$$\def\vepsi{\varepsilon} \def\bold#1{{\bf #1}}$$ $$\def\Hcal{{\mathcal{H}}} \def\bold#1{{\bf #1}}$$ $$\def\RR{{\bf R}} \def\bold#1{{\bf #1}}$$ $$\def\LL{{\bf L}} \def\bold#1{{\bf #1}}$$ $$\def\xv{{\bf x}} \def\bold#1{{\bf #1}}$$ $$\def\uv{{\bf u}} \def\bold#1{{\bf #1}}$$ $$\def\qv{{\bf q}} \def\bold#1{{\bf #1}}$$ $$\def\vv{{\bf v}} \def\bold#1{{\bf #1}}$$ $$\def\Xv{{\bf X}} \def\bold#1{{\bf #1}}$$ $$\def\cv{{\bf c}} \def\bold#1{{\bf #1}}$$ $$\def\gv{{\bf g}} \def\bold#1{{\bf #1}}$$ $$\def\wv{{\bf w}} \def\bold#1{{\bf #1}}$$ $$\def\Omegamat{{\bf \Omega}} \def\bold#1{{\bf #1}}$$ $$\def\Smat{{\bf S}} \def\bold#1{{\bf #1}}$$ $$\def\Pmat{{\bf P}} \def\bold#1{{\bf #1}}$$ $$\def\Hmat{{\bf H}} \def\bold#1{{\bf #1}}$$ $$\def\Imat{{\bf I}} \def\bold#1{{\bf #1}}$$ $$\def\Gmat{{\bf G}} \def\bold#1{{\bf #1}}$$ $$\def\Gammamat{{\bf \Gamma}} \def\bold#1{{\bf #1}}$$ $$\def\Qmat{{\bf Q}} \def\bold#1{{\bf #1}}$$ $$\def\Rmat{{\bf R}} \def\bold#1{{\bf #1}}$$ $$\def\omegas{{\bf \omega^{\star}}} \def\bold#1{{\bf #1}}$$ $$\def\hf{{\frac{1}{2}}}$$ $$\def\nablau{\nabla_{\uv}^{2}} \def\bold#1{{\bf #1}}$$ $$\def\nablav{\nabla_{\vv}^{2}} \def\bold#1{{\bf #1}}$$ $$\def\nablaw{\nabla_{\wv}^{2}} \def\bold#1{{\bf #1}}$$ $$\def\nablavv{\nabla_{\vv\vv}^{2}} \def\bold#1{{\bf #1}}$$ $$\def\xbar{\overset{-}{\xv}} \def\bold#1{{\bf #1}}$$ $$\def\ubar{\overset{-}{\uv}} \def\bold#1{{\bf #1}}$$ $$\def\vbar{\overset{-}{\vv}} \def\bold#1{{\bf #1}}$$

# MGDA Platform

Multiple Gradient Descent Algorithm for Multi Objective Differentiable Optimization.

# Presentation

The MGDA Platform is intended to provide the user with information or software for multi-objective differentiable optimization. The Platform is made of four components of increasing complexity:

### Basic MGDA tool for finding a search direction

This tool permits to compute a descent direction common to an arbitrary set of cost functions whose gradients are provided in situations other than Pareto stationarity.

### Directions for solving a constrained problem

This chapter refers to the research report Inria "Quasi-Riemannian Multiple Gradient Descent Algorithm for constrained multiobjective differential optimization" for solving constrained problems by using the basic MGDA tool.

### Tool for solving a prioritized optimization problem

This tool permits to solve a multi-objective optimization problem in which the cost functions are given in two sets:

• a primary set of cost functions subject to constraints for which a Pareto optimal point is provided by the user (after using the previous tool or any other multiobjective method, possibly an evolutionary algorithm)
• a secondary set of cost functions to be reduced while maintaining quasi Pareto optimality of the first set.

### Stochastic MGDA

This page provides references to SMGDA, an extension of MGDA applicable to certain stochastic formulations.

# Registration

The Platform is freely open to the academic community. The potential user is required to send a request by email to mgda-contact@inria.fr. Please provide brief but accurate answers to the following questions:

• Last name, firstname, affiliation, personal web page
• Valid academic email, address including department
• Position
• Faculty : Is your study part of a subsidizied project ?
• Ph.D. students : Which institution(s) subsidize(s) your research/scholarship ?
• All students : Which degree are you preparing and expected date of graduation ?
• Title and description of your use case (5 lines)
• Technical aspects of your use case:
• nature and number of optimization variables
• parametric or PDE-constrained cost-functions and number
• parametric or PDE-constrained scalar constraint functions and number
• is the evaluation of cost and constraint functions computationally demanding (typical CPU time)
• Operating system (Ubuntu, Fedora, MacOs)
This email should also contain the following phrase: I accept the terms of the MGDA Platform License. I understand that this platform is intended for academic use and research purposes, and I agree to refer to it in any publication that includes results derived from its usage.

# Credits

The computational core of this platform was originally developed by Jean-Antoine Desideri, implementing the algorithms described in the references cited in the synopsis. The web interface and remote execution system were designed and realized by the SED team of Inria Sophia Antipolis.