OBJECTIVES

In certain applications, the desired solution of an optimization problem can be stated as a tensor. Typically, the size of such a tensor grows exponentially in problem dimension. To deal with this so-called curse of dimensionality, low-rank tensor decompositions have been developed and used to provide compressed approximations to the original tensors. An important application area is the simulation of high-dimensional PDEs, in which case the tensor contains a certain discretization of the solution operator of a PDE (i.e. a multivariate function). Having the solution operator in a tensor is different than having the problem data in a tensor: the desired tensor is given only implicitly, and an algorithm for solving such an optimization problem needs to work with the low-rank representation.

In the first phase of this project, we consider constrained optimization problems with variables in the form of a low-rank tensor decomposition. There are a number of studies in the literature that extend optimization algorithms for tensor variables by using tensor algebra. An alternative approach is to input variables into an existing optimization algorithm directly in the form of its low rank decomposition. The challenges of the approach have been studied in the context of unconstrained tensor optimization problems arising in tensor completion; in this project, we aim at the design, implementation and analysis of new algorithms by extending this approach to constrained optimization. In the second phase, we consider optimization problems with constraints evaluated through tensor-based surrogates (such as machine learning models or simulations). The goal of this phase of the project is to extend existing constrained optimization methods to exploit the special structure of the surrogate models, and possibly deal with the error in resulting constraint evaluations.

Supervisor: Bernard Mourrain, INRIA

Co-supervisors:

• Figen Öztoprak Topkaya, Artelys, Gebze Technical University (GTU)
• Michaël Gabay, Artelys
• Cordian Riener, Arctic University of Norway (UiT)

PHD OBJECTIVES

The main expected result during this thesis is practical algorithms with a theoretical basis for solving constrained optimization problems with variables in the form of low-rank tensor decompositions, and problems involving tensor-based surrogate models. Writing conference/journal papers and completion of a PhD degree by the Doctoral Candidate.

PLANNED SECONDMENTS

The PhD will be a double degree, awarded jointly by universities of Nice, France and the Arctic University of Norway (UiT). The candidate will spend 10 to 12 months in Norway, at UiT (Norway).