A Rewriting System for Convex Optimization Problems

Abstract: We describe a modular rewriting system for translating optimization problems written in a domain-specific language to forms compatible with low-level solver interfaces. Translation is facilitated by reductions, which accept a category of problems and transform instances of that category to equivalent instances of another category. Our system proceeds in two key phases: analysis, in which we attempt to find a suitable solver for a supplied problem, and canonicalization, in which we rewrite the problem in the selected solver's standard form. We implement the described system in version 1.0 of CVXPY, a domain-specific language for mathematical and especially convex optimization. By treating reductions as first-class objects, our method makes it easy to match problems to solvers well-suited for them and to support solvers with a wide variety of standard forms.

• The paper presents a modular two-phase rewriting system that automates transforming convex optimization problems from DSLs to standardized solver forms.
• It employs an analysis phase to identify suitable solvers and a canonicalization phase to convert problems, ensuring mathematical accuracy and efficiency.
• Implemented in CVXPY, the system treats reductions as first-class objects, enhancing extensibility and paving the way for broader solver integration.

A Rewriting System for Convex Optimization Problems

The paper "A Rewriting System for Convex Optimization Problems" presents a modular approach for transforming convex optimization problems expressed in domain-specific languages (DSLs) into formats compatible with low-level solver interfaces. This method leverages a structured system of reductions, which simplifies the task of matching problems with suitable solvers and support various solver standard forms.

Overview and Key Contributions

The central contribution of this work is a novel two-phase modular rewriting system that systematically translates optimization problems. This system is articulated around two key phases:

1. Analysis Phase: The system identifies a suitable solver for a given problem by analyzing its structure.
2. Canonicalization Phase: The problem is rewritten into the standard form required by the chosen solver.

This framework is implemented within CVXPY version 1.0, a Python-embedded DSL for convex optimization. By treating reductions as first-class objects, this system enhances extensibility, allowing for easy integration with various solvers and the ability to handle a wide range of problem transformations.

Detailed Examination

The rewriting system leverages the concept of reductions, where an optimization problem is transformed into an equivalent one. The reductions allow problems to be methodically rewritten, supporting modular transformations that are transparent and reversible. This approach ensures that problems can be accurately matched to solvers based on their characteristics, enhancing performance and solver compatibility.

Domain-Specific Languages (DSLs): DSLs streamline problem specification by providing a user-friendly syntax for expressing optimization problems. However, translating these user-friendly forms into solver-compatible forms typically involves intricate canonicalization processes. The paper describes CVXPY's embedding as a DSL within Python, demonstrating how convex problems can be specified and automatically transformed.

Numerical Solvers: Solvers require problems to be in specific mathematical forms. The paper delineates a hierarchy of convex problems, illustrating how various problems are interrelated and can be reduced to simpler forms, such as linear, quadratic, and cone programs.

Implications and Future Directions

The proposed system addresses several needs in the optimization and AI research communities by automating problem transformations and expanding the applicability of existing solvers. The implications are both theoretical and practical:

• Theoretical: By aligning with established principles of convex optimization and reductions, the methodology ensures correctness and facilitates equivalence in problem transformations.
• Practical: The system reduces the overhead for users who manually transform problems, paving the way for broader adoption of optimization techniques across disciplines.

The modularity of the system suggests potential for future enhancements, such as incorporating more complex reduction techniques and increasing the breadth of supported solvers. As AI continues to evolve, the ability to efficiently translate and solve diverse optimization problems will be critical.

In conclusion, this paper systematically advances the field of convex optimization by introducing a robust framework for problem rewriting. By automating the transformation process and ensuring compatibility with a wide range of solvers, this system is set to make a significant impact on the efficiency and accessibility of optimization technologies.