4,173 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
On a Problem of Weighted Low-Rank Approximation of Matrices
We study a weighted low rank approximation that is inspired by a problem of
constrained low rank approximation of matrices as initiated by the work of
Golub, Hoffman, and Stewart (Linear Algebra and Its Applications, 88-89(1987),
317-327). Our results reduce to that of Golub, Hoffman, and Stewart in the
limiting cases. We also propose an algorithm based on the alternating direction
method to solve our weighted low rank approximation problem and compare it with
the state-of-art general algorithms such as the weighted total alternating
least squares and the EM algorithm
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