4,273 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
Alternating Least-Squares for Low-Rank Matrix Reconstruction
For reconstruction of low-rank matrices from undersampled measurements, we
develop an iterative algorithm based on least-squares estimation. While the
algorithm can be used for any low-rank matrix, it is also capable of exploiting
a-priori knowledge of matrix structure. In particular, we consider linearly
structured matrices, such as Hankel and Toeplitz, as well as positive
semidefinite matrices. The performance of the algorithm, referred to as
alternating least-squares (ALS), is evaluated by simulations and compared to
the Cram\'er-Rao bounds.Comment: 4 pages, 2 figure
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