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To improve the receiver performance, unimodular transmit waveform and mismatched filter are jointly designed by using the alternating direction method of multipliers (ADMM). A series of auxiliary variables are introduced to decouple the variables in objective function and additional constrains. Denominator normalization(fractional programming technology) and step function are applied to obtain optimal solution from some subproblems. For given transmit waveform, same-length mismatched filter is designed in receiver. The low sidelobe level performance of the present methods are illustrated with the simulation results.

In this paper, a preconditioned two-step modulus-based matrix splitting iteration method for linear complementarity problems associated with an M-matrix is proposed. The convergence analysis of the presented method is given. In particular, we provide a comparison theorem between preconditioned two-step modulus-based Gauss–Seidel (PTMGS) iteration method and two-step modulus-based Gauss–Seidel (TMGS) iteration method, which shows that PTMGS method improves the convergence rate of original TMGS method for linear complementarity problem. Numerical tested examples are used to illustrate the theoretical analysis.

With the development of neuroimaging techniques, a growing amount of multi-modal brain imaging data are collected, facilitating comprehensive study of the brain. In this paper, we jointly analyzed functional magnetic resonance imaging (fMRI) collected under different paradigms in order to understand cognitive behaviors of an individual. To this end, we proposed a novel multi-view learning algorithm called structure-enforced collaborative regression (SCoRe) to extract co-expressed discriminative brain regions under the guidance of anatomical structure of the brain. An advantage of SCoRe over its predecessor collaborative regression (CoRe) lies in its incorporation of group structures in the brain imaging data, which makes the model biologically more meaningful. Results from real data analysis has confirmed that by incorporating prior knowledge of brain structure, SCoRe can deliver better prediction performance and is less sensitive to hyper-parameters than CoRe. After validation with simulation experiments, we applied SCoRe to fMRI data collected from the Philadelphia Neurodevelopmental Cohort and adopted the scores from the wide range achievement test (WRAT) to evaluate an individual’s cognitive skills. We located 14 relevant brain regions that can efficiently predict WRAT scores and these brain regions were further confirmed by other independent studies.

In this paper, we consider a prototypical convex optimization problem with multi-block variables and separable structures. By adding the Logarithmic Quadratic Proximal (LQP) regularizer with suitable proximal parameter to each of the first grouped subproblems, we develop a partial LQP-based Alternating Direction Method of Multipliers (ADMM-LQP). The dual variable is updated twice with relatively larger stepsizes than the classical region $(0,\frac{1+\sqrt{5}}{2})$. Using a prediction-correction approach to analyze properties of the iterates generated by ADMM-LQP, we establish its global convergence and sublinear convergence rate of $O(1/T)$ in the new ergodic and nonergodic senses, where $T$ denotes the iteration index. We also extend the algorithm to a nonsmooth composite convex optimization and establish {similar convergence results} as our ADMM-LQP., Comment: 22 pages

The proximal term plays a significant role in the literature of proximal Alternating Direction Method of Multipliers (ADMM), since (positive-definite or indefinite) proximal terms can promote convergence of ADMM and further simplify the involved subproblems. However, an overlarge proximal parameter decelerates the convergence numerically, though the convergence can be established with it. In this paper, we thus focus on a Linearized Symmetric ADMM (LSADMM) with proper proximal terms for solving a family of multi-block separable convex minimization models, and we determine an optimal (smallest) value of the proximal parameter while convergence of this LSADMM can be still ensured. Our LSADMM partitions the data into two group variables and updates the Lagrange multiplier twice in different forms with suitable step sizes. The region of the proximal parameter, involved in the second group subproblems, is partitioned into the union of three different sets. We show the global convergence and sublinear ergodic convergence rate of LSADMM for the two cases, while a counter-example is given to illustrate that convergence of LSADMM can not be guaranteed for the remaining case. Theoretically, we obtain the optimal lower bound of the proximal parameter. Numerical experiments on the so-called latent variable Gaussian graphical model selection problems are presented to demonstrate performance of the proposed algorithm and the significant advantage of the optimal lower bound of the proximal parameter.

An inexact accelerated stochastic Alternating Direction Method of Multipliers (AS-ADMM) scheme is developed for solving structured separable convex optimization problems with linear constraints. The objective function is the sum of a possibly nonsmooth convex function and a smooth function which is an average of many component convex functions. Problems having this structure often arise in machine learning and data mining applications. AS-ADMM combines the ideas of both ADMM and the stochastic gradient methods using variance reduction techniques. One of the ADMM subproblems employs a linearization technique while a similar linearization could be introduced for the other subproblem. For a specified choice of the algorithm parameters, it is shown that the objective error and the constraint violation are $\mathcal{O}(1/k)$ relative to the number of outer iterations $k$. Under a strong convexity assumption, the expected iterate error converges to zero linearly. A linearized variant of AS-ADMM and incremental sampling strategies are also discussed. Numerical experiments with both stochastic and deterministic ADMM algorithms show that AS-ADMM can be particularly effective for structured optimization arising in big data applications.

