Randomized Quaternion Matrix UTV Decomposition and its Applications in Quaternion Matrix Optimization
Xu Renjie, Wei Yimin · Pacific Journal of Optimization · 2023
(Communicated by Liqun Qi) Abstract: We study four interior continuous trajectories defined by matrix differential equations, aiming at developing new interior point methods for convex semidefinite programming (SDP). By assuming the boundedness of the level set, we establish the optimality and convergence of the first and the second tra- jectories for linear SDP. For the convex case, we show that, starting from any interior feasible point, the third trajectory converges to an optimal solution that has the maximal rank among all optimal solutions, under the assumptions that an optimal solution exists and the maximal rank of optimal solutions is one. Finally, we obtain the strongest result under the weakest assumption for the fourth trajectory, namely, by only assuming the existence of an optimal solution, we show that the trajectory converges to an optimal solution that has the maximal rank among all optimal solutions. This paper is dedicated to Professor Masao Fukushima in celebration of his seventy-fifth birthday. Masao is a funding co-editor of Pacific Journal of Optimization. Under his leadership, the journal has become a flagship journal of the Pacific Optimization Research Activity Group and an important publication platform for optimization researchers in the world. We have known Masao since 1990s and he has been a long-term colleague and close friend of us. We wish him the best at this glorious moment of his life.