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Semidefinite Optimization and Convex Algebraic Geometry
Published on Jan 16, 20134877 Views
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Chapter list
Semide nite optimization and convex algebraic geometry00:00
This talk00:33
Convex sets: geometry vs. algebra01:38
The polyhedral case (1)04:14
The polyhedral case (2)05:37
Semide nite programming (SDP, LMIs)06:13
Example07:35
Semide nite representations (1)09:43
Semide nite representations (2)10:35
Semide nite representations (3)10:45
Known SDP-representable sets10:47
Existing results (1)11:36
Existing results (2)11:44
Liftings and projections (1)15:34
Liftings and projections (2)17:35
Example: k-ellipse18:46
5-ellipse20:23
Results on exact SDP representations (1)21:06
Results on exact SDP representations (2)21:09
Sum of squares24:30
Checking the SOS condition (1)26:04
Checking the SOS condition (2)26:53
SOS Example26:57
From feasibility to optimization (1)29:48
From feasibility to optimization (2)30:34
Convex hulls of algebraic varieties (1)31:37
Convex hulls of algebraic varieties (2)32:52
Why?33:50
Polynomial optimization (1)34:14
Polynomial optimization (2)34:48
Univariate case35:23
Polynomial optimization (1)36:35
Polynomial optimization (2)37:22
A geometric interlude (1)38:11
A geometric interlude (2)38:36
A geometric interlude (3)39:21
Geometric interpretation39:49
A "polar" viewpoint41:04
Example: orthogonal matrices42:39
Minimum rank and convex relaxations (1)42:42
Minimum rank and convex relaxations (2)43:09
Convex hulls and nuclear norm (1)43:27
Convex hulls and nuclear norm (2)44:14
Rank, sparsity, and beyond: atomic norms44:43
Connections48:43
Algebraic structure49:40
Numerical structure50:41
Summary (1)51:02
Summary (2)52:23
Summary (3)52:31