Spectral Projected Gradient Algorithm Research Articles

A new nonmonotone spectral projected gradient algorithm for box-constrained optimization problems in [formula omitted] real matrix space with application in image clustering

Box-constrained optimization problems in the real m×n matrix space have been widely applied in big data mining. However, efficient solution of them is still a challenge. In this paper, a new nonmonotone line search rule is first proposed by extending the well-known ones and inheriting their advantages. Then, by analyzing and exploiting properties of this rule, a new nonmonotone spectral projected gradient algorithm is developed to solve the box-constrained optimization problems in the matrix space. Global convergence of the developed algorithm is also established. Numerical tests are conducted on a series of randomly generated test problems and those in the set of benchmark test problems. Compared with other existing nonmonotone line search rules, our rule shows its advantages in terms of the significantly reduced number of function evaluations and significantly reduced number of iterations. To further validate applicability of this research, we apply the studied optimization problem and the developed algorithm to solve the problems of image clustering. Numerical results demonstrate that the proposed method can generate better clustering results and is more robust than the similar ones available in the literature.

A Spectral Gradient Projection Method for the Positive Semi-definite Procrustes Problem

This paper addresses the positive semi-deffnite procrustes problem (PSDP). The PSDP corresponds to a least squares problem over the set of symmetric and semi-deffnite positive matrices. These kinds of problems appear in many applications such as structure analysis, signal processing, among others. A non-monotone spectral projected gradient algorithm is proposed to obtain a numerical solution for the PSDP. The proposed algorithm employs the Zhang and Hager's non-monotone technique in combination with the Barzilai and Borwein's step size to accelerate convergence. Some theoretical results are presented. Finally, numerical experiments are performed to demonstrate the effectiveness and efficiency of the proposed method, and comparisons are made with other state-of-the-art algorithms.

Reconstruction of 3D non-uniformly sampled seismic data along two spatial coordinates using non-equispaced curvelet transform

Seismic data acquisition often faces the challenge of non-uniformly sampled data with missing traces. Only a few existing multitrace reconstruction methods can natively handle non-uniformly sampled data with missing traces. In this paper, we propose the non-equispaced fast discrete curvelet transform (NFDCT)-based reconstruction method designed for 3D seismic data that are non-uniformly sampled along two spatial coordinates. By partitioning 3D seismic datasets into time slices along source-receiver coordinates, we introduce 2D non-equispaced fast Fourier transform in the conventional fast discrete curvelet transform and formulate a regularised inversion of operator that links the uniformly sampled curvelet coefficients to non-uniformly sampled data. Numerically, the uniform curvelet coefficients are calculated by solving the L1-norm problem via the spectral projected-gradient algorithm. With the uniform curvelet coefficients, the NFDCT is formed via the conventional inverse curvelet transform and is used to reconstruct 3D non-uniformly sampled seismic data along two spatial coordinates. At the hand of reconstructed results from synthetic and field data, we demonstrate that the proposed method shows significant improvement over the conventional anti-leakage Fourier transform-based reconstruction method. The method we propose, which has a strong anti-aliasing and anti-noise ability, can be used to reconstruct the subset of observed data to a specified uniform grid along two spatial coordinates.

Power allocation scheme for target tracking in clutter with multiple radar system

In this paper, a power allocation (PA) scheme is developed for target tracking in clutter using multiple radar system (MRS). The key idea of the PA scheme is to maximize the tracking performance in clutter for a predetermined power budget of the MRS. We use the Bayesian Cramér-Rao lower bound as the criterion, and formulate the PA scheme as a mathematical optimization problem. We analytically show that this optimization problem is nonconvex, and adopt a spectral projected gradient algorithm to obtain a constrained stationary solution to this problem. Simulation results demonstrate that the PA process can evidently expand the detection range, increase the resource utilization efficiency of the MRS, and improve the target tracking accuracy with a given power budget.

