报告题目：On solution uniqueness and robust recovery for sparse regularization with a gauge: From dual point of view

报告人：Wen Song

单位：School of Mathematical and sciences, Harbin Normal University

时间：2023年3月9日下午4：00-5：00

地点：数学与统计学院格物楼3103

In this talk, we present a unifying approach for exploring solution uniqueness and robust recovery for sparse regularization problem with a closed gauge . We formulate the Lagrange dual problem (which is a convex optimization problem with an inequality constraint) for this problem, but the strong duality could not hold since this problem is degenerate in general. Fortunately, we find that the Lagrange dual problem for this dual problem is the original problem and Slater’s condition hold and hence the strong duality hold. Therefore, the solution set to the original problem is the Lagrange multiplier set for its dual problem and is same for any solution to this dual problem. Based on criteria for uniqueness of the Lagrange multipliers for its dual problem, we obtain several equivalent sufficient conditions for solution uniqueness of the original problem , which are also necessary conditions under some closedness condition. We also prove that these sufficient conditions are sufficient and necessary conditions for sharpness solution to the original problem, which ensures the robust recovery. Some applications to slope sparse regularization and sparse analysis regularization are given.

宋文， 波兰科学院数学研究所博士，龙江学者，哈尔滨师范大学特聘教授，全国五一劳动奖章获得者， 现为中国运筹学会数学规划分会常务理事，黑龙江省数学会副理事长。主要研究方向是变分分析与最优化。在Siam J. Optim, Math. Program., J. Math. Anal Appl., J. Optim.Theory Appl.,等国际重要学术刊物上发表论文50余篇。主持完成国家自然科学基金项目4项,黑龙江省杰出青年基金，教育部优秀青年教师资助计划等项目。