Matrix-Free Convex Optimization Modeling

TL;DR

This paper presents a novel matrix-free convex optimization modeling framework that transforms problems into cone programs using graph-encoded linear functions, enabling efficient solutions with matrix-free cone solvers.

Contribution

It introduces a new modeling approach that preserves fast linear transforms and represents linear functions as graphs, facilitating efficient convex optimization solving.

Findings

Enables solving convex problems with fast linear transforms efficiently.

Represents linear functions as graphs instead of matrices.

Compatible with matrix-free cone solvers for improved performance.

Abstract

We introduce a convex optimization modeling framework that transforms a convex optimization problem expressed in a form natural and convenient for the user into an equivalent cone program in a way that preserves fast linear transforms in the original problem. By representing linear functions in the transformation process not as matrices, but as graphs that encode composition of linear operators, we arrive at a matrix-free cone program, i.e., one whose data matrix is represented by a linear operator and its adjoint. This cone program can then be solved by a matrix-free cone solver. By combining the matrix-free modeling framework and cone solver, we obtain a general method for efficiently solving convex optimization problems involving fast linear transforms.