Sparse Signal Inversion with Impulsive Noise by Dual Spectral Projected Gradient Method

We consider sparse signal inversion with impulsive noise. There are three major ingredients. The first is regularizing properties; we discuss convergence rate of regularized solutions. The second is devoted to the numerical solutions. It is challenging due to the fact that both fidelity and regularization term lack differentiability. Moreover, for ill-conditioned problems, sparsity regularization is often unstable. We propose a novel dual spectral projected gradient (DSPG) method which combines the dual problem of multiparameter regularization with spectral projection gradient method to solve the nonsmooth l1+l1 optimization functional. We show that one can overcome the nondifferentiability and instability by adding a smooth l2 regularization term to the original optimization functional. The advantage of the proposed functional is that its convex duality reduced to a constraint smooth functional. Moreover, it is stable even for ill-conditioned problems. Spectral projected gradient algorithm is used to compute the minimizers and we prove the convergence. The third is numerical simulation. Some experiments are performed, using compressed sensing and image inpainting, to demonstrate the efficiency of the proposed approach.

基于改进谱投影梯度算法的X射线发光断层成像

基于改进谱投影梯度算法的X射线发光断层成像

Sparse Multichannel Blind Deconvolution of Seismic Data via Spectral Projected-Gradient

In this work, an efficient numerical scheme is presented for seismic blind deconvolution in a multichannel scenario. The proposed method iterate with wo steps: first, wavelet estimation across all channels and second, refinement of the reflectivity estimate simultaneously in all channels using sparse deconvolution. The reflectivity update step is formulated as a basis pursuit denoising problem and a sparse solution is obtained with the spectral projected-gradient algorithm - faithfulness to the recorded traces is constrained by the measured noise level. Wavelet re-estimation has a closed form solution when performed in the frequency domain by finding the minimum energy wavelet common to all channels. Nothing is assumed known about the wavelet apart from its time duration. In tests with both synthetic and real data, the method yields sparse reflectivity series and stable wavelet estimates results compared to existing methods with significantly less computational effort.

An effective feature selection approach using hybrid particle swarm optimisation with spectral projected gradient algorithm for an up-flow anaerobic filter

Wastewater treatment plant (WWTP) helps to overcome water from excessive pollution. Chemical oxygen demand (COD) or biological oxygen demand (BOD) is the measure for the wastewater effluents. Up-flow anaerobic filter (UAF) is used for the removal and digestion of organic matter present in wastewater. In this work, the performance of the UAF with cheese-dairy wastewater was considered for modelling the ANN to predict the effluent COD level. The proposed method uses an effective dynamic score normalisation technique with Mahalanobis distance as a preprocessing step. The hybrid particle swarm optimisation (PSO) with spectral projected gradient (SPG2) algorithm selected the relevant features. Adaptive neuro-fuzzy inference system (ANFIS) with Runge-Kutta learning method (RKLM) is used for the prediction of COD effluent. Experiments are conducted on real datasets obtained from cheese-whey wastewater to predict the COD effluent. The experimental results proved that the proposed method achieves better accuracy and execution time. The average accuracy obtained in the proposed method is 92.94.

Application of the restrained optimal perturbation method to study the backward heat conduction problem

In this paper, a restrained optimal perturbation method is firstly proposed to solve the backward heat conduction problem, the initial temperature distribution will be identified from the overspecified data, a regularization term is introduced in the objective functional for overcoming the ill-posedness of this problem, spectral projected gradient algorithm is used to solve the optimal problem, and we give the sensitivity analysis of the initial value. The results of numerical experiments are also presented.

On the optimal design of river fishways

A river fishway is a hydraulic structure enabling fish to overcome stream obstructions such as dams in hydroelectric power plants. The main aim of this paper is to present an application of mathematical modeling and optimal control theory to an ecological engineering problem related to preserve and enhance natural stocks of fish migrating between saltwater and freshwater. Particularly, we improve the optimal shape design of a fishway. This problem is formulated within the framework of the optimal control of partial differential equations, approximated by a discrete optimization problem, and solved by using both a gradient method (a Spectral Projected-Gradient algorithm) and a derivative-free method (the Nelder-Mead algorithm). Finally, numerical results are compared and analyzed for a standard real-world situation.

