Chapter 3 Geometry of convex functions - Meboo Publishing ...
Chapter 3 Geometry of convex functions - Meboo Publishing ...
Chapter 3 Geometry of convex functions - Meboo Publishing ...
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234 CHAPTER 3. GEOMETRY OF CONVEX FUNCTIONS<br />
3.6.0.0.1 Exercise. Epigraph sufficiency.<br />
Prove that converse: Given any (X, u), (Y , v)∈ epi f , if for all µ∈[0, 1]<br />
µ(X, u) + (1−µ)(Y , v)∈ epif holds, then f must be <strong>convex</strong>. <br />
Sublevel sets <strong>of</strong> a real <strong>convex</strong> function are <strong>convex</strong>. Likewise, corresponding<br />
to each and every ν ∈ R M<br />
L ν f {X ∈ dom f | f(X) ≼<br />
ν } ⊆ R p×k (537)<br />
R M +<br />
sublevel sets <strong>of</strong> a vector-valued <strong>convex</strong> function are <strong>convex</strong>. As for real<br />
<strong>convex</strong> <strong>functions</strong>, the converse does not hold. (Figure 73)<br />
To prove necessity <strong>of</strong> <strong>convex</strong> sublevel sets: For any X,Y ∈ L ν f we must<br />
show for each and every µ∈[0, 1] that µX + (1−µ)Y ∈ L ν f . By definition,<br />
f(µX + (1−µ)Y ) ≼<br />
R M +<br />
µf(X) + (1−µ)f(Y ) ≼<br />
R M +<br />
ν (538)<br />
<br />
When an epigraph (534) is artificially bounded above, t ≼ ν , then the<br />
corresponding sublevel set can be regarded as an orthogonal projection <strong>of</strong><br />
epigraph on the function domain.<br />
Sense <strong>of</strong> the inequality is reversed in (534), for concave <strong>functions</strong>, and we<br />
use instead the nomenclature hypograph. Sense <strong>of</strong> the inequality in (537) is<br />
reversed, similarly, with each <strong>convex</strong> set then called superlevel set.<br />
3.6.0.0.2 Example. Matrix pseud<strong>of</strong>ractional function.<br />
Consider a real function <strong>of</strong> two variables<br />
f(A, x) : S n × R n → R = x T A † x (539)<br />
on domf = S+× n R(A). This function is <strong>convex</strong> simultaneously in both<br />
variables when variable matrix A belongs to the entire positive semidefinite<br />
cone S n + and variable vector x is confined to range R(A) <strong>of</strong> matrix A .