学术报告：Applications of the projection technique for two open problems
发布时间：2021-04-24 浏览次数：185
报告人：袁功林（广西大学）
报告时间：2021年4月24日下午2：00-3:00
报告地点：育才数学楼2楼会议室
报告题目：Applications of the projection technique for two open problems
报告人简介：袁功林，教授，博士生导师，现为广西大学数学与信息科学公司副院长，中国数学会理事、中国数学规划分会理事、广西数学会常务理事、广西运筹学会副理事长、广西高等教育学会数学专业委员会副理事长。主要从事优化理论与方法等方面的研究。主持国家自然科学基金项目2项，主持广西杰出青年自然科学基金项目1项、广西自然科学基金重点项目1项、中央主导地方科技发展基金项目1项、广西面上项目1项。广西十百千第二层次人选，广西特聘青年专家，广西高校卓越学者计划人选，广西高校优秀人才计划人选。以第一或通讯作者发表SCI收录论文50篇，如COAP、JOTA、JCAM等优化期刊发表多篇论文、Top 0.1%“热点论文”1篇、Top 1%“高被引论文”3篇、出版学术专著一部；获得广西科学技术二等奖1项、广西自然科学二等奖1项、广西教学成果二等奖1项等。
报告摘要：There are two open problems for nonconvex functions under the weak Wolfe-Powell (WWP) line search technique in unconstrained optimization problems. The first one is the global convergence of the Polak-Ribi\`{e}re-Polyak (PRP) conjugate gradient algorithm and the second one is the global convergence of the BFGS (Broyden, Fletcher, Goldfarb, and Shanno) quasi-Newton method. Many scholars have proven that these two problems do not converge, even if an exact line search is used. Two circle counterexamples were proposed to generate the nonconvergence of the PRP algorithm for the nonconvex functions under the exact line search (see Powell, Lecture Notes in Math. 1066(1984) and Dai, SIAM J. Optim. 13(2003) in detail), which inspired us to define a new technique to jump out of the circle point and obtain the global convergence. Thus, a new PRP conjugate gradient algorithm is designed by the following steps. (i) The current point $x_k$ is defined, and a parabolic surface $P_k$ is designed; (ii) an assistant point $\kappa_k$ is defined by the PRP formula based on $x_k$; (iii) $\kappa_k$ is projected onto the parabolic surface $P_k$ to generate the next point $x_{k+1}$; (iv) the presented PRP conjugate gradient algorithm has the global convergence for nonconvex functions with the WWP line search; (v) a similar technique is used for the BFGS quasi-Newton method to get a new BFGS algorithm and establish its global convergence; and (vi) The numerical results show that the given algorithms are more competitive than those of other similar algorithms. And the well-known hydrologic engineering application problem called parameter estimation problem of nonlinear Muskingum model is also done by the proposed algorithms.