Application of conditional non-linear optimal perturbations to tropical cyclone adaptive observation using theWeather Research Forecasting (WRF) model

Conditional non-linear optimal perturbation (CNOP), which is a natural extension of the linear singular vector into the non-linear regime, has been suggested to identify data-sensitive regions in the adaptive observation strategy. CNOP is the global maximum of a cost function, whereas, local CNOP is the local maximum of the cost function if the local maximum exists. The potential application of CNOPs to tropical cyclone adaptive observation is researched. The CNOPs and the first singular vector (FSV) are numerically obtained by a spectral projected gradient algorithm with the Weather Research Forecasting (WRF) model. This paper examines two tropical cyclone cases, a fast straight moving typhoon Matsa (2005) and a slow moving recurving typhoon Shanshan (2006). The CNOPs and FSVs are obtained using the norms of background error at initial time and total dry energy at final time with a 36-h optimization time interval. The spatial structures of CNOPs, their energies, non-linear evolutions and impacts on track simulations are compared with those of the FSVs. The results show that both the CNOPs and the FSVs are localized, and evolve into the verification area at the final time with the upscale growth of perturbations. However, the CNOPs are different from the FSVs in spatial patterns, wind maximum distribution, growth rate of energy and impact on track simulation. Compared to FSV, CNOP and local CNOP have greater impact on the forecast in the verification region at the final time in terms of total energy, and have larger, at least similar impact on track simulation too. This indicates the CNOP method with constraint of the norm of background error at initial time and total energy norm at final time is a reasonable candidate in tropical cyclone adaptive observation. Therefore, both CNOP and local CNOP are suggested to be considered in tropical cyclone adaptive observation.

An investigation of feasible descent algorithms for estimating the condition number of a matrix

Techniques for estimating the condition number of a nonsingular matrix are developed. It is shown that Hager’s 1-norm condition number estimator is equivalent to the conditional gradient algorithm applied to the problem of maximizing the 1-norm of a matrix-vector product over the unit sphere in the 1-norm. By changing the constraint in this optimization problem from the unit sphere to the unit simplex, a new formulation is obtained which is the basis for both conditional gradient and projected gradient algorithms. In the test problems, the spectral projected gradient algorithm yields condition number estimates at least as good as those obtained by the previous approach. Moreover, in some cases, the spectral gradient projection algorithm, with a careful choice of the parameters, yields improved condition number estimates.

On the use of the Spectral Projected Gradient method for Support Vector Machines

In this work we study how to solve the SVM optimization problem by using the Spectral Projected Gradient (SPG) method with three different strategies for computing the projection onto the constrained set. One of the strategies is based on Dykstra's alternating projection algorithm since there is not a mathematical equation for the projection onto the whole constrained set but the projection on each restriction is easy to compute with exact formulations. We present another strategy based on the Karush-Kunh-Tucker optimality conditions, we call it the Projected-KKT algorithm. We compare these strategies with a third one proposed by Dai and Fletcher. The three schemes are low computational cost and their use within the SPG algorithm leads to a solution of the SVM problem. We study the computational performance of the three strategies when solving randomly generated as well as real life SVM problems. The numerical results show that Projected-KKT is competitive in general with the Dai and Fletcher algorithm, and it is more efficient for some specific problems. They both outperform Dykstra's algorithm in all the tests.

On the solution of the symmetric eigenvalue complementarity problem by the spectral projected gradient algorithm

This paper is devoted to the eigenvalue complementarity problem (EiCP) with symmetric real matrices. This problem is equivalent to finding a stationary point of a differentiable optimization program involving the Rayleigh quotient on a simplex (Queiroz et al., Math. Comput. 73, 1849–1863, 2004). We discuss a logarithmic function and a quadratic programming formulation to find a complementarity eigenvalue by computing a stationary point of an appropriate merit function on a special convex set. A variant of the spectral projected gradient algorithm with a specially designed line search is introduced to solve the EiCP. Computational experience shows that the application of this algorithm to the logarithmic function formulation is a quite efficient way to find a solution to the symmetric EiCP.

Optical properties of an optically rough coating from inversion of diffuse reflectance measurements

A method is developed for determining the optical properties of an optically rough coating on an opaque substrate from reflectance measurements. A modified Kubelka-Munk two- flux model is used to calculate the reflectance of the coating as a function of the refractive index, absorption coefficient, scattering coefficient, and thickness. The calculated reflectance is then fitted to measurements using a spectral projected gradient algorithm, allowing the optical properties to be obtained. The technique is applied to titanium dioxide coatings on a titanium substrate. Realistic values of refractive index and absorption coefficients are generally obtained. Quantities that are useful for solar water-splitting applications are calculated, including the depth profile of absorption and the proportion of the incident photon flux absorbed in the coating under solar illumination.