Nonquadratic Penalty Functions Research Articles

Distributed optimal consensus with obstacle avoidance algorithm of mixed-order UAVs–USVs–UUVs systems

This article addresses the formation control with obstacle avoidance for a mixed-order linear multi-agent in the ocean environment via a distributed optimal control approach. The considered system is comprised of unmanned aerial vehicles (UAVs), unmanned surface vehicles (USVs) and unmanned underwater vehicles (UUVs). A consensus control law for horizontal direction and height direction is then designed uniformly, and the stability is ensured using block Kronecker product and matrix transformation theory. A linear quadratic regulator method is further designed to achieve distributed optimization. Moreover, for obstacle avoidance in the UUVs, a non-quadratic penalty function is constructed based on the seamount threat model by adopting an inverse optimal control approach. Simulation results confirm the efficiency of the consensus and obstacle avoidance.

Solving large-scale reactive optimal power flow problems by a primal–dual $$\hbox {M}^{2}\hbox {BF}$$ approach

In this paper, we propose a predictor–corrector primal–dual approach for the doubly modified logarithmic barrier function ($$\hbox {M}^{2}\hbox {BF}$$) method in order to solve Optimal Reactive Power Flow (ORPF) problems. The $$\hbox {M}^{2}\hbox {BF}$$ is a modification of the Polyak’s modified logarithmic barrier function (MBF) and is also a particular element of a recent family of nonquadratic penalty functions for augmented Lagrangian methods for handling convex problems only with inequality constraints. We also propose a global convergence strategy to be inserted in the proposed algorithm, which is developed over weak assumptions concerning the primal Hessian matrix. The resulting predictor–corrector primal–dual $$\hbox {M}^{2}\hbox {BF}$$ approach is applied for solving ORPF problems involving power systems with 57, 89, 118, 200, 300, 1354, 2007 and 2869 buses. A comparison with two state-of-the-art methods is performed. Numerical results show that the proposed approach is competitive and capable of solving ORPF problems for small to large-scale power systems.

A primal-dual integrated nonlinear rescaling approach applied to the optimal reactive dispatch problem

The objective of this paper is to propose a novel augmented Lagrangian approach for solving the Optimal Reactive Dispatch (ORD) problem in electric power systems, which consists in a large nonconvex Nonlinear Programming (NLP) problem. Given an NLP problem, its associated dual problem is solved by a proximal point method, whose distance between two points is calculated by means of a convex combination involving Bregman’s distance and Csiszar’s entropic φ-divergence distance pondered by a weighting factor, namely Integrated Distance Measure (IDM). Such dual proximal point method with the IDM leads to an equivalent augmented Lagrangian method with a new family of nonquadratic penalty functions for the primal problem, namely Integrated Nonlinear Rescaling (INLR) method. The concepts of integrated logarithmic modified barrier and integrated exponential functions, which are elements of the new family of penalty functions, are defined. A Primal-Dual Integrated Nonlinear Rescaling (PDINLR) method is proposed to solve convex and nonconvex NLPs. A small convex NLP problem is used to depict how such algorithm attains optima, and a small nonconvex NLP problem is used to show how the proposed approach outperforms other known methods in the area when a suitable value for the weighting factor is chosen. The effectiveness, efficiency and the robustness of the proposed PDINLR method with the integrated logarithmic modified barrier function are shown by the resolution of the ORD problem for the IEEE 30 and 300-bus test-systems.

Nonquadratic penalization improves near-infrared diffuse optical tomography

A new approach that can easily incorporate any generic penalty function into the diffuse optical tomographic image reconstruction is introduced to show the utility of nonquadratic penalty functions. The penalty functions that were used include quadratic (ℓ2), absolute (ℓ1), Cauchy, and Geman-McClure. The regularization parameter in each of these cases was obtained automatically by using the generalized cross-validation method. The reconstruction results were systematically compared with each other via utilization of quantitative metrics, such as relative error and Pearson correlation. The reconstruction results indicate that, while the quadratic penalty may be able to provide better separation between two closely spaced targets, its contrast recovery capability is limited, and the sparseness promoting penalties, such as ℓ1, Cauchy, and Geman-McClure have better utility in reconstructing high-contrast and complex-shaped targets, with the Geman-McClure penalty being the most optimal one.

