Affine invariant convergence rates of the conditional gradient method

We show that the conditional gradient method for the convex composite problem \[\min_x\{f(x) + \Psi(x)\}\] generates primal and dual iterates with a duality gap converging to zero provided a suitable growth property holds and the algorithm makes a judicious choice of stepsizes. The rate of convergence of the duality gap to zero ranges from sublinear … Read more

Projection and rescaling algorithm for finding most interior solutions to polyhedral conic systems

We propose a simple projection and rescaling algorithm that finds {\em most interior} solutions to the pair of feasibility problems \[ \text{find} x\in L\cap \R^n_{+} \text{ and } \text{find} \; \hat x\in L^\perp\cap\R^n_{+}, \] where $L$ is a linear subspace of $\R^n$ and $L^\perp$ is its orthogonal complement. The algorithm complements a basic procedure that … Read more

Equivalences among the chi measure, Hoffman constant, and Renegar’s distance to ill-posedness

We show the equivalence among the following three condition measures of a full column rank matrix $A$: the chi measure, the signed Hoffman constant, and the signed distance to ill-posedness. The latter two measures are constructed via suitable collections of matrices obtained by flipping the signs of some rows of $A$. Our results provide a … Read more

New characterizations of Hoffman constants for systems of linear constraints

We give a characterization of the Hoffman constant of a system of linear constraints in $\R^n$ relative to a reference polyhedron $R\subseteq\R^n$. The reference polyhedron $R$ represents constraints that are easy to satisfy such as box constraints. In the special case $R = \R^n$, we obtain a novel characterization of the classical Hoffman constant. More … Read more

Generalized conditional subgradient and generalized mirror descent: duality, convergence, and symmetry

We provide new insight into a generalized conditional subgradient algorithm and a generalized mirror descent algorithm for the convex minimization problem \[\min_x \; \{f(Ax) + h(x)\}.\] As Bach showed in [SIAM J. Optim., 25 (2015), pp. 115–129], applying either of these two algorithms to this problem is equivalent to applying the other one to its … Read more

The condition number of a function relative to a set

The condition number of a differentiable convex function, namely the ratio of its smoothness to strong convexity constants, is closely tied to fundamental properties of the function. In particular, the condition number of a quadratic convex function is the square of the aspect ratio of a canonical ellipsoid associated to the function. Furthermore, the condition … Read more

A unified framework for Bregman proximal methods: subgradient, gradient, and accelerated gradient schemes

We provide a unified framework for analyzing the convergence of Bregman proximal first-order algorithms for convex minimization. Our framework hinges on properties of the convex conjugate and gives novel proofs of the convergence rates of the Bregman proximal subgradient, Bregman proximal gradient, and a new accelerated Bregman proximal gradient algorithm under fairly general and mild … Read more

A data-independent distance to infeasibility for linear conic systems

We offer a unified treatment of distinct measures of well-posedness for homogeneous conic systems. To that end, we introduce a distance to infeasibility based entirely on geometric considerations of the elements defining the conic system. Our approach sheds new light into and connects several well-known condition measures for conic systems, including {\em Renegar’s} distance to … Read more

An algorithm to compute the Hoffman constant of a system of linear constraints

We propose a combinatorial algorithm to compute the Hoffman constant of a system of linear equations and inequalities. The algorithm is based on a characterization of the Hoffman constant as the largest of a finite canonical collection of easy-to-compute Hoffman constants. Our algorithm and characterization extend to the more general context where some of the … Read more

Computational performance of a projection and rescaling algorithm

This paper documents a computational implementation of a {\em projection and rescaling algorithm} for finding most interior solutions to the pair of feasibility problems find $x\in L\cap\mathbb{R}^n_{+} $ and find $x\in L^\perp\cap\mathbb{R}^n_{+},$ where $L$ denotes a linear subspace in $\mathbb{R}^n$ and $L^\perp$ denotes its orthogonal complement. The projection and rescaling algorithm is a recently developed … Read more