SymbolicOptimization.jl

A Julia package for multi-objective symbolic optimization using grammar-guided genetic programming and NSGA-II.

What is Symbolic Optimization?

Symbolic optimization evolves mathematical expressions to optimize arbitrary objectives. The result is an interpretable formula, not a black-box model.

This is not (just) symbolic regression. While symbolic regression finds $f(x) \approx y$ by minimizing prediction error, symbolic optimization is the general framework for searching expression space with any objectives:

Application	Objective(s)	Output
Aggregator discovery	Calibration, accuracy on crowd predictions	Formula combining forecaster estimates
Belief update rules	Match normative Bayesian updates	Heuristic for updating credences
Scoring rule design	Proper scoring, discrimination	Formula evaluating forecaster quality
Curve fitting	MSE on (x, y) data	Regression formula

Quick Start

The simplest way to use SymbolicOptimization is via the DSL Interface:

using SymbolicOptimization

# Generate some data
X = reshape(-3:0.2:3, :, 1)
y = vec(X.^2 .- 1)  # Target: x² - 1

# Define and solve the problem in one call
result = solve(symbolic_problem(
    X = X,
    y = y,
    variables = [:x],
    binary_operators = [+, -, *, /],
    population = 200,
    generations = 50
))

# Get the best solution
best_sol = best(result)
println(best_sol.expression)  # Something like "(x * x) - 1.0"
println(best_sol.objectives)  # [MSE, complexity]

For more control, see the Model API which provides a JuMP-inspired macro-based interface.

Installation

using Pkg
Pkg.add("SymbolicOptimization")

Package Overview

Component	Description
DSL Interface	High-level `symbolic_problem()` and builder pattern
Model API	JuMP-style `SymbolicModel` with macros
Optimization	NSGA-II engine, objectives, results
Core Types & Trees	`AbstractNode`, tree utilities
Grammar System	Typed/untyped grammars, safe operations
Evaluation	Expression evaluation, context-aware evaluation
Genetic Operators	Generation, mutation, crossover, simplification
Advanced Topics	Symbolics.jl integration, constraints, policy problems

Limitations

For high-dimensional symbolic regression (e.g., 100+ variables), consider using SymbolicRegression.jl which implements multi-population island models optimized for that use case.

SymbolicOptimization.jl excels at:

Custom objective functions beyond MSE
Domain-specific operators and grammars
Context-aware evaluation (belief updating, aggregation)
Problems with smaller search spaces but complex objectives