arXiv subjects

David H. Gutman

Publications and source records attributed to David H. Gutman.

6 recordsLinked to original sources

arXiv2026

An Adaptive Proximal ADMM for Nonconvex Linearly Constrained Composite Programs

This paper develops an adaptive proximal alternating direction method of multipliers (ADMM) for solving linearly constrained, composite optimization problems under the assumption that the smooth component of the objective is weakly convex, while the non-smooth component is convex and block-separable. The proposed method is adaptive to all problem parameters, including smoothness and weak convexity constants, and allows each of its block proximal subproblems to be inexactly solved. Each iteration of our adaptive proximal ADMM consists of two steps: the sequential solution of each block proximal subproblem; and adaptive tests to decide whether to perform a full Lagrange multiplier and/or penalty parameter update(s). Without any rank assumptions on the constraint matrices, it is shown that the adaptive proximal ADMM obtains an approximate first-order stationary point of the constrained problem in a number of iterations that matches the state-of-the-art complexity for the class of proximal ADMM's. The three proof-of-concept numerical experiments that conclude the paper suggest our adaptive proximal ADMM enjoys significant computational benefits.

math.OC↗

arXiv2021

Perturbed Fenchel duality and first-order methods

We show that the iterates generated by a generic first-order meta-algorithm satisfy a canonical perturbed Fenchel duality inequality. The latter in turn readily yields a unified derivation of the best known convergence rates for various popular first-order algorithms including the conditional gradient method as well as the main kinds of Bregman proximal methods: subgradient, gradient, fast gradient, and universal gradient methods.

math.OC↗

arXiv2020

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 number of a function bounds the linear rate of convergence of the gradient descent algorithm for unconstrained convex minimization. We propose a condition number of a differentiable convex function relative to a reference convex set and distance function pair. This relative condition number is defined as the ratio of a relative smoothness to a relative strong convexity constants. We show that the relative condition number extends the main properties of the traditional condition number both in terms of its geometric insight and in terms of its role in characterizing the linear convergence of first-order methods for constrained convex minimization. When the reference set $X$ is a convex cone or a polyhedron and the function $f$ is of the form $f = g\circ A$, we provide characterizations of and bounds on the condition number of $f$ relative to $X$ in terms of the usual condition number of $g$ and a suitable condition number of the pair $(A,X)$.

math.OC↗

arXiv2018

Enhanced Basic Procedures for the Projection and Rescaling Algorithm

Using an efficient algorithmic implementation of Caratheodory's theorem, we propose three enhanced versions of the Projection and Rescaling algorithm's basic procedures each of which improves upon the order of complexity of its analogue in [Mathematical Programming Series A, 166 (2017), pp. 87-111].

math.OC↗

arXiv2018

The condition of a function relative to a polytope

The condition number of a smooth 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 precisely the square of the diameter-to-width ratio of a canonical ellipsoid associated to the function. Furthermore, the condition number of a function bounds the linear rate of convergence of the gradient descent algorithm for unconstrained minimization. We propose a condition number of a smooth convex function relative to a reference polytope. This relative condition number is defined as the ratio of a relative smooth constant to a relative strong convexity constant of the function, where both constants are relative to the reference polytope. The relative condition number extends the main properties of the traditional condition number. In particular, we show that the condition number of a quadratic convex function relative to a polytope is precisely the square of the diameter-to-facial-distance ratio of a scaled polytope for a canonical scaling induced by the function. Furthermore, we illustrate how the relative condition number of a function bounds the linear rate of convergence of first-order methods for minimization of the function over the polytope.

math.OC↗

arXiv2018

Convergence rates of proximal gradient methods via the convex conjugate

We give a novel proof of the $O(1/k)$ and $O(1/k^2)$ convergence rates of the proximal gradient and accelerated proximal gradient methods for composite convex minimization. The crux of the new proof is an upper bound constructed via the convex conjugate of the objective function.

math.OC↗