Source-linked AI summary
CVXPY: A Python-Embedded Modeling Language for Convex Optimization
Steven Diamond, Stephen Boyd
TL;DR
Applying convex optimization often requires expertise to convert problems into solver-ready standard form. This paper presents CVXPY, a Python-embedded modeling language that automates this conversion and has been downloaded by thousands of users and used in courses.
Problem
Using convex optimization requires converting problems into solver-specific standard form, a time-consuming and error-prone task requiring expertise.
Method
CVXPY lets users construct convex problems naturally, verifies convexity with signed DCP, supports parameters, and converts problems into equivalent conic form.
Results
CVXPY has been downloaded by thousands of users, used to teach multiple courses, and extended with tools such as stochastic optimization.
Takeaways & Limitations
CVXPY supports practical convex-optimization workflows in Python, including repeated parameterized solves and parallel trade-off-curve computation.
Abstract
from arXiv · showhide
CVXPY is a domain-specific language for convex optimization embedded in Python. It allows the user to express convex optimization problems in a natural syntax that follows the math, rather than in the restrictive standard form required by solvers. CVXPY makes it easy to combine convex optimization with high-level features of Python such as parallelism and object-oriented design. CVXPY is available at http://www.cvxpy.org/ under the GPL license, along with documentation and examples.
1. Introduction
CVXPY is introduced as a Python-embedded domain-specific language that simplifies expressing convex optimization problems while integrating with Python features. Its adoption includes thousands of downloads, use in multiple courses, and extensions such as stochastic optimization.
- Motivation: Domain-specific languages avoid the expertise, time, and error risks of custom solvers or manual conversion to standard form.Convex optimization supports applications including machine learning, control, finance, and signal and image processing.
- Contribution: CVXPY is a new convex-optimization DSL based on CVX, adding signed disciplined convex programming analysis and parameters.It is implemented as an ordinary Python library.
- Contribution: CVXPY integrates convex optimization with Python features including parallelism and object-oriented design.
- Adoption: Thousands of users have downloaded CVXPY, and it has been used to teach multiple courses.
- Adoption: Tools built on CVXPY include an extension for stochastic optimization.
2. CVXPY Syntax
CVXPY provides readable, CVX-inspired Python syntax for expressing constrained optimization problems, while separating variables, objectives, and constraints before combining them into a problem.
- Problem construction: A least-squares problem uses x = Variable(n), minimizes sum_squares(A*x - b), and constrains 0 <= x and x <= 1.Problem data A and b can be represented as NumPy ndarrays or other common Python matrix types.
- Solving: prob.solve() returns the optimal objective value, while x.value stores the optimal value of x.The example retrieves the solution with result = prob.solve() and prints x.value.
- Problem construction: CVXPY’s variable, objective, and constraints are constructed separately, then combined in a final Problem object.This isolation supports high-level code that constructs optimization problems.
3. Solvers
CVXPY converts optimization problems into equivalent conic form using graph implementations of convex functions, then solves them with cone solvers. It interfaces with CVXOPT, ECOS, and SCS, which differ in supported cones and algorithms.
- Conic form: CVXPY converts problems into equivalent conic form through graph implementations of convex functions, preserving the original solution.Solving the resulting cone program yields a solution to the original problem.
- Cone solvers: CVXPY interfaces with the open-source cone solvers CVXOPT, ECOS, and SCS, implemented in combinations of Python and C.Cone solvers handle combinations of several cone types.
- Cone solvers: The interfaced cone solvers differ in the cone types they support and the algorithms they employ.CVXOPT and ECOS are interior-point solvers.
4. Signed DCP
Signed DCP extends standard disciplined convex programming by tracking expression signs, enabling verification of additional convex compositions whose monotonicity depends on argument sign.
- DCP foundation: CVXPY uses DCP rules based on known function curvature and monotonicity to verify problem convexity.Problems use a fixed function library, and compositions must follow rules ensuring known curvature.
- Signed DCP: Signed DCP tracks expression signs, allowing more compositions to be verified as convex when function monotonicity depends on argument sign.This extends the DCP rules used in CVX.
- Signed DCP: Under standard DCP, square(square(x)) is not verified as convex because square is nonmonotonic.Signed DCP verifies this composition because square is increasing for nonnegative arguments.
5. Parameters
CVXPY introduces parameters whose fixed symbolic properties but changeable numeric values enable repeated solves without recomputing parameter-independent work. This supports parallel computation of trade-off curves, such as LASSO’s error–regularization curve, using Python multiprocessing.
- Parameters: Parameters fix symbolic properties such as dimensions and sign while allowing numeric values to change across repeated problem solves.CVXPY avoids redoing computations that do not depend on parameter values.
- Parameters: In the LASSO example, positive γ trades off sum-of-squares error against the regularization term.The problem data are A ∈Rm×n and b ∈Rm, and positivity is required by DCP rules.
- Parallel computation: Trade-off curves are trivially parallelizable because each parameterized problem can be solved independently.CVXPY can distribute these computations across many processes through Python multiprocessing or another parallelism library.
6. Object-Oriented Convex Optimization
CVXPY supports object-oriented construction of convex optimization problems by encapsulating local variables, constraints, and objective terms in vertex and edge objects. These objects can be connected into a graph, whose local problems are summed and solved as one flow problem.
- Object-oriented problem model: CVXPY models a directed-graph flow problem with edge flows, vertex source variables, and convex edge and vertex costs.Negative flow or source values represent opposite-direction flow or a sink, respectively.
- Object-oriented problem model: Vertex and edge objects store local optimization variables, constraints, and associated objective terms, then export each local component as a CVXPY problem.Vertex objects include source variables and costs, while edge objects maintain flow connections.
- Graph composition: Edges connect vertices by appending negative flow to the input vertex and positive flow to the output vertex.This implements flow direction through the edge’s connect method.
- Graph composition: CVXPY composes the graph by summing vertex and edge local problems, with overloaded addition combining objectives and concatenating constraints.The resulting single commodity flow problem is then solved with prob.solve().