v2010.10.26 - Convex Optimization

3.5. EPIGRAPH, SUBLEVEL SET 243Sublevel sets of a convex real function are convex. Likewise, correspondingto each and every ν ∈ R ML ν f {X ∈ dom f | f(X) ≼ν } ⊆ R p×k (559)R M +sublevel sets of a convex vector-valued function are convex. As for convexreal functions, the converse does not hold. (Figure 75)To prove necessity of convex sublevel sets: For any X,Y ∈ L ν f we mustshow for each and every µ∈[0, 1] that µX + (1−µ)Y ∈ L ν f . By definition,f(µX + (1−µ)Y ) ≼R M +µf(X) + (1−µ)f(Y ) ≼R M +ν (560)When an epigraph (556) is artificially bounded above, t ≼ ν , then thecorresponding sublevel set can be regarded as an orthogonal projection ofepigraph on the function domain.Sense of the inequality is reversed in (556), for concave functions, and weuse instead the nomenclature hypograph. Sense of the inequality in (559) isreversed, similarly, with each convex set then called superlevel set.3.5.0.0.2 Example. Matrix pseudofractional function.Consider a real function of two variablesf(A, x) : S n × R n → R = x T A † x (561)on domf = S+× n R(A). This function is convex simultaneously in bothvariables when variable matrix A belongs to the entire positive semidefinitecone S n + and variable vector x is confined to range R(A) of matrix A .To explain this, we need only demonstrate that the function epigraph is