Optimization

Optimization is a field of mathematics where the goal is to select the best element with respect to criteria from a set of available alternatives. Put this way, it resembles a mathematician’s playground. However, optimization is at the heart of many algorithms. It is used to answer direct questions like: What is the best financial investment that I can make? What is the best shape for this component? Where is the safest place to live? It is also used indirectly. Many machine-learning algorithms require solving an optimization problem and many problems can be reformulated as optimization problem. For instance, “ How to recover missing values?” can be expressed as “What is the best way to recover the missing values?”

Personal competences

I would summarize my knowledge as skills for the two important steps in solving a problem with convex optimization. Figure 1 illustrates the two steps.

  • Modelization : writing the mathematical form.
  • Optimization : using the right algorithm.

Graph de-noising example
Figure 1: Modelization of a question as an optimization problem.

Modelization(formulation of the problem)

The conception of the model is a creative step where we transform our requirements into mathematical equations. There is usually a trade-off between the accuracy and the complexity (the more accurate the model, the more complex). Modelization also includes the final design of the problem. The same model can usually be expressed with different mathematical formulations. However, some may be much easier to solve than others. A clever choice can make the difference between what is solvable and unsolvable. This competence is mostly acquired with experience.

Optimization (solving the problem) - UNLocBoX

Optimization was so omnipresent in my research that I have developed my own software. The UNLocBoX is a convex optimization toolbox written in MALAB and ported to Python (PyUNLocBoX). It is capable of solving large non-smooth problems, is based on proximal-splitting methods and includes the most recent efficient algorithms.