Nonconvex Function

Nonconvex functions, characterized by multiple local minima and saddle points, pose significant challenges in optimization, hindering the efficient discovery of global optima. Current research focuses on developing and analyzing algorithms, such as stochastic gradient descent (SGD) and its variants, augmented Lagrangian methods, and accelerated gradient methods, to navigate these complex landscapes, often incorporating techniques like smoothing and variance reduction. These efforts are crucial for advancing machine learning, particularly deep learning, where nonconvex optimization is ubiquitous, and for improving the efficiency and theoretical understanding of various optimization problems across diverse scientific domains. The development of robust and efficient methods for handling nonconvexity is essential for progress in many fields.