Disciplined Biconvex Programming

2025-11-03

AI Summary

Biconvex optimization problems are traditionally modeled and solved using manually designed alternating convex search (ACS) algorithms—processes that are error-prone and require substantial expertise. This paper introduces DBCP, the first structured modeling paradigm specifically designed for biconvex optimization. Extending the convex optimization programming paradigm, DBCP enables users to declaratively specify biconvex problems using a concise, intuitive syntax and automatically decomposes them into a sequence of provably convex subproblems. DBCP tightly integrates syntactic parsing, structural problem analysis, and ACS solver code generation—implemented as an open-source Python extension to CVXPY (dbcp)—to achieve end-to-end automation from model specification to solver execution. Experimental evaluation demonstrates that DBCP efficiently handles diverse biconvex problem classes, substantially lowers the barrier to entry for non-expert users, and improves modeling accuracy and development productivity.

Technology Category

• Search and Optimization: Non-convex Optimization
• Constraint Satisfaction and Optimization: Constraint Programming
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• User Modeling, Personalization and Recommendation: On-Device user modeling, personalization, and recommendation

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.

Problem

• Automating heuristic methods for biconvex optimization problems
• Reducing user effort in designing alternate convex search solvers
• Extending disciplined convex programming to biconvex problems

Innovation

• DBCP extends disciplined convex programming to biconvex problems
• Automatically splits problems into solvable convex subproblems
• Generates customized ACS solvers for biconvex optimization formulations