Ghost-penalties are an innovative tool, recently intorduced by members of this research project, that can be used for the analysis of the convergence properties of optimization methods. While they share several characteristics with traditional Lyapunov functions classically used to analyze optimization algorithms, they have some distinctive features that make them extremely flexible and suitable for applications in domains where more traditional methods have not brought significant results. For example, by using ghost-penalties it was possible to give the first convergence and complexity results for diminishing stepsize methods in nonconvex optimization. Building on the experties accumulated in the past few years, this project aims at uncovering new applications for the ghost-penalty technique. In particular we plan to investigate the following topics
1) Development of the first provably convergent algorithm for nonconvex stochastic optimization problems with stochastic constraints
2) Development of the the first provably convergent distributed algorithm for nonconvex problems with nonconvex constraints
3) A complexity analysis for sequential quadratic programming methiods under realistic assumptions.
4) Applications to games and bilevel optimization
5) Application in sciences and engineering
6) Development of a computer code and its release
Traditionally, convergence results in optimziation are all based on either a fixd-point argument or on a Lyapunov function techniques. The ghost penalty technique is a new method a represents a breaktrhough in the field. They can be viewed as a generalization of the Lyapunov function technique but it permits to deal with convergence to generalized stationary points (a feature that is essential when dealing with problems that have no feasibility or regularity guarantees) and is based on a different priciple. This has already allowed to obtain some innovative result in the optimization field and we expect the approach to yeld further dividends and allow for the development of algorithms for classes of problems that were so far untractable.