We establish polynomial complexity bounds of the Mehrotra-type predictorcorrector algorithms for linear programming over symmetric cones. We first slightly modify the maximum step size in the predictor step of the safeguard based Mehrotra-type algorithm for linear programming, that was proposed by Salahi et al[18]. Then, using the machinery of Euclidean Jordan algebras, we extend the modified algorithm to symmetric cones. Based on the Nesterov-Todd direction, we obtain O(r log ε-1) iteration complexity bound of this algorithm, where r is the rank of the Jordan algebras and ε is the required precision. We also present a new variant of Mehrotra-type algorithm using a new adaptive updating scheme of centering parameter and show that this algorithm enjoys the same order of complexity bound as the safeguard algorithm. We illustrate the numerical behaviour of the methods on some small examples.
The nonlocal means( NLM) has been widely used in image processing. In this paper,we introduce a modified weight function for NLM denoising, which will compute the nonlocal similarities among the pre-processing pixel patches instead of the commonly used similarity measure based on noisy observations. By the law of large number,the norm for the pre-processing pixel patches is closer to the norm of the original clean pixel patches,so the proposed weight functions are more optimized and the selected similar patches are more accurate. Experimental results indicate the proposed algorithm achieves better restored results compared to the classical NLM's method.
