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This paper aims at solving the sparse reconstruction (SR) problem via a multiobjective evolutionary algorithm. Existing multiobjective evolutionary algorithms for the SR problem have high computational complexity, especially in scenarios of high-dimensional reconstruction. Furthermore, these algorithms focus on estimating the whole Pareto front rather than the knee region, thus leading to solutions with limited diversity in the knee region and causing a waste of computational effort. To tackle these issues, this paper proposes an adaptive decomposition-based evolutionary approach (ADEA) for the SR problem. Firstly, we employ the decomposition-based evolutionary paradigm to guarantee a high computational efficiency and the diversity of solutions in the whole objective space. Then, we propose a two-stage iterative soft-thresholding (IST)-based local search operator for improving the convergence. Finally, we develop an adaptive decomposition-based environmental selection strategy, by which the decomposition in the knee region can be adjusted dynamically. This strategy makes it possible to focus the selection effort on the knee region, hence involving low computational complexity. Experimental results on images, and simulated and benchmark signals demonstrate the superiority of ADEA in terms of reconstruction accuracy and computational efficiency, compared to five stateof-the-art algorithms.

This paper is concerned with a sequentially semidefinite programming (SSDP) algorithm for solving nonlinear semidefinite programming problems (NLSDP), which does not use a penalty function or a filter. This method, inspired by the classic SQP method, calculates a trial step by a quadratic semidefinite programming subproblem at each iteration. The trial step is determined such that either the value of the objective function or the measure of constraint violation is sufficiently reduced. In order to guarantee global convergence, the measure of constraint violation in each iteration is required not to exceed a progressively decreasing limit. We prove the global convergence properties of the algorithm under mild assumptions. We also analyze the local behaviour of the proposed method while using a second order correction strategy to avoid Maratos effect. We prove that, under the strict complementarity and the strong second order sufficient conditions with the sigma term, the rate of local convergence is superlinear. Finally, some numerical results with nonlinear semidefinite programming formulation of control design problem with the data contained in COM P l e ib are given.

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