# Anders Forsgren's research concerns large-scale nonlinear optimiztion

Optimization problems arise almost everywhere. They may the "natural", exemplified by current flowing through an electrical circuit in such a way that the total power is minimized. They may also be "man made". An example is by making choice of route in a trip by public transport so as to minimize the travel time, although the traveler may not think about it. My research is directed towards mathematical optimization problems, where values of a set of optimization variables are to be determined. The values of the variables are chosen such that the value of an objective function is minimized. There are often restrictions on the variables, referred to as constraints. An example is a box of which the space diagonal is to be minimized. The variables are then breadth, width and height. The objective function is the space diagonal as a function of the variables (the sum of their squares) and the constraints are given by nonnegativity of breadth, width and depth.

The optimization problem that we study are typically large-scale, several thousands of variables, and the functions are in general nonlinear. The optimization problems then become hard to solve. The solution methods are typically constructed so that they solve a sequence of simpler subproblems that eventually give the solution to the original problem. The specific research concerns the design and the solution of the subproblems. We want to utilize problem structure to solve them approximately fast. Although the research concerns the fine details of the optimization problems, these are fine details that are of significant importance for the behavior of the overall methods.

The methods which we study are generally applicable and the type of optimization problem which we consider have many application areas. A particular application area where we are active concerns optimization of radiation therapy, where we study the optimization problems that arise when best possible radiation strategy is desired. These optimization problems have demonstrated very interesting properties that we have managed to understand and take advantage of. In particular, there are many conflicting goals that make the optimization problem complicated. This application area has also given important impulses to our development of new general optimization methods.

**Watch a movie from a talk on optimization of radiation therapy, which Anders gave when KTH celebrated the 100th anniversary of its campus in 2017.**