Noise‐resolution tradeoffs in x‐ray CT imaging: A comparison of penalized alternating minimization and filtered backprojection algorithms

In comparison with conventional filtered backprojection (FBP) algorithms for x-ray computed tomography (CT) image reconstruction, statistical algorithms directly incorporate the random nature of the data and do not assume CT data are linear, noiseless functions of the attenuation line integral. Thus, it has been hypothesized that statistical image reconstruction may support a more favorable tradeoff than FBP between image noise and spatial resolution in dose-limited applications. The purpose of this study is to evaluate the noise-resolution tradeoff for the alternating minimization (AM) algorithm regularized using a nonquadratic penalty function. Idealized monoenergetic CT projection data with Poisson noise were simulated for two phantoms with inserts of varying contrast (7%-238%) and distance from the field-of-view (FOV) center (2-6.5 cm). Images were reconstructed for the simulated projection data by the FBP algorithm and two penalty function parameter values of the penalized AM algorithm. Each algorithm was run with a range of smoothing strengths to allow quantification of the noise-resolution tradeoff curve. Image noise is quantified as the standard deviation in the water background around each contrast insert. Modulation transfer functions (MTFs) were calculated from six-parameter model fits to oversampled edge-spread functions defined by the circular contrast-insert edges as a metric of local resolution. The integral of the MTF up to 0.5 1p/mm was adopted as a single-parameter measure of local spatial resolution. The penalized AM algorithm noise-resolution tradeoff curve was always more favorable than that of the FBP algorithm. While resolution and noise are found to vary as a function of distance from the FOV center differently for the two algorithms, the ratio of noises when matching the resolution metric is relatively uniform over the image. The ratio of AM-to-FBP image variances, a predictor of dose-reduction potential, was strongly dependent on the shape of the AM's nonquadratic penalty function and was also strongly influenced by the contrast of the insert for which resolution is quantified. Dose-reduction potential, reported here as the fraction (%) of FBP dose necessary for AM to reconstruct an image with comparable noise and resolution, for one penalty parameter value of the AM algorithm was found to vary from 70% to 50% for low-contrast and high-contrast structures, respectively, and from 70% to 10% for the second AM penalty parameter value. However, the second penalty, AM-700, was found to suffer from poor low-contrast resolution when matching the high-contrast resolution metric with FBP. The results of this simulation study imply that penalized AM has the potential to reconstruct images with similar noise and resolution using a fraction (10%-70%) of the FBP dose. However, this dose-reduction potential depends strongly on the AM penalty parameter and the contrast magnitude of the structures of interest. In addition, the authors' results imply that the advantage of AM can be maximized by optimizing the nonquadratic penalty function to the specific imaging task of interest. Future work will extend the methods used here to quantify noise and resolution in images reconstructed from real CT data.

Multi-agent consensus algorithm with obstacle avoidance via optimal control approach

Multi-agent consensus problem in an obstacle-laden environment is addressed in this study. A novel optimal control approach is proposed for the multi-agent system to reach consensus as well as avoid obstacles with a reasonable control effort. An innovative nonquadratic penalty function is constructed to achieve obstacle avoidance capability from an inverse optimal control perspective. The asymptotic stability and optimality of the consensus algorithm are proven. In addition, the optimal control law only requires local information from the communication topology to guarantee the proposed behaviour, rather than all agents’ information. The consensus and obstacle avoidance are validated through various simulations.

Serial turbo decoder for robust communication

A new serial turbo decoder for robust communication in the presence of impulsive noise is proposed and examined. The new decoder iteratively performs robust maximum a posteriori (MAP) decoding by cascading a nonlinear MAP inner decoder incorporating a non-quadratic penalty function with an MAP outer decoder. Results show that the proposed serial turbo decoder facilitates considerable performance improvement.

Further results on linear nonquadratic optimal control

Continues an investigation by the authors of minimizing the transient response of a linear system as measured by nonquadratic penalty functions, in particular, penalty functions which have linear growth. First, this note shows that the optimal state feedback which minimizes the transient response in the case of no exogenous inputs also minimizes the induced l/sup 1/ norm in case exogenous inputs are present. Second, it considers the case of constrained systems and derives bounds which establish the stability and performance of receding horizon control laws. Finally, this note illustrates the results for scheduling of reliable manufacturing systems.

Conjugate-gradient preconditioning methods for shift-variant PET image reconstruction

Gradient-based iterative methods often converge slowly for tomographic image reconstruction and image restoration problems, but can be accelerated by suitable preconditioners. Diagonal preconditioners offer some improvement in convergence rate, but do not incorporate the structure of the Hessian matrices in imaging problems. Circulant preconditioners can provide remarkable acceleration for inverse problems that are approximately shift-invariant, i.e., for those with approximately block-Toeplitz or block-circulant Hessians. However, in applications with nonuniform noise variance, such as arises from Poisson statistics in emission tomography and in quantum-limited optical imaging, the Hessian of the weighted least-squares objective function is quite shift-variant, and circulant preconditioners perform poorly. Additional shift-variance is caused by edge-preserving regularization methods based on nonquadratic penalty functions. This paper describes new preconditioners that approximate more accurately the Hessian matrices of shift-variant imaging problems. Compared to diagonal or circulant preconditioning, the new preconditioners lead to significantly faster convergence rates for the unconstrained conjugate-gradient (CG) iteration. We also propose a new efficient method for the line-search step required by CG methods. Applications to positron emission tomography (PET) illustrate the method.