more than 200% variation in the gait frequency, which implies a considerable level of flexibility in locomotion planning for further user studies.

KD-LBA: a Kernighan Lin-driven logarithmic barrier approach to solve the many-to-many assignment problem and its application in CPU/FPGA scheduling

KD-LBA: a Kernighan Lin-driven logarithmic barrier approach to solve the many-to-many assignment problem and its application in CPU/FPGA scheduling

A Full-Newton Step Interior Point Method for Fractional Programming Problem Involving Second Order Cone Constraint

Some efficient interior-point methods (IPMs) are based on using a self-concordant barrier function related to the feasibility set of the underlying problem.Here, we use IPMs for solving fractional programming problems involving second order cone constraints. We propose a logarithmic barrier function to show the self concordant property and present an algorithm to compute $\varepsilon-$solution of a fractional programming problem. Finally, we provide a numerical example to illustrate the approach.

WITHDRAWN: Logarithmic barrier methods and linear connections on manifolds

This article presented several main stages of timeline of evolutionary history of logarithmic barrier functions along with interior point methods and affine connections on manifolds. The crucial idea is many optimization problems can be better stated on manifolds rather than Euclidean space, such as interior point methods, which in turns based on self-concordant functions (logarithmic barrier functions). This led us to the truth, that there is a logical evidence to study the relationships between self-concordant functions and Riemannian manifolds including affine connections.

A second-order corrector wide neighborhood infeasible interior-point method for linear optimization based on a specific kernel function

In this paper, we present a second-order corrector infeasibleinterior-point method for linear optimization in a largeneighborhood of the central path. The innovation of our method is tocalculate the predictor directions using a specific kernel functioninstead of the logarithmic barrier function. We decompose thepredictor direction induced by the kernel function to two orthogonaldirections of the corresponding to the negative and positivecomponent of the right-hand side vector of the centering equation.The method then considers the new point as a linear combination ofthese directions along with a second-order corrector direction. Theconvergence analysis of the proposed method is investigated and itis proved that the complexity bound is Ο(n5/4 log e-1).

Target Tracking With Equality/Inequality Constraints Based on Trajectory Function of Time

This letter addresses the constrained target tracking problem based on the approach of the trajectory function of time (T-FoT). Both state equality and inequality constraints such as the trajectory geometry and width are considered, respectively. The penalty function is used in the T-FoT framework to account for the constraint. Specifically, the logarithmic barrier function is used to solve the challenging inequality constraint. Different from existing works which are mostly based on constrained Bayesian filters, our approaches make full use of constraints with little approximation. Three representative scenarios have been considered in the simulation for demonstrating the performance of our proposed approaches in comparison with existing constrained approaches based on Bayes filters.

Special Section on the 48th Annual ACM Symposium on Theory of Computing (STOC 2016)

Special Section on the 48th Annual ACM Symposium on Theory of Computing (STOC 2016)

A log –barrier approach for airborne gamma-ray spectrometry inversion

The standard processing of airborne gamma-ray spectrometry (AGRS) data provides useful preliminary information to interpretation in several contexts, such as environmental studies, geological mapping, and analysis of mineral deposits. For optimal results, the acquisition conditions should be nearly constant, and the flight height should be uniform. However, abrupt changes in flight height (often originated from mountainous terrain) are common and lead to spurious variations and incorrect anomaly interpretation. Moreover, the commonly used corrections applied to radiometric data do not consider the effective sampling area of a survey, especially the overlap between successive fields of view. As a consequence, the concentrations estimates of potassium (K), uranium (eU), and thorium (eTh) are not sharp. A solution to deal with this problem is to compute the concentration from the AGRS data through an inversion algorithm. Inversion of AGRS data has shown to be an effective approach to suppress the significant overlap between successive fields of view. We introduce a logarithmic barrier approach for the radiometric inversion to avoid spurious negative values in the radioelement concentration models. We choose calibration ranges from Brazil and Canada to test the proposed methodology. We produced concentration models for K, eU and eTh and compared the standard approach model with the recovered model from both ranges. The predicted data from inversion of both calibration ranges are mostly consistent with the survey data, not over smoothing the data or fitting noise. The predicted data from inversion showed more consistency with the observed data than the one predicted from the standard procedure.

A modified Nelder-Mead barrier method for constrained optimization

<p style='text-indent:20px;'>An interior point modified Nelder Mead method for nonlinearly constrained optimization is described. This method neither uses nor estimates objective function or constraint gradients. A modified logarithmic barrier function is used. The method generates a sequence of points which converges to KKT point(s) under mild conditions including existence of a Slater point. Numerical results are presented that show the algorithm performs well in practice.</p>

An interior point-proximal method of multipliers for convex quadratic programming

In this paper we combine an infeasible Interior Point Method (IPM) with the Proximal Method of Multipliers (PMM). The resulting algorithm (IP-PMM) is interpreted as a primal-dual regularized IPM, suitable for solving linearly constrained convex quadratic programming problems. We apply few iterations of the interior point method to each sub-problem of the proximal method of multipliers. Once a satisfactory solution of the PMM sub-problem is found, we update the PMM parameters, form a new IPM neighbourhood and repeat this process. Given this framework, we prove polynomial complexity of the algorithm, under standard assumptions. To our knowledge, this is the first polynomial complexity result for a primal-dual regularized IPM. The algorithm is guided by the use of a single penalty parameter; that of the logarithmic barrier. In other words, we show that IP-PMM inherits the polynomial complexity of IPMs, as well as the strict convexity of the PMM sub-problems. The updates of the penalty parameter are controlled by IPM, and hence are well-tuned, and do not depend on the problem solved. Furthermore, we study the behavior of the method when it is applied to an infeasible problem, and identify a necessary condition for infeasibility. The latter is used to construct an infeasibility detection mechanism. Subsequently, we provide a robust implementation of the presented algorithm and test it over a set of small to large scale linear and convex quadratic programming problems. The numerical results demonstrate the benefits of using regularization in IPMs as well as the reliability of the method.

