MOSolvers.jl

MOSolvers.jl is a Julia package that provides the following implementations of algorithms for Multi-Objective Optimization (MO):

• CondG: doi.org/10.1080/02331934.2023.2257709
• ProxGrad: doi.org/10.1007/s10589-018-0043-x
• PDFPM: arXiv:2508.20071.

This package was originally developed to support the research presented in the paper cited below.

Reference

If you use this package in your research, please cite the following paper:

• PDFPM: arXiv:2508.20071

Installation

This package is currently not registered in the General Julia registry, and neither is its dependency MultiObjectiveLineSearches.jl. Add the dependency first, then the package, using the package manager mode (press ]) or Pkg:

import Pkg
Pkg.add(url="https://github.com/VectorOptimizationGroup/MultiObjectiveLineSearches.jl")
Pkg.add(url="https://github.com/VectorOptimizationGroup/MOSolvers.jl")

If installation via URL fails, you can install the package from a local clone of the repository.

First, clone the repository to a directory of your choice:

git clone https://github.com/VectorOptimizationGroup/MOSolvers.jl

This command will create a local directory named MOSolvers.jl.

Next, decide whether you want to install the package in a specific Julia project (recommended) or in the global environment. In both cases, install it by explicitly pointing to the local path of the cloned repository.

import Pkg
Pkg.add(url="https://github.com/VectorOptimizationGroup/MultiObjectiveLineSearches.jl")
Pkg.add(path="/path/to/MOSolvers.jl")

Important

• Replace /path/to/MOSolvers.jl with the actual location where the repository was cloned on your system.

Problem Classes

These solvers are specialized for composite problems of the form $F(x) = f(x) + h(x)$, where $h(x)$ has a specific max-linear structure. For a detailed mathematical description of the problems solved, please see Solvers and Problem Structure.

Usage Example

Here is a basic example of how to solve a multi-objective problem (BK1) using the Partially Derivative-Free Proximal Method (PDFPM) solver.

using MOSolvers
using LinearAlgebra

# 1. Define the Problem Data (BK1)
function bk1_evalf(x)
    return [x[1]^2 + x[2]^2, (x[1] - 5.0)^2 + (x[2] - 5.0)^2]
end

function bk1_evalJf(x)
    J = zeros(Float64, 2, 2)
    J[1, 1] = 2.0 * x[1]
    J[1, 2] = 2.0 * x[2]
    J[2, 1] = 2.0 * (x[1] - 5.0)
    J[2, 2] = 2.0 * (x[2] - 5.0)
    return J
end

# Initial guess and bounds
x0 = [5.0, 10.0]
lb = [-5.0, -5.0]
ub = [10.0, 10.0]

# Problem parameters for PDFPM (Partially Derivative-Free Proximal Method)
# A and delta correspond to the linearization model parameters.
# For standard problems, A_j is typically Identity and delta is 0.
A = [Matrix{Float64}(I, 2, 2) for _ in 1:2]
delta = 0.0

# 2. Configure Solver Options
options = PDFPM_options(
    verbose=1,
    max_iter=100,
    opt_tol=1e-6,
    sigma=1.0,
    alpha=0.1,
    print_interval=1
)

# 3. Solve the Problem
# Note: evalJf is optional for PDFPM, but can be provided if available.
result = PDFPM(bk1_evalf, A, delta, x0, options; evalJf=bk1_evalJf, lb=lb, ub=ub)

# 4. Inspect Results
println("Success: ", result.success)
println("Final x: ", result.x)
println("Final Objectives: ", result.Fval)
println("Iterations: ", result.iter)

Package Structure

The source code is organized as follows:

• src/solvers/: Contains the main solver implementations.

  • condg.jl: Conditional Gradient Method.
  • proxgrad.jl: Proximal Gradient Method.
  • pdfpm.jl: Partially Derivative-Free Proximal Method.

• src/subproblems/: Solvers for the auxiliary subproblems required by each method.

  • subproblem_condg.jl
  • subproblem_proxgrad.jl
  • subproblem_pdfpm.jl

• src/utils/: Utility modules.

  • linesearch.jl: Multi-objective line search strategies (e.g., Armijo).
  • evalh.jl: Routines for evaluating supporting functions.
  • solver_utils.jl: Data structures like OptimResult and helper functions.

Contributing

Contributions are welcome! Please feel free to submit a Pull Request or open an Issue to discuss improvements or report bugs.