Related papers: Horoballs and the subgradient method
A parametric class of trust-region algorithms for constrained nonconvex optimization is analyzed, where the objective function is never computed. By defining appropriate first-order stationarity criteria, we are able to extend the Adagrad…
Convex optimization encompasses a wide range of optimization problems that contain many efficiently solvable subclasses. Interior point methods are currently the state-of-the-art approach for solving such problems, particularly effective…
We introduce a notion of halfspace for Hadamard manifolds that is natural in the context of convex optimization. For this notion of halfspace, we generalize a classic result of Gr\"unbaum, which itself is a corollary of Helly's theorem.…
This paper proposes novel algorithm for non-convex multimodal constrained optimisation problems. It is based on sequential solving restrictions of problem to sections of feasible set by random subspaces (in general, manifolds) of low…
The performance of optimization methods is often tied to the spectrum of the objective Hessian. Yet, conventional assumptions, such as smoothness, do often not enable us to make finely-grained convergence statements -- particularly not for…
In this work, we consider constrained stochastic optimization problems under hidden convexity, i.e., those that admit a convex reformulation via non-linear (but invertible) map $c(\cdot)$. A number of non-convex problems ranging from…
We introduce a Riemannian optimistic online learning algorithm for Hadamard manifolds based on inexact implicit updates. Unlike prior work, our method can handle in-manifold constraints, and matches the best known regret bounds in the…
In recent years, various subspace algorithms have been developed to handle large-scale optimization problems. Although existing subspace Newton methods require fewer iterations to converge in practice, the matrix operations and full…
In recent years, important progress has been made in applying methods and techniques of convex optimization to many fields of applications such as location science, engineering, computational statistics, and computer science. In this paper,…
Second-order methods, which utilize gradients as well as Hessians to optimize a given function, are of major importance in mathematical optimization. In this work, we prove tight bounds on the oracle complexity of such methods for smooth…
In this paper, we propose two algorithms for solving convex optimization problems with linear ascending constraints. When the objective function is separable, we propose a dual method which terminates in a finite number of iterations. In…
We study a general class of bilevel problems, consisting in the minimization of an upper-level objective which depends on the solution to a parametric fixed-point equation. Important instances arising in machine learning include…
We study the query complexity of geodesically convex (g-convex) optimization on a manifold. To isolate the effect of that manifold's curvature, we primarily focus on hyperbolic spaces. In a variety of settings (smooth or not; strongly…
Hadamard spaces have traditionally played important roles in geometry and geometric group theory. More recently, they have additionally turned out to be a suitable framework for convex analysis, optimization and nonlinear probability…
It is well-known that accelerated gradient first order methods possess optimal complexity estimates for the class of convex smooth minimization problems. In many practical situations, it makes sense to work with inexact gradients. However,…
We introduce and study various algorithms for solving nonconvex minimization with inequality constraints, based on the construction of convex surrogate envelopes that majorize the objective and the constraints. In the case where the…
We propose new sequential simulation-optimization algorithms for general convex optimization via simulation problems with high-dimensional discrete decision space. The performance of each choice of discrete decision variables is evaluated…
This paper explores variants of the subspace iteration algorithm for computing approximate invariant subspaces. The standard subspace iteration approach is revisited and new variants that exploit gradient-type techniques combined with a…
Minimizing a convex function of a measure with a sparsity-inducing penalty is a typical problem arising, e.g., in sparse spikes deconvolution or two-layer neural networks training. We show that this problem can be solved by discretizing the…
Consider convex optimization problems subject to a large number of constraints. We focus on stochastic problems in which the objective takes the form of expected values and the feasible set is the intersection of a large number of convex…