Matrix-Free Convex Optimization Modeling

@article{Diamond2015MatrixFreeCO,
  title={Matrix-Free Convex Optimization Modeling},
  author={Steven Diamond and Stephen P. Boyd},
  journal={arXiv: Optimization and Control},
  year={2015},
  pages={221-264}
}
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… 
View PDF on arXiv
28 Citations
Background Citations
11
Methods Citations
12

28 Citations

Differentiating through a cone program

We consider the problem of efficiently computing the derivative of the solution map of a convex cone program, when it exists. We do this by implicitly differentiating the residual map for its…

Differentiating Through a Conic Program

We consider the problem of efficiently computing the derivative of the solution map of a convex cone program, when it exists. We do this by implicitly differentiating the residual map for its…

O C ] 5 J un 2 01 9 Differentiating Through a Cone Program

We consider the problem of efficiently computing the derivative of the solution map of a convex cone program, when it exists. We do this by implicitly differentiating the residual map for its…

The Symbolic Interior Point Method

This work develops an engine that can fully leverage the structure of symbolic representations to solve convex linear and quadratic optimization problems and demonstrates the flexibility of the resulting symbolic-numeric optimizer on decision making and compressed sensing tasks with millions of non-zero entries.

O C ] 2 7 M ay 2 01 9 Differentiating Through a Cone Program

An open-source Python implementation of the method that solves a cone program and returns the derivative and its adjoint as abstract linear maps is presented, which can be easily integrated into software systems for automatic differentiation.

A Rewriting System for Convex Optimization Problems

A modular rewriting system for translating optimization problems written in a domain-specific language to forms compatible with low-level solver interfaces, which makes it easy to match problems to solvers well-suited for them and to support solvers with a wide variety of standard forms.

A First-Order Numerical Algorithm without Matrix Operations

An easy-to-compute decomposition method to solve sparse linear systems that arise in conic optimization problems and is compared with the state-of-the-art first-order solvers.

Explorer The Symbolic Interior Point Method

This work develops an engine that can fully leverage the structure of symbolic representations to solve convex linear and quadratic optimization problems and demonstrates the flexibility of the resulting symbolic-numeric optimizer on decision making and compressed sensing tasks with millions of non-zero entries.

Stochastic Matrix-Free Equilibration

We present a novel method for approximately equilibrating a matrix using only multiplication by the matrix and its transpose. Our method is based on convex optimization and projected stochastic…

A New Architecture for Optimization Modeling Frameworks

We propose a new architecture for optimization modeling frameworks in which solvers are expressed as computation graphs in a framework like TensorFlow rather than as standalone programs built on a…

References

SHOWING 1-10 OF 172 REFERENCES

Interior-point methods for large-scale cone programming

In this chapter, an overview of ad hoc techniques that can be used to exploit non-sparse structure in specific classes of applications are given.

Convex Optimization in Julia

Convex translates problems from a user-friendly functional language into an abstract syntax tree describing the problem, which allows Convex to infer whether the problem complies with the rules of disciplined convex programming (DCP), and to pass the problem to a suitable solver.

Conic formulation of a convex programming problem and duality

We study a special universal “conic” formulation of a convex program where the problem is in minimizing a linear objective over the intersection of a convex cone and an affine plane. We focus on the…

Conic Optimization via Operator Splitting and Homogeneous Self-Dual Embedding

We introduce a first-order method for solving very large convex cone programs. The method uses an operator splitting method, the alternating directions method of multipliers, to solve the homogeneous…

Interior-point polynomial algorithms in convex programming

This book describes the first unified theory of polynomial-time interior-point methods, and describes several of the new algorithms described, e.g., the projective method, which have been implemented, tested on "real world" problems, and found to be extremely efficient in practice.

Parameter Selection and Preconditioning for a Graph Form Solver

This chapter addresses the critical practical issues of how to select the proximal parameter in each iteration, and how to scale the original problem variables, so as to achieve reliable practical performance.

Convergence Analysis of an Inexact Feasible Interior Point Method for Convex Quadratic Programming

  • J. Gondzio
  • Mathematics, Computer Science
    SIAM J. Optim.
  • 2013
This paper discusses two variants of an inexact feasible interior point algorithm for convex quadratic programming and provides conditions for the level of error acceptable in the Newton equation and establishes the worst-case complexity results.

A Primal-Dual Operator Splitting Method for Conic Optimization

A simple operator splitting method for solving a primal conic optimization problem is developed and it is shown that the iterates also solve the dual problem.

Graph Implementations for Nonsmooth Convex Programs

We describe graph implementations, a generic method for representing a convex function via its epigraph, described in a disciplined convex programming framework. This simple and natural idea allows a…

Solving semidefinite-quadratic-linear programs using SDPT3

Computational experiments with linear optimization problems involving semidefinite, quadratic, and linear cone constraints (SQLPs) are discussed and computational results on problems from the SDPLIB and DIMACS Challenge collections are reported.
...