In the literature, there are a few researches to design some parameters in the Proximal Point Algorithm (PPA), especially for the multi-objective convex optimizations. Introducing some parameters to PPA can make it more flexible and attractive. Mainly motivated by our recent work (Bai et al., A parameterized proximal point algorithm for separable convex optimization, Optim. Lett. (2017) doi: 10.1007/s11590-017-1195-9), in this paper we develop a general parameterized PPA with a relaxation step for solving the multi-block separable structured convex programming. By making use of the variational inequality and some mathematical identities, the global convergence and the worst-case $\mathcal{O}(1/t)$ convergence rate of the proposed algorithm are established. Preliminary numerical experiments on solving a sparse matrix minimization problem from statistical learning validate that our algorithm is more efficient than several state-of-the-art algorithms.

We design a novel preconditioned alternating direction method for solving a class of bilinear programming problems, where each subproblem is solved by adding a positive-definite regularization term with a proximal parameter. By the aid of the variational inequality, the global convergence of the proposed method is analyzed and a worst-case O(1/t) convergence rate in an ergodic sense is established. Several preliminary numerical examples, including the Markowitz portfolio optimization problem, are also tested to verify the performance of the proposed method.

This work is devoted to designing a unified alternating update method for solving a class of structured low rank approximations under the convex and unitarily invariant norm. By the aid of the variational inequality and monotone operator, the proposed method is proved to converge to the solution point of an equivalent variational inequality with a worst-case O ( 1 / t ) convergence rate in a nonergodic sense. We also analyze that the involved subproblems under the Frobenius norm are respectively equivalent to the structured least-squares problem and low rank least-squares problem, where the explicit solutions to some special cases are derived. In order to investigate the efficiency of the proposed method, several examples in system identification are tested to validate that the proposed method can outperform some state-of-the-art methods.

In this paper, we first present a general preconditioner P for solving linear complementarity problem (LCP) associated with an M -matrix A and a vector f , and prove that the LCP ( A , f ) is equivalent to the LCP ( P A , P f ) . Then based on this general preconditioner P , two preconditioned SSOR methods for solving the linear complementarity problems are proposed. We show that this general preconditioner P accelerates the convergence of two SSOR methods under the assumption that P A is a Z -matrix. In addition, we also give a practically concrete choice for the preconditioner P satisfying aforementioned assumption. Numerical examples are used to illustrate the theoretical results obtained.

In this paper, a class of minimization problems over density matrices arising in the quantum state estimation is investigated. By making use of the Nesterov’s accelerated strategies, we introduce a modified augmented Lagrangian method to solve it, where the subproblem is tackled by the projected Barzilai–Borwein method with nonmonotone line search. Compared with the existing projected gradient method, several numerical examples are tested to show that the proposed method is efficient and promising.

This work is devoted to studying the problem of approximating a given matrix by the product of two low-rank structured matrix which arises in the material processing. Firstly, we analyse some prope...

The robust principal component analysis (RPCA) model is a popular method for solving problems with the nuclear norm and l1 norm. However, it is time-consuming since in general one has to use the singular value decomposition in each iteration. In this paper, we introduce a novel model to reformulate the existed model by making use of low-rank matrix factorization to surrogate the nuclear norm for the sparse and low-rank decomposition problem. In such case we apply the Penalty Function Method (PFM) and Augmented Lagrangian Multipliers Method (ALMM) to solve this new nonconvex optimization problem. Theoretically, corresponding to our methods, the convergence analysis is given respectively. Compared with classical RPCA, some practical numerical examples are simulated to show that our methods are much better than RPCA.

In this paper, a multi-parameterized proximal point algorithm combining with a relaxation step is developed for solving convex minimization problem subject to linear constraints. We show its global convergence and sublinear convergence rate from the prospective of variational inequality. Preliminary numerical experiments on testing a sparse minimization problem from signal processing indicate that the proposed algorithm performs better than some well-established methods, 7 pages

In this note, we show a sublinear nonergodic convergence rate for the algorithm developed in [Bai, et al. Generalized symmetric ADMM for separable convex optimization. Comput. Optim. Appl. 70, 129-170 (2018)], as well as its linear convergence under assumptions that the sub-differential of each component objective function is piecewise linear and all the constraint sets are polyhedra. These remaining convergence results are established for the stepsize parameters of dual variables belonging to a special isosceles triangle region, which aims to strengthen our understanding for convergence of the generalized symmetric ADMM., 16 pages

In this paper, we give new error bounds for linear complementarity problems when the involved matrices are S-Nekrasov matrices and B–S-Nekrasov matrices, respectively. We prove that the obtained bounds are better than those of Gao et al. (J Comput Appl Math 336:147–159, 2018) in some cases.

In this paper, we consider the generalized low rank approximation of the correlation matrices problem which arises in the asset portfolio. We first characterize the feasible set by using the Gramian representation together with a special trigonometric function transform, and then transform the generalized low rank approximation of the correlation matrices problem into an unconstrained optimization problem. Finally, we use the conjugate gradient algorithm with the strong Wolfe line search to solve the unconstrained optimization problem. Numerical examples show that our new method is feasible and effective.