Random Projections for Semidefinite Programming and Polynomial Optimization with Applications in Adversarial Machine Learning

Bi- and more general multilevel optimization problems are a very versatile tool to model problems with multiple competing actors. A classical example are bilevel problems that occur in the design of pricing schemes in eg energy markets. Solution methods for bilevel problems with convex lower level are well understood, nevertheless they remain challenging. Introducing nonconvexities into the lower level (eg integer variables or nonlinear equalities) makes these problems computationally intractable for practical applications. For specific applications, ad hoc methods are known. The goal is to further the theory and develop algorithms for subclasses of these problems that can be applied to pricing problems in smart-grids.