报告题目：An Inner-outer Iteration Methods for Solving Convex Optimization Problems Involving the Sum of Three Convex Functions
主 讲 人：唐 玉 超
腾 讯 ID：329 234 645
In this paper, we consider solving a class of convex optimization problem which minimizes the sum of three convex functions $f(x)+g(x)+h(Bx)$, where $f(x)$ is differentiable with a Lipschitz continuous gradient, $g(x)$ and $h(x)$ have a closed-form expression of their proximity operators and $B$ is a bounded linear operator. This type of optimization problem has wide application in signal recovery and image processing. To make full use of the differentiability function in the optimization problem, we take advantage of two operator splitting methods: the forward-backward splitting method and the three operator splitting method. In the iteration scheme derived from the two operator splitting methods, we need to compute the proximity operator of $g+h \circ B$ and $h \circ B$, respectively. Although these proximity operators do not have a closed-form solution in general, they can be solved very efficiently. We mainly employ two different approaches to solve these proximity operators: one is dual and the other is primal-dual.
Following this way, we fortunately find that three existing iterative algorithms including Condat and Vu algorithm, primal-dual fixed point (PDFP) algorithm and primal-dual three operator (PD3O) algorithm are a special case of our proposed iterative algorithms. Moreover, we discover a new kind of iterative algorithm to solve the considered optimization problem, which is not covered by the existing ones. Under mild conditions, we prove the convergence of the proposed iterative algorithms. Numerical experiments applied on fused Lasso problem, constrained total variation regularization in computed tomography (CT) image reconstruction and low-rank total variation image super-resolution problem demonstrate the effectiveness and efficiency of the proposed iterative algorithms.
唐玉超，南昌大学理学院数学系。2013年西安交通大学金沙赌城网站博士毕业。主要研究方向图像反问题中的优化问题。在研国家自然科学基金地区项目一项，主持完成国家自然科学基金青年项目、江西省自然科学基金青年项目和江西省教育厅青年科学基金项目各一项。已在《中国科学-数学》，《Journal of Computational Mathematics》、《Applied Mathematics Letters》、《Mathematical and Computer Modelling》、《Nonlinear Analysis: Theory, Methods & Applications》和《Numerical Algorithms》等国内外期刊发表论文30余篇。