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Convex Optimization in Julia
Madeleine Udell, Karanveer Mohan, David Zeng, Jenny Hong, Steven Diamond, Stephen Boyd
TL;DR
Convex addresses the challenge of translating user-friendly convex optimization models into solver-oriented representations while retaining useful problem structure. It uses abstract syntax trees and Julia’s multiple dispatch to verify disciplined convexity and transform problems into conic form. The paper reports performance as fast as or faster than more mature convex-optimization modeling codes, while noting limits for extremely large-scale problems.
Problem
Optimization modeling requires bridging user-friendly problem descriptions and solver-oriented representations while preserving structural information useful for specialized methods.
Method
Convex represents expressions as abstract syntax trees and uses them to verify disciplined convexity and transform problems into conic form, with Julia multiple dispatch supporting the implementation.
Results
Convex parses disciplined-convex-programming problems into conic form as fast as or faster than other more mature convex-optimization modeling codes.
Takeaways & Limitations
The framework combines high-level abstraction with performance comparable to or better than similar systems and supports adding new AST-based transformations.
Takeaways & Limitations
Convex currently performs computations serially, limiting its handling of extremely large-scale problems until parallelism can be leveraged.
Abstract
from arXiv · showhide
This paper describes Convex, a convex optimization modeling framework in Julia. Convex translates problems from a user-friendly functional language into an abstract syntax tree describing the problem. This concise representation of the global structure of the problem allows Convex to infer whether the problem complies with the rules of disciplined convex programming (DCP), and to pass the problem to a suitable solver. These operations are carried out in Julia using multiple dispatch, which dramatically reduces the time required to verify DCP compliance and to parse a problem into conic form. Convex then automatically chooses an appropriate backend solver to solve the conic form problem.
1. INTRODUCTION
Convex bridges user-friendly optimization modeling and solver-oriented computation by representing problems structurally and exploiting Julia’s multiple dispatch. This representation supports disciplined convexity checking, conic transformation, and efficient solver use for structured convex programs.
- Motivation: Convex translates optimization problems from a user-friendly language into a solver-friendly representation while preserving their structural form.The framework targets the gap between human-readable problem specification and solver-oriented computation.
- Representation: Convex represents expressions as an abstract syntax tree with variables or constants at leaves and atoms at internal nodes.This recursive representation makes the problem’s global structure computationally explicit.
- Capabilities: The AST lets Convex check disciplined convexity and transform problems into conic form with a concise global description for solvers.The same representation supports both DCP verification and conic-form conversion.
- Implementation: Julia’s multiple dispatch separates mathematical functions from operations on them, enabling performant high-level code and easier implementation of new AST methods.The paper reports performance that rivals or surpasses similar systems and identifies extensibility as another benefit.
- Scope: Convex focuses on cone programs, including LPs, QPs, QCQPs, SOCPs, SDPs, and exponential cone programs.Cone programs occur frequently and can be solved quickly in theory and practice.
2. FUNCTIONAL REPRESENTATION
Convex represents optimization problems as abstract syntax trees built from variables, constants, and composable atoms. This representation supports structural analysis, expression reuse, disciplined-convexity checking, conic conversion, and solver interaction.
- Expressions and ASTs: Convex provides atoms for functions such as addition, multiplication, absolute value, and norms, applied through overloaded Julia operators and language features.Composite atoms can be defined by composing existing atoms; for example, norm(x, 1) is implemented as sum(abs(x)).
- Atom methods: Each atom exposes methods for sign, evaluation, curvature, monotonicity, and conic-form conversion, dispatched according to the expression’s top-level atom.Defining these five methods is sufficient to add a new atom that Convex can use for solving problems.
- Atom methods: Local combination rules make atom-method classifications sound, but they can miss stronger properties that require nonlocal information.For example, Convex returns no sign for square(x) + x - x even though the expression is always nonnegative.
