Spectral Projected Gradient Method Research Articles

Dielectric function of palladium capped zirconium thin films as a function of absorbed hydrogen

We report measurements of the optical transmission, between 240 and 1040 nm, and electrical resistivity of polycrystalline zirconium thin films as they absorb hydrogen. Both are measured as H2 pressure is increased up to 880 mbar, at room temperature. Films, 20–22 nm thick, are deposited on fused quartz substrates by e-beam evaporation at 5.3 × 10−7 mbar base pressure and covered with a 8.0 nm Pd over-coat. The morphology of the films is studied by means of AFM images. The complex refractive indices of Zr and Pd are extracted numerically from the transmission spectra by using a spectral projected gradient method for different hydrogen pressures. The corresponding dielectric functions for various Zr hydrogen concentrations are described with the parametric Drude-Lorentz and Brendel-Bormann (DL & BB) models. The Acceptance-Probability-Controlled Simulated Annealing approach is applied to calculate the parameters of the DL & BB model. This allows us to describe the effect of increasing hydrogen absorption on these parameters and in derived quantities, like the relaxation time and the effective mass of conduction electrons, the electrical resistance, the Fermi energy, and the electronic density of states at the Fermi level.

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.

Signal Processing in Massive MIMO With IQ Imbalances and Low-Resolution ADCs

This paper proposes a massive multiple-input multiple-output (MIMO) with low-precision in-phase quadrature-phase (IQ) (de-)modulators and low-resolution analog-to-digital convertors, and develops uplink baseband algorithms for the proposed MIMO system. The IQ imbalance parameters are recovered by a two-stage method. At the first stage, the base station (BS) antennas acquire phase-shifted versions of their imbalance parameters after receiving some predefined training sequence emitted by a specific transceiver that is near to the BS and shares flat-fading channels between itself and the BS. At the second stage, the phase shifts from the first stage are estimated by the specific transceiver through receiving a common sequence transmitted by the BS antennas alternatively. In order to jointly handle the IQ imbalance and nonlinear quantization loss, the IQ-imbalanced multiuser (MU)-MIMO with single-antenna users is first converted into an IQ-balanced MU-MIMO with dual-antenna users. Then, two kinds of channel estimator and multiuser detectors are constructed for the equivalent IQ-balanced system based on the spectral projected gradient method and vectorized message passing de-quantization algorithm. Our numerical results validate the effectiveness of the above baseband algorithms, and show that the proposed massive MIMO is more energy-efficient than the conventional one when the channel coherence time is long enough.

A block active set algorithm with spectral choice line search for the symmetric eigenvalue complementarity problem

In this paper, we address the solution of the symmetric eigenvalue complementarity problem (EiCP) by treating an equivalent reformulation of finding a stationary point of a fractional quadratic program on the unit simplex. The spectral projected-gradient (SPG) method has been recommended to this optimization problem when the dimension of the symmetric EiCP is large and the accuracy of the solution is not a very important issue. We suggest a new algorithm which combines elements from the SPG method and the block active set method, where the latter was originally designed for box constrained quadratic programs. In the new algorithm the projection onto the unit simplex in the SPG method is replaced by the much cheaper projection onto a box. This can be of particular advantage for large and sparse symmetric EiCPs. Global convergence to a solution of the symmetric EiCP is established. Computational experience with medium and large symmetric EiCPs is reported to illustrate the efficacy and efficiency of the new algorithm.

Computing the generalized eigenvalues of weakly symmetric tensors

Tensor is a hot topic in the past decade and eigenvalue problems of higher order tensors become more and more important in the numerical multilinear algebra. Several methods for finding the Z-eigenvalues and generalized eigenvalues of symmetric tensors have been given. However, the convergence of these methods when the tensor is not symmetric but weakly symmetric is not assured. In this paper, we give two convergent gradient projection methods for computing some generalized eigenvalues of weakly symmetric tensors. The gradient projection method with Armijo step-size rule (AGP) can be viewed as a modification of the GEAP method. The spectral gradient projection method which is born from the combination of the BB method with the gradient projection method is superior to the GEAP, AG and AGP methods. We also make comparisons among the four methods. Some competitive numerical results are reported at the end of this paper.

Global and local R-linear convergence of a spectral projected gradient method for convex optimization with singular solution

Global and local R-linear convergence of a spectral projected gradient method for convex optimization with singular solution

On the low rank solution of the Q‐weighted nearest correlation matrix problem

SummaryThe low rank solution of the Q‐weighted nearest correlation matrix problem is studied in this paper. Based on the property of Q‐weighted norm and the Gramian representation, we first reformulate the Q‐weighted nearest correlation matrix problem as a minimization problem of the trace function with quadratic constraints and then prove that the solution of the minimization problem the trace function is the stationary point of the original problem if it is rank‐deficient. Finally, the nonmonotone spectral projected gradient method is constructed to solve them. Numerical examples illustrate that the new method is feasible and effective. Copyright © 2015 John Wiley & Sons, Ltd.

