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. You can install it directly from GitHub using the package manager mode (press ]) or Pkg:

import Pkg
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(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.