Global optimization solver based on copositive optimization

The project deals with solving optimization problems, i.e. problems of choosing a certain input in order to optimize a certain output. For example, finding a traveling plan from A to B that minimizes things like traveling time and cost (output). For that you have to make a decision on certain variables (input) like what roads to travel and which to avoid, at what times to travel and by which means etc. Another very different example would be optimizing the output of networks of power plants. The questions is, which plant should produce how much energy at which point in time (input), in order to produce as cheaply as possible (output), while respecting physical limits of the network and meeting customer demand, which puts restrictions on the possible inputs. In mathematics, we distinguish between convex and nonconvex problems. Convex problems are like finding the deepest point in a valley or a bathtub: as long as you go down, you will eventually find the deepest point. Nonconvex problems are like finding the deepest point in the Sahara: if you just go downwards, you will find a sink the is deeper than where you started, hence a local optimum, but probably not the deepest one in the dessert, the global optimum. You would have to check all the valleys in the Sahara to find the deepest one, so just walking downwards alone is not going to cut it. Thus, nonconvex problems are much harder to solve than convex ones. Fortunately, there is a theory called copositive optimization theory, that allows you to turn a nonconvex problem into a convex one. However, this doesnt come for free since these reformulations contain much more variables to make a decision on than before the transformation, along with other technical complications. This makes copositive reformulations notoriously hard to solve, in spite of their convexity. For this reason copositive optimization has had little practical relevance despite its mathematical elegance. Our project is concerned with closing this gap. We have good reasons to expect that the technical difficulties, such as the explosion of variables, can be dealt with by cleverly exploiting certain features of the copositive reformulations. In addition, we plan to expand the range of problems that can be reformulated via copositive techniques. Preliminary results give us hope that we can establish a new independent paradigm for practical nonconvex, global optimization.