- Expressions and ASTs: Expressions are recursively formed from variables, constants, or atoms applied to other expressions, and represented as ASTs with leaves and internal atom nodes.The AST is a directed acyclic graph whose top-level atom is its head.
- Expression identity: Convex hashes each AST’s head atom and child identifiers to detect repeated expressions, memoize computations, reuse results, and reduce solver problem complexity.Expressions with the same structure automatically receive the same unique identifier.
- Problems and solving: A Convex problem contains a sense, objective, and constraints; solve! checks DCP compliance, converts the problem to conic form, and passes it to a solver.After solving, the optimal value and expression values are accessible, and solver-computed dual variables may populate constraints.
3. DISCIPLINED CONVEX PROGRAMMING
Convex uses disciplined convex programming to verify convexity recursively, while multiple dispatch encodes the rules and makes extensions and verification efficient. DCP compliance also enables conversion to conic form and access to fast solvers.
- Checking convexity is difficult in general, motivating Convex’s use of disciplined convex programming for transparent verification.
- DCP verification is transparent and permits equivalent conic-form conversion, enabling fast and reliable solvers for verifiable problems.
- Disciplined convex expressions: Convex classifies expressions using constant, affine, convex, concave, and not DCP curvature categories.Constants are affine, and affine expressions are both convex and concave.
- Disciplined convex expressions: Convex determines expression curvature recursively from each atom’s curvature, argument monotonicity, and child curvature.The rules allow nondecreasing atoms to take convex arguments, nonincreasing atoms to take concave arguments, or affine arguments.
- Disciplined convex expressions: The recognized DCP expression set depends on available atoms and their monotonicity, signs, and argument ranges.Convex can add atoms and implements signed monotonicity, allowing it to recognize functions such as log(Σ exp x_i) that basic atoms alone cannot derive.
- Verifying DCP using multiple dispatch: Multiple dispatch implements DCP arithmetic through curvature and monotonicity types, recursively computing the curvature of expression trees.This design follows the mathematics, simplifies new-atom implementation, and uses lookup-table dispatch instead of slower if statements.
- Disciplined convex programs: A DCP problem has a valid objective sense and only DCP-compliant constraints, and Convex checks these rules recursively when forming the problem.Minimization requires a convex DCP objective, maximization a concave DCP objective, and every constraint must be compliant.
4. CONIC FORM OPTIMIZATION
Convex recursively transforms DCP expressions and problems into conic form using atom-specific templates, then selects a compatible cone solver automatically. The transformation preserves primal-dual solutions for DCP problems.
- 4. CONIC FORM OPTIMIZATION: Conic-form optimization problems are specified by data A, b, c, and a cone K, with convex cones including second-order, semidefinite, and exponential cones.Convex extends the definition to affine expressions with conic constraints.
- 4. CONIC FORM OPTIMIZATION: Convex automatically rewrites DCP problems into standard cone format and selects a solver based on required cones and available backends.For example, it can choose an SOCP solver when exponential and semidefinite cones are absent.
- 4.1 Example: Convex constructs an expression’s conic form recursively by processing child expressions and applying the top-level atom’s conic-form template.Affine expressions are returned directly, while non-affine expressions combine child conic forms through the atom template.
- 4.1 Example: A function is cone-representable when its value equals the optimal value of a conic optimization problem, typically using auxiliary variables.The paper gives |x| as an example of a cone-representable function.
- 4.1 Example: A conic-form template maps function arguments to a conic problem whose objective and constraints are affine in those arguments.For convex functions, feasible points lie in the function’s epigraph; for concave functions, they lie in its hypograph.
- 4.1 Example: For each constraint, Convex combines left- and right-hand conic forms and adds cone membership constraints according to the constraint sense.The resulting objective and constraint set are returned as the conic form of the optimization problem.
- 4.1 Example: The transformation preserves solutions: any primal-dual solution of the conic form yields a primal-dual solution of the original DCP problem.The theorem accounts for auxiliary primal and dual variables introduced during conic conversion.
