Scientific Project

During the last decades, a lot of progress has been made in the optimization research field. New challenges in data science and operational research have not only motivated the development of new optimization tools, but also many (old) theoretical studies have become relevant in these areas.  In this project, we propose to study diverse optimization models, deterministic and stochastic, and to investigate matrix inverse eigenvalue problems, some with potential applications in data science. More specifically, we propose to develop numerical methods for computing critical angles between cones, which are formulated as an optimization problem, and to apply them to the image set classification problem. To summarize the main features of an image set data, it will be necessary a matrix analysis and inverse eigenvalue problems will take part on it. In addition, we will address the Extremal EIP for symmetric and non-symmetric pentadiagonal matrices and some structured Hermitian matrices. We propose also to study a (nonsmooth) joint chance constrained optimization problem and to propose numerical methods to solve it based on a sequence of smooth chance constrained optimization problems. This problem has applications in resource management, electricity network expansion, telecommunications, etc. Another optimization model that we propose to study is the nonlinear second-order cone programming. For this, we propose to develop a numerical method for its resolution with the specific purpose of reducing the numerical cost compared to existing methods. In this way, we expect to obtain a tool to deal with large-scale classification problems.  Last but not least, we propose to develop and use Euclidean Jordan algebra techniques to solve optimization problems under conic constraints, like the problem of computing the maximal angle between LISC cones (Linear Image of Symmetric Cones) and the problem of computing the conic intrinsic volumes of LISC cones. Finally, we propose to introduce new proximal algorithms to solve Separable Symmetric Cone Minimization and analyze their properties: convergence, rate of convergence, finite termination and complexity of the iterations. Then we give some applications of these algorithms to the above mentioned problems.