## A Newton-CG based barrier-augmented Lagrangian method for general nonconvex conic optimization

 In this paper we consider finding an approximate second-order stationary point (SOSP) of general nonconvex conic optimization that minimizes a twice differentiable function subject to nonlinear equality constraints and also a convex conic constraint. In particular, we propose a Newton-conjugate gradient (Newton-CG) based barrier-augmented Lagrangian method for finding an approximate SOSP of this problem. … Read more

## A Newton-CG based augmented Lagrangian method for finding a second-order stationary point of nonconvex equality constrained optimization with complexity guarantees

 In this paper we consider finding a second-order stationary point (SOSP) of nonconvex equality constrained optimization when a nearly feasible point is known. In particular, we first propose a new Newton-CG method for finding an approximate SOSP of unconstrained optimization and show that it enjoys a substantially better complexity than the Newton-CG method [56]. … Read more

## A Newton-CG based barrier method for finding a second-order stationary point of nonconvex conic optimization with complexity guarantees

In this paper we consider finding an approximate second-order stationary point (SOSP) of nonconvex conic optimization that minimizes a twice differentiable function over the intersection of an affine subspace and a convex cone. In particular, we propose a Newton-conjugate gradient (Newton-CG) based barrier method for finding an $(\epsilon,\sqrt{\epsilon})$-SOSP of this problem. Our method is not … Read more

## SPIDER: Near-Optimal Non-Convex Optimization via Stochastic Path Integrated Differential Estimator

In this paper, we propose a new technique named \textit{Stochastic Path-Integrated Differential EstimatoR} (SPIDER), which can be used to track many deterministic quantities of interest with significantly reduced computational cost. We apply SPIDER to two tasks, namely the stochastic first-order and zeroth-order methods. For stochastic first-order method, combining SPIDER with normalized gradient descent, we propose … Read more