5. SPEED
Convex is evaluated against CVXPY and CVX on parsing and solving representative problems using ECOS with matched parameters. Its compiled Julia execution substantially outperforms the alternatives after the first evaluation, benefiting repeated inner-loop use.
- Convex is compared with CVXPY and CVX on representative problems that are easy to solve but difficult to parse.
- The evaluation measures both conic-form conversion time and solve time, using ECOS with identical solver parameters across frameworks.
- Parse times are obtained by subtracting solve time from the total problem-formation-and-solve time.
- Convex’s first evaluation is comparable to CVX and CVXPY, while its second, compiled evaluation substantially outperforms them.Julia compiles code on first evaluation and uses the faster compiled version subsequently.
- After the initial call, repeated convex-optimization routines in an inner loop can use Convex’s faster compiled performance on every iteration.The paper gives sequential convex programming as an example of such an inner-loop application.
6. DISCUSSION
Convex’s parsing and modeling methodology can extend beyond conic problems to other structured problem classes with specialized solvers. Its AST-based framework supports detecting and transforming such problems, but large-scale and automatic-detection capabilities remain incomplete.
- 6. DISCUSSION: Convex’s methodology is not restricted to conic-form problems and could support several other structured optimization classes with specialized solvers.Examples include integer linear, biconvex, sums of prox-capable functions, difference-of-convex, and sigmoidal programming problems.
- 6. DISCUSSION: The AST framework makes it easy to add transformations that detect these problem types and convert them into standard formats.
- 6. DISCUSSION: Automatic detection of some proposed problem types remains ongoing research.
- 6. DISCUSSION: Convex currently performs computations serially, limiting its ability to handle extremely large-scale problems.The paper identifies Julia’s developing threading capabilities as a route toward leveraging parallelism.
7. CONCLUSIONS
The paper presents Convex as a Julia library that uses multiple dispatch and JIT compilation to model DCP problems and transform them into conic form. It reports parsing performance at least as fast as, and often faster than, more mature convex-optimization modeling codes.
- 7. CONCLUSIONS: Convex is efficiently and simply implemented as a convex optimization modeling library using multiple dispatch.
- 7. CONCLUSIONS: Convex uses Julia’s multiple dispatch and JIT compilation to parse DCP problems into conic form.
- 7. CONCLUSIONS: Convex parses DCP problems into conic form as fast as or faster than other, significantly more mature, convex-optimization modeling codes.
CVX.
The CVX example introduces a summation-based optimization model using a scalar variable and an iterative accumulation expression.
- CVX.: The CVX example sets n = 10000, declares a scalar variable x, and initializes the expression e to zero.
A.1 Summation
The appendix compares Julia, MATLAB, and Python formulations of optimization examples, including summation and matrix problems solved with conic optimization models.
- A.1 Summation: The Julia summation example creates a scalar variable, accumulates e over 10,000 iterations, and solves a norm-minimization problem with x >= 0 using ECOS.
- A.1 Summation: The Python summation example uses a scalar variable, accumulates e over 10,000 iterations, and solves the corresponding norm-minimization problem with ECOS.
- A.1 Summation: The Julia vector formulation replaces repeated scalar addition with indexed access to a length-10,000 variable while retaining the ECOS norm-minimization model.
- A.1 Summation: The MATLAB summation formulation uses a length-10,000 variable and expresses the same norm-minimization setup with a nonnegativity constraint.
- A.1 Summation: The Python vector formulation uses indexed access to a length-10,000 variable and solves the norm-minimization problem with x >= 0.
- A.1 Summation: The matrix examples set n = 500 and compare Julia, MATLAB, and Python formulations of Frobenius-norm objectives with matrix constraints or assignments.
- A.1 Summation: A second matrix example sets X equal to B while minimizing the Frobenius norm of X - A in Julia, MATLAB, and Python.