報告題目：Globally solving Tikhonov regularized total least squares problem
報 告 人：夏勇 教授 北京航空航天大學
報告時間：2020年12月7日 上午 9:50-10:30
報告地點：騰訊會議 ID：191 170 890
會議密碼：9999
校内聯系人：李欣欣 xinxinli@jlu.edu.cn
報告摘要：The well-known total least squares problem with the general Tikhonov regularization can be reformulated as a one-dimensional parametric minimization problem (PM), where each parameterized function evaluation corresponds to solving an n-dimensional trust region subproblem. Under a mild assumption, the parametric function is differentiable and then an efficient bisection method has been proposed for solving (PM) in literature. In the first part, we show that the bisection algorithm can be greatly improved by reducing the initially estimated interval covering the optimal parameter. It is observed that the bisection method cannot guarantee to find the globally optimal solution since the nonconvex (PM) could have a local non-global minimizer. The main contribution of this talk is to propose an efficient branch-and-bound algorithm for globally solving (PM), based on a new underestimation of the parametric function over any given interval using only the information of the parametric function evaluations at the two endpoints. We can show that the new algorithm (BTD Algorithm) returns a global \epsilon-approximation solution in a computational effort of at most O (n^3/\sqrt{\epsilon}) under the same assumption as in the bisection method. The numerical results demonstrate that our new global optimization algorithm performs even much faster than the improved version of the bisection heuristic algorithm. For a special case, the Tikhonov identical regularized total least squares, we propose a more efficient algorithm based on the hidden convexity.
報告人簡介：夏勇，北京航空航天大學數學科學學院教授、副院長。2002年本科畢業于北京大學數學科學學院，2007年博士畢業于中國科學院數學與系統科學研究院，同年入職北航。研究方向為非凸優化，在MP、SIOPT、MOR等期刊發表SCI論文54篇，2018獲批國家自然科學基金優秀青年科學基金項目。代表性工作：與合作者建立了完整的等式型S-引理；與導師袁業湘院士合作在經典二次指派問題上提出的模型被國内外同行稱為Xia-Yuan線性化。