Towards a Proof of the Fourier-Entropy Conjecture?

The total influence of a function is a central notion in analysis of Boolean functions, and characterizing functions that have small total influence is one of the most fundamental questions associated with it. The KKL theorem and the Friedgut junta theorem give a strong characterization of such functions whenever the bound on the total influence is \(o(\log n)\). However, both results become useless when the total influence of the function is \(\omega (\log n)\). The only case in which this logarithmic barrier has been broken for an interesting class of functions was proved by Bourgain and Kalai, who focused on functions that are symmetric under large enough subgroups of \(S_n\). In this paper, we build and improve on the techniques of the Bourgain–Kalai paper and establish new concentration results on the Fourier spectrum of Boolean functions with small total influence. Our results include: 1. A quantitative improvement of the Bourgain–Kalai result regarding the total influence of functions that are transitively symmetric. 2. A slightly weaker version of the Fourier-Entropy Conjecture of Friedgut and Kalai. Our result establishes new bounds on the Fourier entropy of a Boolean function f, as well as stronger bounds on the Fourier entropy of low-degree parts of f. In particular, it implies that the Fourier spectrum of a constant variance, Boolean function f is concentrated on \(2^{O(I[f]\log I[f])}\) characters, improving an earlier result of Friedgut. Removing the \(\log I[f]\) factor would essentially resolve the Fourier-Entropy Conjecture, as well as settle a conjecture of Mansour regarding the Fourier spectrum of polynomial size DNF formulas. Our concentration result for the Fourier spectrum of functions with small total influence also has new implications in learning theory. More specifically, we conclude that the class of functions whose total influence is at most K is agnostically learnable in time \(2^{O(K\log K)}\) using membership queries. Thus, the class of functions with total influence \(O(\log n/\log \log n)\) is agnostically learnable in \(\mathsf{poly}(n)\) time.

Reconfigurable Intelligent Surfaces Assisted Secure Multicast Communications

Reconfigurable Intelligent surface (RIS) can enhance the desired signals and suppress the undesired signals by adjusting the phase shifts of its reflector elements. This letter considers secure multicast communication with the help of RIS. A multiple-antennas transmitter multicasts the common messages to a group of single-antenna receivers in the presence of a single-antenna eavesdropper, where an RIS is set up to enhance the desired signals for the receivers and suppress the signals for the eavesdropper. First, the equivalent multicast and wiretap channels are analyzed, and a secrecy capacity maximization problem is formulated. Next, this letter obtains the necessary conditions that the optimal solution needs to satisfy. Finally, a logarithmic barrier algorithm is proposed to obtain a local optimal solution.

A log-barrier Newton-CG method for bound constrained optimization with complexity guarantees

Abstract We describe an algorithm based on a logarithmic barrier function, Newton’s method and linear conjugate gradients that seeks an approximate minimizer of a smooth function over the non-negative orthant. We develop a bound on the complexity of the approach, stated in terms of the required accuracy and the cost of a single gradient evaluation of the objective function and/or a matrix-vector multiplication involving the Hessian of the objective. The approach can be implemented without explicit calculation or storage of the Hessian.

IPO: Interior-Point Policy Optimization under Constraints

In this paper, we study reinforcement learning (RL) algorithms to solve real-world decision problems with the objective of maximizing the long-term reward as well as satisfying cumulative constraints. We propose a novel first-order policy optimization method, Interior-point Policy Optimization (IPO), which augments the objective with logarithmic barrier functions, inspired by the interior-point method. Our proposed method is easy to implement with performance guarantees and can handle general types of cumulative multi-constraint settings. We conduct extensive evaluations to compare our approach with state-of-the-art baselines. Our algorithm outperforms the baseline algorithms, in terms of reward maximization and constraint satisfaction.

Deep unfolding of a proximal interior point method for image restoration

Variational methods are widely applied to ill-posed inverse problems for they have the ability to embed prior knowledge about the solution. However, the level of performance of these methods significantly depends on a set of parameters, which can be estimated through computationally expensive and time-consuming methods. In contrast, deep learning offers very generic and efficient architectures, at the expense of explainability, since it is often used as a black-box, without any fine control over its output. Deep unfolding provides a convenient approach to combine variational-based and deep learning approaches. Starting from a variational formulation for image restoration, we develop iRestNet, a neural network architecture obtained by unfolding a proximal interior point algorithm. Hard constraints, encoding desirable properties for the restored image, are incorporated into the network thanks to a logarithmic barrier, while the barrier parameter, the stepsize, and the penalization weight are learned by the network. We derive explicit expressions for the gradient of the proximity operator for various choices of constraints, which allows training iRestNet with gradient descent and backpropagation. In addition, we provide theoretical results regarding the stability of the network for a common inverse problem example. Numerical experiments on image deblurring problems show that the proposed approach compares favorably with both state-of-the-art variational and machine learning methods in terms of image quality.