165 research outputs found
Coordinate shadows of semi-definite and Euclidean distance matrices
We consider the projected semi-definite and Euclidean distance cones onto a
subset of the matrix entries. These two sets are precisely the input data
defining feasible semi-definite and Euclidean distance completion problems. We
classify when these sets are closed, and use the boundary structure of these
two sets to elucidate the Krislock-Wolkowicz facial reduction algorithm. In
particular, we show that under a chordality assumption, the "minimal cones" of
these problems admit combinatorial characterizations. As a byproduct, we record
a striking relationship between the complexity of the general facial reduction
algorithm (singularity degree) and facial exposedness of conic images under a
linear mapping.Comment: 21 page
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
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