Characterizations, Dynamical Systems and Gradient Methods for Strongly Quasiconvex Functions
研究了可微强拟凸函数的新性质，证明了一阶和二阶梯度系统在无需Lipschitz连续假设下的指数收敛，并分析了梯度下降和重球法的线性收敛性。
Abstract We study differentiable strongly quasiconvex functions for providing new properties for algorithmic and monotonicity purposes. Furthermore, we provide insights into the decreasing behaviour of strongly quasiconvex functions, applying this for establishing exponential convergence for first- and second-order gradient systems without relying on the usual Lipschitz continuity assumption on the gradient of the function. The explicit discretization of the first-order dynamical system leads to the gradient descent method while discretization of the second-order dynamical system with viscous damping recovers the heavy ball method. We establish the linear convergence of both methods under suitable conditions on the parameters as well as numerical experiments for supporting our theoretical findings.