v2006.03.09 - Convex Optimization

3.1. CONVEX FUNCTION 209Sublevel sets of a matrix-valued convex function (confer (408))L V g ∆ = {X ∈ dom g | g(X) ≼V } ⊆ R p×k (437)S M +are convex. There is no converse.3.1.2.3 Second-order convexity condition, matrix-valued function3.1.2.3.1 Theorem. Line theorem. [38,3.1.1]Matrix-valued function g(X) : R p×k →S M is convex in X if and only if itremains convex on the intersection of any line with its domain. ⋄Now we assume a twice differentiable function and drop the subscript S M +from an inequality when apparent.3.1.2.3.2 Definition. Differentiable convex matrix-valued function.Matrix-valued function g(X) : R p×k →S M is convex in X iff domg is anopen convex set, and its second derivative g ′′ (X+ t Y ) : R→S M is positivesemidefinite on each point along every line X+ t Y that intersects domg ;id est, iff for each and every X, Y ∈ R p×k such that X+ t Y ∈ domg oversome open interval of t ∈ Rd 2g(X+ t Y ) ≽ 0 (438)dt2 Similarly, ifd 2g(X+ t Y ) ≻ 0 (439)dt2 then g is strictly convex; the converse is generally false. [38,3.1.4] 3.8 △3.8 Quadratic forms constitute a notable exception where the strict-case converse isreliably true.