This paper introduces a two-stage model for multi-channel image segmentation, which is motivated by minimal surface theory. Indeed, in the first stage, we acquire a smooth solution u from a convex variational model related to minimal surface property and different data fidelity terms are considered. This minimization problem is solved efficiently by the classical primal-dual approach. In the second stage, we adopt thresholding to segment the smoothed image u. Here, instead of using K-means to determine the thresholds, we propose a more stable hill-climbing procedure to locate the peaks on the 3D histogram of u as thresholds, in the meantime, this algorithm can also detect the number of segments. Finally, numerical results demonstrate that the proposed method is very robust against noise and superior to other image segmentation approaches.

Recently, [Y.Q. Bai, M. El Ghami and C. Roos,SIAM J. Opt. 15 (2004) 101–128]investigated a new class of kernel functions which differs from theclass of self-regular kernel functions. The class is defined by somesimple conditions on the growth and the barrier behavior of thekernel function. In this paper we generalize theanalysis presented in the above paper for P * (κ) LinearComplementarity Problems (LCPs). The analysis for LCPs deviates significantly from the analysisfor linear optimization. Several new tools and techniques are derived in this paper.

Recently, Y.Q. Bai, M. El Ghami and C. Roos [3] introduced a new class ofso-called eligible kernel functions which are defined by somesimple conditions.The authors designed primal-dual interior-point methods for linear optimization (LO)based on eligible kernel functionsand simplified the analysis of these methods considerably.In this paper we consider the semidefinite optimization (SDO) problemand we generalize the aforementioned results for LO to SDO.The iteration bounds obtained are analogous to the results in [3] for LO.

Interior-Point Methods (IPMs) are not only very effective in practice for solving linear optimization problems but also have polynomial-time complexity. Despite the practical efficiency of large-update algorithms, from a theoretical point of view, these algorithms have a weaker iteration bound with respect to small-update algorithms. In fact, there is a significant gap between theory and practice for large-update algorithms. By introducing self-regular barrier functions, Peng, Roos and Terlaky improved this gap up to a factor of log n. However, checking these self-regular functions is not simple and proofs of theorems involving these functions are very complicated. Roos el al. by presenting a new class of barrier functions which are not necessarily self-regular, achieved very good results through some much simpler theorems. In this paper we introduce a new kernel function in this class which yields the best known complexity bound, both for large-update and small-update methods.