On the Quadratic Eigenvalue Complementarity Problem over a general convex cone

The solution of the Conic Quadratic Eigenvalue Complementarity Problem (CQEiCP) is first investigated without assuming symmetry on the matrices defining the problem. A new sufficient condition for existence of solutions of CQEiCP is presented, extending to arbitrary pointed, closed and convex cones a condition known to hold when the cone is the nonnegative orthant. We also address the symmetric CQEiCP where all its defining matrices are symmetric. We show that, assuming that two of its defining matrices are positive definite, this symmetric CQEiCP reduces to the computation of a stationary point of an appropriate merit function on a convex set. Furthermore, we discuss the use of the so called Spectral Projected Gradient (SPG) algorithm for solving CQEiCP when the cone of interest is the second-order cone (SOCQEiCP). A new algorithm is designed for the computation of the projections required by the SPG method to deal with SOCQEiCP. Numerical results are included to illustrate the efficiency of the SPG method and the new projection technique in practice.

Minimizing quadratic functions with semidefinite Hessian subject to bound constraints

The MPRGP (modified proportioning with reduced gradient projections) algorithm for minimization of the strictly convex quadratic function subject to bound constraints is adapted to the solution of problems with a semidefinite Hessian A. The adapted algorithm accepts the decrease directions that belong to the null space of A and generates the iterates that are proved to minimize the cost function. The paper examines specific features of the solution of the problems with convex, but not necessarily strictly convex Hessian. The performance of the algorithm is demonstrated by the solution of a semi-coercive contact problem of elasticity and a 3D particle dynamics problem. The results are compared with those obtained by the spectral projected gradient method and the projected-Jacobi method.

On the Application of the Spectral Projected Gradient Method in Image Segmentation

We investigate the application of the nonmonotone spectral projected gradient (SPG) method to a region-based variational model for image segmentation. We consider a "discretize-then-optimize" approach and solve the resulting nonlinear optimization problem by an alternating minimization procedure that exploits the SPG2 algorithm by Birgin et al. (SIAM J Optim 10(4):1196---1211, 2000). We provide a convergence analysis and perform numerical experiments on several images, showing the effectiveness of this procedure.

An alternating direction method for solving a class of inverse semi-definite quadratic programming problems

In this paper, we propose an alternating-direction-type numerical method to solvea class of inverse semi-definite quadratic programming problems. Anexplicit solution to one direction subproblem is given and the otherdirection subproblem is proved to be a convex quadratic programmingproblem over positive semi-definite symmetric matrix cone. We designa spectral projected gradient method for solving the quadraticmatrix optimization problem and demonstrate its convergence.Numerical experiments illustrate that our method can solve inversesemi-definite quadratic programming problems efficiently.

Reconstruction of fluorescence molecular tomography via a nonmonotone spectral projected gradient pursuit method.

Fluorescence molecular tomography (FMT) is a promising imaging technique in preclinical research, enabling three-dimensional location of the specific tumor position for small animal imaging. However, FMT presents a challenging inverse problem that is quite ill-posed and ill-conditioned. Thus, the reconstruction of FMT faces various challenges in its robustness and efficiency. We present an FMT reconstruction method based on nonmonotone spectral projected gradient pursuit (NSPGP) with /₁-norm optimization. At each iteration, a spectral gradient-projection method approximately minimizes a least-squares problem with an explicit one-norm constraint. A nonmonotone line search strategy is utilized to get the appropriate updating direction, which guarantees global convergence. Additionally, the Barzilai-Borwein step length is applied to build the optimal step length, further improving the convergence speed of the proposed method. Several numerical simulation studies, including multisource cases as well as comparative analyses, have been performed to evaluate the performance of the proposed method. The results indicate that the proposed NSPGP method is able to ensure the accuracy, robustness, and efficiency of FMT reconstruction. Furthermore, an in vivo experiment based on a heterogeneous mouse model was conducted, and the results demonstrated that the proposed method held the potential for practical applications of FMT.

