| 21century is an age of knowledge and information, of which digital signal communication and processingbecome important research subjects. Especially compressed sensing and image restoration are key technologiesin the field of information processing, and have wide application in many fields, bringing much economic effect.Some fast optimization algorithms are studied in this thesis, which includes:(1)Compressed sensing is an important communication mean, to which reconstructing signals is a key technique.A primal-dual proximal point algorithm with adaptive stepsize is proposed for the problem of reconstruction ofsparse signals. The proposed algorithm shows better numerical performance compared with proximal pointalgorithm with constant stepsize, and the convergence of the proposed algorithm is justified in this work.(2)Digital images or any digital signals are unavoidably subjected to degradation (including blurring and beingnoised) during the processed of sensing, restore, reconstruction etc., thus image restoration becomes an essentialprocedure for image analysis. Image restoration operates in both space domain and wavelet domain, in bothcases we proposed fixed and adaptive primal-dual algorithm for image restoration problems. Numerical resultshows that our algorithm is competitive with some popular algorithms, and even outperforms them in someaspects. The convergence proof of the proposed algorithms are included in this thesis as well.(3)The proposing of the algorithms aforementioned is impossible without powerful mathematical tools, in whichtwo extremely important notions are monotonic variational inequality and proximal point algorithm. This thesisdiscussed linear proximal point algorithms for monotonic variational inequalities, and proposed a nonlinearproximal point algorithm which can include the linear algorithm as one of its special cases. The convergenceand computation complexity of the proposed nonlinear algorithm were analyzed. |
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