MIT6_079F09_lec08

MIT6_079F09_lec08 - Convex Optimization Boyd Vandenberghe 8...

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Convex Optimization — Boyd & Vandenberghe 8. Geometric problems extremal volume ellipsoids centering classifcation placement and Facility location 8–1

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Minimum volume ellipsoid around a set owner-John ellipsoid of a set C : minimum volume ellipsoid E s.t. C ⊆ E parametrize E as E | Av + b 2 1 } ; w.l.o.g. assume A S n = { v ++ vol E is proportional to det A 1 ; to compute minimum volume ellipsoid, minimize (over A , b ) log det A 1 subject to sup v C Av + b 2 1 convex, but evaluating the constraint can be hard (for general C ) fnite set C = { x 1 , . . . , x m } : minimize (over A , b ) log det A 1 subject to Ax i + b 2 1 , i = 1 , . . . , m also gives L¨owner-John ellipsoid for polyhedron conv { x 1 , . . . , x m } Geometric problems 8–2
Maximum volume inscribed ellipsoid maximum volume ellipsoid E inside a convex set C R n parametrize E as E = { Bu + d | u 2 1 } ; w.l.o.g. assume B S n ++ vol E is proportional to det B ; can compute E by solving maximize log det B subject to sup b u b 2 1 I C ( Bu + d ) 0 (where I C ( x ) = 0 for x C and I C ( x ) = for x ±∈ C ) convex, but evaluating the constraint can be hard (for general C ) polyhedron { x | a i T x b i , i = 1 , . . . , m } : maximize log det B subject to Ba i 2 + a i T d b i , i = 1 , . . . , m (constraint follows from sup b u b 2 1 a T ( Bu + d ) = Ba i 2 + a T d ) i i Geometric problems 8–3

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Eﬃciency of ellipsoidal approximations C R n convex, bounded, with nonempty interior L¨owner-John ellipsoid, shrunk by a factor n , lies inside C maximum volume inscribed ellipsoid, expanded by a factor
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MIT6_079F09_lec08 - Convex Optimization Boyd Vandenberghe 8...

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