Matrix-Form Neural Networks for Complex-Variable Basis Pursuit Problem With Application to Sparse Signal Reconstruction.

In this article, a continuous-time complex-valued projection neural network (CCPNN) in a matrix state space is first proposed for a general complex-variable basis pursuit problem. The proposed CCPNN is proved to be stable in the sense of Lyapunov and to be globally convergent to the optimal solution under the condition that the sensing matrix is not row full rank. Furthermore, an improved discrete-time complex projection neural network (IDCPNN) is proposed by discretizing the CCPNN model. The proposed IDCPNN consists of a two-step stop strategy to reduce the calculational cost. The proposed IDCPNN is theoretically guaranteed to be global convergent to the optimal solution. Finally, the proposed IDCPNN is applied to the reconstruction of sparse signals based on compressed sensing. Computed results show that the proposed IDCPNN is superior to related complex-valued neural networks and conventional basis pursuit algorithms in terms of solution quality and computation time.

Solving Nonlinear Optimization Problems of Real Functions in Complex Variables by Complex-Valued Iterative Methods.

Much research has been devoted to complex-variable optimization problems due to their engineering applications. However, the complex-valued optimization method for solving complex-variable optimization problems is still an active research area. This paper proposes two efficient complex-valued optimization methods for solving constrained nonlinear optimization problems of real functions in complex variables, respectively. One solves the complex-valued nonlinear programming problem with linear equality constraints. Another solves the complex-valued nonlinear programming problem with both linear equality constraints and an -norm constraint. Theoretically, we prove the global convergence of the proposed two complex-valued optimization algorithms under mild conditions. The proposed two algorithms can solve the complex-valued optimization problem completely in the complex domain and significantly extend existing complex-valued optimization algorithms. Numerical results further show that the proposed two algorithms have a faster speed than several conventional real-valued optimization algorithms.

Two Fast Complex-Valued Algorithms for Solving Complex Quadratic Programming Problems.

In this paper, we propose two fast complex-valued optimization algorithms for solving complex quadratic programming problems: 1) with linear equality constraints and 2) with both an l1 -norm constraint and linear equality constraints. By using Brandwood's analytic theory, we prove the convergence of the two proposed algorithms under mild assumptions. The two proposed algorithms significantly generalize the existing complex-valued optimization algorithms for solving complex quadratic programming problems with an l1 -norm constraint only and unconstrained complex quadratic programming problems, respectively. Numerical simulations are presented to show that the two proposed algorithms have a faster speed than conventional real-valued optimization algorithms.

A Complex-Valued Projection Neural Network for Constrained Optimization of Real Functions in Complex Variables.

In this paper, we present a complex-valued projection neural network for solving constrained convex optimization problems of real functions with complex variables, as an extension of real-valued projection neural networks. Theoretically, by developing results on complex-valued optimization techniques, we prove that the complex-valued projection neural network is globally stable and convergent to the optimal solution. Obtained results are completely established in the complex domain and thus significantly generalize existing results of the real-valued projection neural networks. Numerical simulations are presented to confirm the obtained results and effectiveness of the proposed complex-valued projection neural network.

A complex-valued neural dynamical optimization approach and its stability analysis.

In this paper, we propose a complex-valued neural dynamical method for solving a complex-valued nonlinear convex programming problem. Theoretically, we prove that the proposed complex-valued neural dynamical approach is globally stable and convergent to the optimal solution. The proposed neural dynamical approach significantly generalizes the real-valued nonlinear Lagrange network completely in the complex domain. Compared with existing real-valued neural networks and numerical optimization methods for solving complex-valued quadratic convex programming problems, the proposed complex-valued neural dynamical approach can avoid redundant computation in a double real-valued space and thus has a low model complexity and storage capacity. Numerical simulations are presented to show the effectiveness of the proposed complex-valued neural dynamical approach.