Chapter 3 Geometry of convex functions - Meboo Publishing ...

218 CHAPTER 3. GEOMETRY OF CONVEX FUNCTIONS
Over some convex set C given vector constant y or matrix constant Y
arg inf
x∈C ‖x − y‖ 2 = arg inf
x∈C ‖x − y‖2 2 (490)
arg inf
X∈ C ‖X − Y ‖ F = arg inf
X∈ C ‖X − Y ‖2 F (491)
are unconstrained convex quadratic problems. Equality does not hold for a
sum of norms. (5.4.2.3.2) Optimal solution is norm dependent: [59, p.297]
minimize
x∈R n ‖x‖ 1
subject to x ∈ C
≡
minimize 1 T t
x∈R n , t∈R n
subject to −t ≼ x ≼ t
x ∈ C
(492)
minimize
x∈R n ‖x‖ 2
subject to x ∈ C
≡
minimize
x∈R n , t∈R
subject to
t
[ tI x
x T t
x ∈ C
]
≽
S n+1
+
0
(493)
minimize ‖x‖ ∞
x∈R n
subject to x ∈ C
≡
minimize t
x∈R n , t∈R
subject to −t1 ≼ x ≼ t1
x ∈ C
(494)
In R n these norms represent: ‖x‖ 1 is length measured along a grid
(taxicab distance), ‖x‖ 2 is Euclidean length, ‖x‖ ∞ is maximum |coordinate|.
minimize
x∈R n ‖x‖ 1
subject to x ∈ C
≡
minimize 1 T (α + β)
α∈R n , β∈R n
subject to α,β ≽ 0
x = α − β
x ∈ C
(495)
These foregoing problems (486)-(495) are convex whenever set C is. Their
equivalence transformations make objectives smooth.