Many problems in economics, natural sciences and engineering can be formulated in terms of convex optimization (CO) problems. The development of efficient algorithms for solving CO problems is then of paramount importance to handle these applications.
In this project, the investigation of two important classes of algorithms for solving CO problems will be continued, namely: interior-point methods (also, called second-order methods) and first-order methods. The latter methods are particularly important for solving CO problems that, due to their huge size, cannot be solved by the first ones. More specifically, the purpose of this project is to investigate the following topics: 1) development of new algorithms and implementations for first-order smooth and non-smooth methods for cone programming (CP) problems; 2) development and implementation of first-order methods for geometric programming problems; 3) investigation of the complexity of first-order algorithms for solving CO problems based on the augmented Lagrangian penalty methods; 4) development of new insights on the geometry of the central path and examination of their impact into the polynomial complexity analysis of second-order methods for CP problems.
If successful, this project will lead to the development of new software packages, which will increase and improve the existing tools available to practitioners to handle CO problems originating from many areas of economics, natural sciences, and engineering. For example, geometric programming is a tool commonly used to model many circuit design problems. The development of fast algorithms for geometric programming will have strong impact in the development of higher performance computers, which are important to several applications (e.g., weather prediction).
