In recent years, compressive sensing (CS) problem is being popularly applied in the fields of signal processing and statistical inference. The alternating direction method of multipliers (ADMM) is applicable to the equivalent forms of basis pursuit denoising (BPDN) in CS problem. However, the solving speed and accuracy are adversely affected when the dimension increases greatly. In this paper, a new iterative format of proximal ADMM, which has fast solving speed and pinpoint accuracy when the dimension increases, is proposed to solve BPDN problem. Global convergence of the new type proximal ADMM is established in detail, and we exhibit a $R-$ linear convergence rate under suitable condition. Moreover, we apply this new algorithm to solve different types of BPDN problems. Compared with the state-of-the-art of algorithms in BPDN problem, the proposed algorithm is more accurate and efficient.

For sparse signal reconstruction (SSR) problem in compressive sensing (CS), by the splitting technique, we first transform it into a continuously differentiable convex optimization problem, and then a new self-adaptive gradient projection algorithm is proposed to solve the SSR problem, which has fast solving speed and pinpoint accuracy when the dimension increases. Global convergence of the proposed algorithm is established in detail. Without any assumptions, we establish global $R-$linear convergence rate of the proposed algorithm, which is a new result for constrained convex (rather than strictly convex) quadratic programming problem. Furthermore, we can also obtain an approximate optimal solution in a finite number of iterations. Some numerical experiments are made on the sparse signal recovery and image restoration to exhibit the efficiency of the proposed algorithm. Compared with the state-of-the-art algorithms in SSR problem, the proposed algorithm is more accurate and efficient.

For the positive semi-definite tensor complementarity problem (TCP), based on the natural residual function, we first established an error bound estimation for the positive semi-definite TCP without the fractional term of the residual function. Compared with the existing results, the requirements imposed on the TCP such as being an $m$-uniform $P$-function and being $m$-monotone were removed. As an application of the error bound obtained, we showed the global $R$-linear convergence rate of the proposed self-adaptive projection algorithm for solving the TCP via an equivalent transformation of this problem. Meanwhile, we also obtained an $ϵ$-optimal solution in a finite number of iterations. Finally, numerical results were reported to demonstrate the efficiency of the proposed method.