## An Improved Unconstrained Approach for Bilevel Optimization

In this paper, we focus on the nonconvex-strongly-convex bilevel optimization problem (BLO). In this BLO, the objective function of the upper-level problem is nonconvex and possibly nonsmooth, and the lower-level problem is smooth and strongly convex with respect to the underlying variable $y$. We show that the feasible region of BLO is a Riemannian manifold. … Read more

## Retraction based Direct Search Methods for Derivative Free Riemannian Optimization

Direct search methods represent a robust and reliable class of algorithms for solving black-box optimization problems. In this paper, we explore the application of those strategies to Riemannian optimization, wherein minimization is to be performed with respect to variables restricted to lie on a manifold. More specifically, we consider classic and line search extrapolated variants … Read more

## Spectral Residual Method for Nonlinear Equations on Riemannian Manifolds

In this paper, the spectral algorithm for nonlinear equations (SANE) is adapted to the problem of finding a zero of a given tangent vector field on a Riemannian manifold. The generalized version of SANE uses, in a systematic way, the tangent vector field as a search direction and a continuous real–valued function that adapts this … Read more

## Primal-Dual Optimization Algorithms over Riemannian Manifolds: an Iteration Complexity Analysis

In this paper we study nonconvex and nonsmooth multi-block optimization over Riemannian manifolds with coupled linear constraints. Such optimization problems naturally arise from machine learning, statistical learning, compressive sensing, image processing, and tensor PCA, among others. We develop an ADMM-like primal-dual approach based on decoupled solvable subroutines such as linearized proximal mappings. First, we introduce … Read more

## Vector Transport-Free SVRG with General Retraction for Riemannian Optimization: Complexity Analysis and Practical Implementation

In this paper, we propose a vector transport-free stochastic variance reduced gradient (SVRG) method with general retraction for empirical risk minimization over Riemannian manifold. Existing SVRG methods on manifold usually consider a specific retraction operation, and involve additional computational costs such as parallel transport or vector transport. The vector transport-free SVRG with general retraction we … Read more

## Optimality conditions for the nonlinear programming problems on Riemannian manifolds

In recent years, many traditional optimization methods have been successfully generalized to minimize objective functions on manifolds. In this paper, we first extend the general traditional constrained optimization problem to a nonlinear programming problem built upon a general Riemannian manifold $\mathcal{M}$, and discuss the first-order and the second-order optimality conditions. By exploiting the differential geometry … Read more

## Central Paths in Semidefinite Programming, Generalized Proximal Point Method and Cauchy Trajectories in Riemannian Manifolds

The relationships among central path in the context of semidefinite programming, generalized proximal point method and Cauchy trajectory in Riemannian manifolds is studied in this paper. First it is proved that the central path associated to the general function is well defined. The convergence and characterization of its limit point is established for functions satisfying … Read more

## Perturbations and metric regularity

A point x is an approximate solution of a generalized equation [b lies in F(x)] if the distance from the point b to the set F(x) is small. Metric regularity of the set-valued mapping F means that, locally, a constant multiple of this distance bounds the distance from x to an exact solution. The smallest … Read more