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Disciplined Biconvex Programming

Hao Zhu, Joschka Boedecker

TL;DR

The paper introduces disciplined biconvex programming (DBCP), a modeling framework that extends disciplined convex programming to biconvex problems. By enforcing a DCP-like syntax, DBCP guarantees biconvexity and automatically splits models into convex subproblems, enabling automated ACS-based solving. The authors implement DBCP in the open-source Python package dbcp, as an extension of CVXPY, including biconvex atoms, a product rule, and support for generalized inequality constraints. They validate the approach with examples spanning nonnegative matrix factorization, bilinear logistic regression, k means clustering, dictionary learning, blind deconvolution, and IO-HMM fitting, illustrating rapid prototyping and effective solutions for practical biconvex problems.

Abstract

We introduce disciplined biconvex programming (DBCP), a modeling framework for specifying and solving biconvex optimization problems. Biconvex optimization problems arise in various applications, including machine learning, signal processing, computational science, and control. Solving a biconvex optimization problem in practice usually resolves to heuristic methods based on alternate convex search (ACS), which iteratively optimizes over one block of variables while keeping the other fixed, so that the resulting subproblems are convex and can be efficiently solved. However, designing and implementing an ACS solver for a specific biconvex optimization problem usually requires significant effort from the user, which can be tedious and error-prone. DBCP extends the principles of disciplined convex programming to biconvex problems, allowing users to specify biconvex optimization problems in a natural way based on a small number of syntax rules. The resulting problem can then be automatically split and transformed into convex subproblems, for which a customized ACS solver is then generated and applied. DBCP allows users to quickly experiment with different biconvex problem formulations, without expertise in convex optimization. We implement DBCP into the open source Python package dbcp, as an extension to the famous domain specific language CVXPY for convex optimization.

Disciplined Biconvex Programming

TL;DR

The paper introduces disciplined biconvex programming (DBCP), a modeling framework that extends disciplined convex programming to biconvex problems. By enforcing a DCP-like syntax, DBCP guarantees biconvexity and automatically splits models into convex subproblems, enabling automated ACS-based solving. The authors implement DBCP in the open-source Python package dbcp, as an extension of CVXPY, including biconvex atoms, a product rule, and support for generalized inequality constraints. They validate the approach with examples spanning nonnegative matrix factorization, bilinear logistic regression, k means clustering, dictionary learning, blind deconvolution, and IO-HMM fitting, illustrating rapid prototyping and effective solutions for practical biconvex problems.

Abstract

We introduce disciplined biconvex programming (DBCP), a modeling framework for specifying and solving biconvex optimization problems. Biconvex optimization problems arise in various applications, including machine learning, signal processing, computational science, and control. Solving a biconvex optimization problem in practice usually resolves to heuristic methods based on alternate convex search (ACS), which iteratively optimizes over one block of variables while keeping the other fixed, so that the resulting subproblems are convex and can be efficiently solved. However, designing and implementing an ACS solver for a specific biconvex optimization problem usually requires significant effort from the user, which can be tedious and error-prone. DBCP extends the principles of disciplined convex programming to biconvex problems, allowing users to specify biconvex optimization problems in a natural way based on a small number of syntax rules. The resulting problem can then be automatically split and transformed into convex subproblems, for which a customized ACS solver is then generated and applied. DBCP allows users to quickly experiment with different biconvex problem formulations, without expertise in convex optimization. We implement DBCP into the open source Python package dbcp, as an extension to the famous domain specific language CVXPY for convex optimization.

Paper Structure

This paper contains 34 sections, 40 equations, 6 figures, 3 algorithms.

Table of Contents

  1. Introduction
  2. Previous and related work
  3. Biconvex analysis.
  4. Solution methods for biconvex problems.
  5. Biconvex optimization applications.
  6. Disciplined convex programming.
  7. Outline
  8. Biconvex programming
  9. Biconvex sets
  10. Biconvex functions
  11. Biconvex optimization problems
  12. Solving biconvex problems
  13. Alternate convex search
  14. Proximal regularizations
  15. Initialization
  16. ...and 19 more sections

Figures (6)

  • Figure 1: Examples of biconvex sets.
  • Figure 2: Graph of the biconvex function $f(x, y) = x^2 y^2$.
  • Figure 3: Results of the $k$-means clustering example.
  • Figure 4: Trade off curve of the sparse dictionary learning example.
  • Figure 5: Results of the blind deconvolution example.
  • ...and 1 more figures