**Student in Nonlinear Inverse Problems**

Huawei R&D Sites in Belgium and the Netherlands

Thesis Projecten Vlaams-Brabant, België Vlaams-Brabant, België

#### Student in Nonlinear Inverse Problems

**Loonopties**

**Thesis Projecten**

Inverse problems seek to reconstruct the unknown model parameters characterizing the system under investigation from measured data. More precisely, nonlinear inverse problems start by considering a nonlinear operator between the parameters, and acquired information / observation about them, i.e., data or measurements. This operator constructs a model for the available data, assumed to depend only on these parameters. Since the inverse problem is the inverse of the forward problem, this operator is often referred to as the forward operator. n nonlinear inverse problems, the forward operator being nonlinear, the model parameters depend in a nonlinear way of the state of the system and therefore on the observations available on it.

However, inverse problems are typically ill-posed, as opposed to the well-posed problems (verifying existence, uniqueness, and stability of the solution(s)). Also inverse problems are also typically ill-conditioned, i.e., a small change in the input induce a large change in the response. Hence, to tackles them, nonlinear regularization functionals are often introduced in the objective function to prevent overfitting. The specific context of this internship sits when the resulting minimization problem also involves inequality constraints on the model parameters or some functions of them. These constraints are important to avoid invalid or non-significant values for the model parameters.

**Task**:

- Design and compare different numerical solving algorithms associated to different constrained nonlinear optimization methods.
- Evaluate and compare these algorithms both formally and numerically using various (synthetic or real) benchmark cases including network tomography, network properties inference, etc. These tasks will be realized under supervision of senior (postdoc-level) researcher.

**Duration**: from 3 months to 1 year (max.)

**Candidate profile**:

- if MSc thesis: the candidate must be following the last year of the curriculum in, e.g., Applied/ Numerical mathematics, Math/Mechanical engineering, Theoretical computer science, Computer science engineering. Detailed coordinates of MSc promotor and his/her academic affiliation must be provided in the CV application form.
- if internship: the candidate must have completed his MSc (in one of these disciplines). Copy of the MSc diploma/certificate shall be included in annex of the CV. The internship can also be considered as part of post-MSc graduation or PhD graduation program.
- Good knowledge of networked systems is considered as a plus.

Note well: candidate must have obtained their University degree from an academic institution of one of the EU country.

**Student in Fast Converging Solvers**

Huawei R&D Sites in Belgium and the Netherlands

Thesis Projecten Vlaams-Brabant, België Vlaams-Brabant, België

#### Student in Fast Converging Solvers

**Loonopties**

**Thesis Projecten**

Despite recent advances in automatic differentiation (AD) algorithms and software, incorporating automatic differentiation tools in optimization solvers remains largely unexploited. Commonly, the solving of bound-constrained nonlinear optimization problems with, e.g., the Newton method or the Augmented Lagrangian Method (ALM), require the evaluation of the objective function, its gradient and the (sparsity pattern of the) Hessian matrix. Additionally, constrained optimization problems also involve providing the sparsity pattern and the Jacobian matrix of the constraints. The time and resource required to obtain this information and verify their correctness can be relatively large even for simple problems. Nowadays, optimization problem-solving environments provide modeling language and state-of-the-art optimization solvers together with packages that are capable to compute first-order information, e.g., derivatives, gradients. However, AD remains largely unexploited for generating these quantities; requiring consequently to run the entire framework and its packages altogether to solve nonlinear optimization problems. In this context, the main goal is to blend AD tools that automatically generate the required first- and second-order quantities together with nonlinear optimization methods such as ALM for constrained problems.

**Objective**: instigate and investigate automatic and numerical differentiation tools for nonconvex optimization solvers that can rapidly converge with few function and derivative evaluations as well as provide and improve second-order information with the same efficiency and reliability as available for first-order information.

**Task(s)**: design/formulate, develop and numerically evaluate enhancements of existing computational methods/algorithms such as ALM for solving constrained nonconvex optimization problems. To address the computational performance objectives, the candidate will also actively participate to the design (extension) and evaluation of (existing) automatic differentiation tools for solving such optimization problems. These tasks will be realized under the supervision of a senior (postdoc-level) researcher.

**Candidate profile**:

- If MSc thesis : the candidate must be following the last year of the curriculum in, e.g., Applied mathematics, Math. engineering, Theoretical computer science, Computer science engineering. Detailed coordinates of MSc promotor and his/her academic affiliation must be provided in the CV application form.
- If PhD student : the candidate must have completed his/her MSc (in one of these disciplines) and ideally realized a thesis in one of the following domain(s): nonlinear programming, computational methods and algorithms for nonconvex optimization (incl. second order iterative methods, subspace methods, etc.), automatic differentiation. Copy of the MSc diploma/ certificate shall be included in annex of the CV. The internship can also be considered as part of post-MSc graduation or PhD graduation program.
- Good knowledge of functional programming language (LISP, Julia, etc.) is considered as a plus.

**Note well: candidate must have obtained their University degree from an academic institution of one of the EU country.**