Section: New Software and Platforms
MGDA
Multiple Gradient Descent Algorithm
Keywords: Descent direction  Multiple gradients  Multiobjective differentiable optimization  Prioritized multiobjective optimization
Scientific Description: The software relies upon a basic MGDA tool which permits to calculate a descent direction common to an arbitrary set of cost functions whose gradients at a computational point are provided by the user, as long as a solution exists, that is, with the exclusion of a Paretostationarity situation.
More specifically, the basic software computes a vector d whose scalar product with each of the given gradients (or directional derivative) is positive. When the gradients are linearly independent, the algorithm is direct following a GramSchmidt orthogonalization. Otherwise, a subfamily of the gradients is identified according to a hierarchical criterion as a basis of the spanned subspace associated with a cone that contains almost all the gradient directions. Then, one solves a quadratic programming problem formulated in this basis.
This basic tool admits the following extensions:  constrained multiobjective optimization  prioritized multiobjective optimization  stochastic multiobjective optimization.
Functional Description: Chapter 1: Basic MGDA tool Software to compute a descent direction common to an arbitrary set of cost functions whose gradients are provided in situations other than Pareto stationarity.
Chapter 2: Directions for solving a constrained problem Guidelines and examples are provided according the Inria research report 9007 for solving constrained problems by a quasiRiemannian approach and the basic MGDA tool.
Chapter 3: Tool for prioritized optimization Software permitting to solve a multiobjective optimization problem in which the cost functions are defined by two subsets:  a primary subset 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 subset of cost functions to be reduced while maintaining quasi Pareto optimality of the first set. Procedures defining the cost and constraint functions, and a small set of numerical parameters are uploaded to the platform by an external user. The site returns an archive containing datafiles of results including graphics automatically generated.
Chapter 4: Stochastic MGDA Information and bibliographic references about SMGDA, an extension of MGDA applicable to certain stochastic formulations.
Concerning Chapter 1, the utilization of the platform can be made via two modes : – the interactive mode, through a web interface that facilitates the data exchange between the user and an Inria dedicated machine, – the iterative mode, in which the user downloads the object library to be included in a personal optimization software. Concerning Chapters 2 and 3, the utilizer specifies cost and constraint functions by providing procedures compatible with Fortran 90. Chapter 3 does not require the specification of gradients, but only the functions themselves that are approximated by the software by quadratic metamodels.

Publications: Revision of the MultipleGradient Descent Algorithm (MGDA) by Hierarchical Orthogonalization  Parametric optimization of pulsating jets in unsteady flow by MultipleGradient Descent Algorithm (MGDA)  A quasiRiemannian approach to constrained optimization  Platform for prioritized multiobjective optimization by metamodelassisted Nash games  Direct and adaptive approaches to multiobjective optimization

URL: http://mgda.inria.fr