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

216 CHAPTER 3. GEOMETRY OF CONVEX FUNCTIONS
3.2 Conic origins of Lagrangian
The cone of convex functions, membership relation (476), provides a
foundation for what is known as a Lagrangian function. [246, p.398] [274]
Consider a conic optimization problem, for proper cone K and affine
subset A
minimize
x
f(x)
subject to g(x) ∈ K
h(x) ∈ A
A Cartesian product of convex functions remains convex, so we may write
[ fg
h
]
⎡
convex w.r.t ⎣ RM +
K
A
(482)
⎤
[ ] ⎡ ⎤ ⎡ ⎤
fg w
⎦ ⇔ [ w T λ T ν T ] convex ∀⎣
λ ⎦∈ G⎣ RM∗ +
K ∗ ⎦
h ν A ⊥
(483)
where A ⊥ is the normal cone to A . This holds because of equality for h
in convexity criterion (475) and because membership relation (429), given
point a∈A , becomes
h ∈ A ⇔ 〈ν , h − a〉=0 for all ν ∈ G(A ⊥ ) (484)
When A = 0, for example, a minimal set of generators G for A ⊥ is a superset
of the standard basis for R M (Example E.5.0.0.7) the ambient space of A .
A real Lagrangian arises from the scalar term in (483); id est,
L [w T λ T ν T ]
[ fg
h
]
= w T f + λ T g + ν T h (485)
3.3 Practical norm functions, absolute value
To some mathematicians, “all norms on R n are equivalent” [155, p.53];
meaning, ratios of different norms are bounded above and below by finite
predeterminable constants. But to statisticians and engineers, all norms
are certainly not created equal; as evidenced by the compressed sensing
revolution, begun in 2004, whose focus is predominantly 1-norm.
because each summand is considered to be an entry from a vector-valued function.