Source-linked AI summary
A Rewriting System for Convex Optimization Problems
Akshay Agrawal, Robin Verschueren, Steven Diamond, Stephen Boyd
TL;DR
Existing optimization DSLs automate some canonicalization, but their solver-selection and canonicalization procedures are ad hoc and difficult to extend. The paper proposes a modular rewriting system with analysis and canonicalization phases, implemented in CVXPY 1.0, and uses reductions as first-class objects to connect problems with suitable solvers.
Problem
Existing solver interfaces make translation and solver choice onerous, while DSL procedures for solver selection and canonicalization are difficult to modify or extend.
Method
The paper introduces a rewriting system between optimization DSLs and numerical solvers, using modular reductions plus analysis and canonicalization phases.
Results
The system is implemented in CVXPY 1.0, where an analyzer selects a suitable back end and reductions support translation to solver-compatible standard forms.
Takeaways & Limitations
Treating reductions as first-class rewriting units supports modular separation of front ends and back ends and matching problems to specific solver classes.
Takeaways & Limitations
Existing DSLs, including CVXPY, have solver-selection and canonicalization procedures that are implemented ad hoc and cannot easily be modified or extended.
Abstract
from arXiv · showhide
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.
1. Introduction
Convex-optimization DSLs let users state problems declaratively, while rewriting systems connect those problems to suitable numerical solvers through analysis and canonicalization. The paper proposes treating these rewritings as modular, first-class software components.
- Domain-specific languages: Convex-optimization DSLs provide human-readable problem specifications and invoke numerical solvers on users’ behalf.CVXPY is presented as a Python-embedded DSL for convex optimization.
- Domain-specific languages: CVXPY’s solve method assigns values that minimize the objective subject to constraints, returning the objective value at those assignments.For the toy problem, alice.value == -0.5, bob.value == -0.5, and opt == 1.0.
- Numerical solvers: Numerical solvers require rigidly encoded problem classes, making solver selection and translation difficult for users.Different convex-problem classes form a hierarchy, and more specific solvers are preferable when appropriate.
- Canonicalization: CVXPY canonicalizes the toy problem to a solver-compatible linear-program form, introducing an auxiliary variable alongside the original variables.The transformed variable x has three components: alice, bob, and a canonicalization auxiliary variable.
- Canonicalization: Existing DSLs automate some canonicalization, but their solver-selection and canonicalization procedures are ad hoc and difficult to extend to new problem classes.Manual canonicalization also becomes tedious, laborious, and error-prone as problems grow larger and more complex.
- This paper: The proposed rewriting system translates DSL problems into solver-compatible forms and uses reductions as modular units for canonicalization and retrieval.The system’s rewritings can support solver targeting, new solver interfaces, and a unified treatment of transformations and presolves.
2. An architecture for rewriting systems
The architecture separates optimization rewriting into front end, analysis, and back end phases. This modular design supports solver selection, target-compatible rewriting, tolerable rewriting time, and manageable maintenance.
- Optimization rewriting systems should produce equivalent target-compatible problems, select suitable solvers, keep rewriting time tolerable, and manage maintenance effort.The first three principles are necessary for usefulness; the fourth also benefits engineers and solver developers.
- Separating front ends from back ends and using modular reductions lets new solvers connect to existing DSLs more easily.Researchers interfacing solvers with popular DSLs gain access to a broad ecosystem of problems for testing and tuning.
- Reductions automatically preserve equivalence, while phase decomposition addresses solver selection, rewriting time, and engineering manageability.The analysis phase satisfies solver-selection requirements, best-effort analysis addresses time, and modular front/back ends support maintenance.
- Phases: A three-phase architecture separates parsing, problem analysis and solver selection, and translation into a solver-compatible form.The front end creates an intermediate representation, the analyzer selects a target, and the back end performs the final translation.
- Phases: Back ends are sequences of reductions, so each supported solver has its own canonicalization procedure.The front end parses rather than reduces, while back ends apply reductions to produce target-compatible forms.
3. Reductions
The section presents reductions as modular transformations between equivalent optimization problems, including simple algebraic changes, variable transformations, presolves, and canonicalization steps. It also shows that structural conditions on expression trees can identify broader classes reducible to quadratic programs.
- Reductions transform optimization problems into equivalent problems, supporting both routine canonicalization operations and reductions to simpler problem classes.
- Simple reductions: Flipping objectives, moving expressions across relations, and introducing slack variables convert problem representations without changing the relevant solution set.Maximization becomes minimization of the negated objective, zero right-hand sides can be enforced, and affine inequalities become equalities with nonnegative slack variables.
- Transformations: Monotone transformations and one-to-one changes of variables preserve equivalence while potentially removing exponentials and logarithms or converting nonconvex geometric programs into convex problems.For geometric programs, xi = exp(zi) followed by logarithmic transformation yields a convex problem, and monomial-only cases yield a linear program.
- Domain reductions: Complex-domain problems can be reduced to real-domain problems by separating real and imaginary parts, including corresponding transformations for norms, equalities, and semidefinite constraints.Recovering a complex-domain solution requires bookkeeping to map real-domain variable values back to real and imaginary components.
- Other reductions: Relaxing a convex equality to a nonpositive inequality is equivalent under monotonicity conditions on the objective, inequalities, and equality function.The relaxation is tight when the objective and inequality functions are nondecreasing in one variable and the equality function is decreasing in that variable.
- Quadratic reductions: DCP-compliant problems whose objective-tree paths match A∗QP∗|P+ can be reduced to quadratic programs by graph-expanding piecewise-linear atoms and then forming a quadratic objective.The broader reduction extends beyond the initially apparent A∗QA∗|P+ class, provided the constraints meet the stated admissibility conditions.
4. Implementation
The implementation represents rewriting operations as composable Reduction objects and uses an analyzer to select suitable solver back ends. CVXPY 1.0 applies the resulting chain, invokes the solver, and retrieves a solution for the original problem.
- CVXPY 1.0 implements reductions as first-class Reduction objects in an open-source three-phase rewriting system.
- Reduction objects: Each reduction exposes accepts, apply, and retrieve methods for testing applicability, producing an equivalent problem, and mapping solutions back to the source problem.
- Solver selection: Chain objects compose reductions and solver back ends, while the analyzer checks back ends by decreasing specificity and short-circuits after finding a suitable one.Analysis may prepend simple reductions, after which the complete chain is applied, the solver is invoked, and the solution is retrieved.