Random search methods are widely used for global optimization due to their theoretical generality and implementation simplicity. This paper proposes a depth-first directional search (DFDS) algorithm for globally solving nonconvex optimization problems. Motivated by the penetrating beam of a searchlight, DFDS performs a complete stepping line search along each sampled direction before proceeding to the next, contrasting with existing directional search methods that prioritize broad exploratory coverage. We establish the convergence and computational complexity of DFDS through a novel geometric framework that models the success probability of finding a global optimizer as the surface area of a spherical cap. Numerical experiments on benchmark problems demonstrate that DFDS achieves significantly higher accuracy in locating the global optimum compared to other random search methods under the same function evaluation budget.
Nonconvex weak sharp minima on Riemannian manifolds
我们要建立黎曼流形上非凸最优化问题的弱锐极小集的（原始和对偶类型的）必要条件。在这里，我们将提供由Li等人（SIAM J. Optim.，21（2011），pp. 1523- 1560）的非凸问题。利用Fréchet理论和黎曼流形上的极限次微分给出了对偶型的必要条件。我们还考虑了黎曼流形上的相依方向导数理论和相依锥的概念，给出了原型的必要条件。关于黎曼流形上的相依锥，已有几种定义。我们表明，这些定义，与一些修改，是等价的。我们建立了一个关于距离函数局部性态的引理。利用这个引理，我们将黎曼流形上的距离函数的Fréchet次微分（即偶然的方向导数）表示为正规锥（即偶然锥），从而建立了它的必要条件.作为一个应用，我们展示了如何可以使用弱尖锐极小属性模型的一个图的Cheeger型常数作为一个最优化问题的Stiefel流形。
On the Regularization of Primal-Dual Dynamics in Time-Dependent Nonconvex Optimization
School of Science and Technology, IE University（IE大学科学与技术学院）；Department of Mechanical and Aerospace Engineering, University of California San Diego（加州大学圣地亚哥分校机械与航空航天工程系）
Directional first order approach for a class of bilevel programs
School of Mathematics, Hunan University（湖南大学数学学院）；School of Artificial Intelligence, Southern University of Science and Technology（南方科技大学人工智能学院）；Department of Mathematics and Statistics, University of Victoria（维多利亚大学数学与统计系）；Department of Mathematics, Southern University of Science and Technology（南方科技大学数学系）
Radial Epiderivative Based Fritz John and KKT Conditions in Nonsmooth Nonconvex Optimization
Chongqing Normal University（重庆师范大学）；Eskisehir Technical University（埃斯基谢希尔理工大学）；China Medical University（中国医药大学）；Academy of Romanian Scientists（罗马尼亚科学家学院）
A stochastic smoothing framework for nonconvex-nonconcave minEmax problems with applications to Wasserstein distributionally robust optimization
Department of Applied Mathematics, Hong Kong Polytechnic University, Hong Kong, China（应用数学系，香港理工大学）；Institute for Math & AI, Wuhan (IMAI), Wuhan university, China（数学与人工智能研究院，武汉大学）；Department of Mathematical Sciences Rensselaer Polytechnic Institute, Troy, NY, USA（数学科学系，伦斯勒理工学院）
Adversary-Robust Learning from Fully Asynchronous Directional Derivative Estimates
Computer Science and Automation, Indian Institute of Science（印度科学研究院计算机科学与自动化系）；Edwardson School of Industrial Engineering, Purdue University（普渡大学埃德沃兹工业工程学院）；Department of Computer Science, IMT Atlantique（IMT阿登提大学计算机科学系）
