4 research outputs found
Conic Optimization Theory: Convexification Techniques and Numerical Algorithms
Optimization is at the core of control theory and appears in several areas of
this field, such as optimal control, distributed control, system
identification, robust control, state estimation, model predictive control and
dynamic programming. The recent advances in various topics of modern
optimization have also been revamping the area of machine learning. Motivated
by the crucial role of optimization theory in the design, analysis, control and
operation of real-world systems, this tutorial paper offers a detailed overview
of some major advances in this area, namely conic optimization and its emerging
applications. First, we discuss the importance of conic optimization in
different areas. Then, we explain seminal results on the design of hierarchies
of convex relaxations for a wide range of nonconvex problems. Finally, we study
different numerical algorithms for large-scale conic optimization problems.Comment: 18 page
A relaxed interior point method for low-rank semidefinite programming problems with applications to matrix completion
A new relaxed variant of interior point method for low-rank semidefinite
programming problems is proposed in this paper. The method is a step outside of
the usual interior point framework. In anticipation to converging to a low-rank
primal solution, a special nearly low-rank form of all primal iterates is
imposed. To accommodate such a (restrictive) structure, the first order
optimality conditions have to be relaxed and are therefore approximated by
solving an auxiliary least-squares problem. The relaxed interior point
framework opens numerous possibilities how primal and dual approximated Newton
directions can be computed. In particular, it admits the application of both
the first- and the second-order methods in this context. The convergence of the
method is established. A prototype implementation is discussed and encouraging
preliminary computational results are reported for solving the
SDP-reformulation of matrix-completion problems