Robust least square semidefinite programming with applications

In this paper, we consider a least square semidefinite programming problem under ellipsoidal data uncertainty. We show that the robustification of this uncertain problem can be reformulated as a semidefinite linear programming problem with an additional second-order cone constraint. We then provide an explicit quantitative sensitivity analysis on how the solution under the robustification depends on the size/shape of the ellipsoidal data uncertainty set. Next, we prove that, under suitable constraint qualifications, the reformulation has zero duality gap with its dual problem, even when the primal problem itself is infeasible. The dual problem is equivalent to minimizing a smooth objective function over the Cartesian product of second-order cones and the Euclidean space, which is easy to project onto. Thus, we propose a simple variant of the spectral projected gradient method (Birgin et al. in SIAM J. Optim. 10:1196---1211, 2000) to solve the dual problem. While it is well-known that any accumulation point of the sequence generated from the algorithm is a dual optimal solution, we show in addition that the dual objective value along the sequence generated converges to a finite value if and only if the primal problem is feasible, again under suitable constraint qualifications. This latter fact leads to a simple certificate for primal infeasibility in situations when the primal feasible set lies in a known compact set. As an application, we consider robust correlation stress testing where data uncertainty arises due to untimely recording of portfolio holdings. In our computational experiments on this particular application, our algorithm performs reasonably well on medium-sized problems for real data when finding the optimal solution (if exists) or identifying primal infeasibility, and usually outperforms the standard interior-point solver SDPT3 in terms of CPU time.

Spectral Projected Gradient Methods: Review and Perspectives

Over the last two decades, it has been observed that using the gradient vector as a search direction in large-scale optimization may lead to efficient algorithms. The effectiveness relies on choosing the step lengths according to novel ideas that are related to the spectrum of the underlying local Hessian rather than related to the standard decrease in the objective function. A review of these so-called spectral projected gradient methods for convex constrained optimization is presented. To illustrate the performance of these low-cost schemes, an optimization problem on the set of positive definite matrices is described.

Multivariate spectral gradient projection method for nonlinear monotone equations with convex constraints

In this paper, we present a multivariate spectral gradient projection method for nonlinear monotone equations with convex constraints, which can be viewed as an extension of multivariate spectral gradient method for solving unconstrained optimization problems. The proposed method does not need the computation of the derivative as well as the solution of some linear equations. Under some suitable conditions, we can establish its global convergence results. Preliminary numerical results show that the proposed method is efficient and promising.

A hybrid-optimization method for large-scale non-negative full regularization in image restoration

We describe a new hybrid-optimization method for solving the full-regularization problem of computing both the regularization parameter and the corresponding regularized solution in 1-norm and 2-norm Tikhonov regularizations with additional non-negativity constraints. The approach combines the simulated annealing technique for global optimization and the low-cost spectral projected gradient method for the minimization of large-scale smooth functions on convex sets. The new method is matrix-free in the sense that it relies on matrix–vector multiplications only. We describe the method and discuss some of its properties. Numerical results indicate that the new approach is a promising tool for solving large-scale image restoration problems.

Energy Minimization of Point Charges on a Sphere with a Spectral Projected Gradient Method

Energy Minimization of Point Charges on a Sphere with a Spectral Projected Gradient Method

A nonmonotone spectral projected gradient method for large-scale topology optimization problems

An efficient gradient-based method to solve the volume constrained topology optimization problems is presented. Each iterate of this algorithm is obtained by the projection of a Barzilai-Borwein step onto the feasible set consisting of box and one linear constraints (volume constraint). To ensure the global convergence, an adaptive nonmonotone line search is performed along the direction that is given by the current and projection point. The adaptive cyclic reuse of the Barzilai-Borwein step is applied as the initial stepsize. The minimum memory requirement, the guaranteed convergence property, and almost only one function and gradient evaluations per iteration make this new method very attractive within common alternative methods to solve large-scale optimal design problems. Efficiency and feasibility of the presented method are supported by numerical experiments.

Compressed Sensing Reconstruction Algorithm Based on Spectral Projected Gradient Pursuit

摘要: 为了改进方向追踪法的重建精度和算法效率, 提出了一种基于谱投影梯度(Spectral projected gradient, SPG)追踪的压缩感知(Compressed sensing, CS) 重建算法. 该算法采用方向追踪法框架, 运用谱投影梯度方法计算更新方向和步长, 引进非单调线性搜索策略使算法避免收敛至局部最优解. 实验结果证明了该算法的有效性, 通过设定合适的阈值参数可以取得重建精度和算法效率之间的平衡. 关键词: 压缩感知 / 信号重建 / 谱投影梯度 / 方向追踪

Coverage segmentation based on linear unmixing and minimization of perimeter and boundary thickness

We present a method for coverage segmentation, where the, possibly partial, coverage of each image element by each of the image components is estimated. The method combines intensity information with spatial smoothness criteria. A model for linear unmixing of image intensities is enhanced by introducing two additional conditions: (i) minimization of object perimeter, leading to smooth object boundaries, and (ii) minimization of the thickness of the fuzzy object boundary, and to some extent overall image fuzziness, to respond to a natural assumption that imaged objects are crisp, and that fuzziness is mainly due to the imaging and digitization process. The segmentation is formulated as an optimization problem and solved by the Spectral Projected Gradient method. This fast, deterministic optimization method enables practical applicability of the proposed segmentation method. Evaluation on both synthetic and real images confirms very good performance of the algorithm.