v2009.01.01 - Convex Optimization
v2009.01.01 - Convex Optimization v2009.01.01 - Convex Optimization
240 CHAPTER 3. GEOMETRY OF CONVEX FUNCTIONS Figure 69: Iconic unimodal differentiable quasiconvex function of two variables graphed in R 2 × R on some open disc in R 2 . Note reversal of curvature in direction of gradient.
3.3. QUASICONVEX 241 Unlike convex functions, quasiconvex functions are not necessarily continuous; e.g., quasiconcave rank(X) on S M + (2.9.2.6.2) and card(x) on R M + . Although insufficient for convex functions, convexity of each and every sublevel set serves as a definition of quasiconvexity: 3.3.0.0.2 Definition. Quasiconvex multidimensional function. Scalar-, vector-, or matrix-valued function g(X) : R p×k →S M is a quasiconvex function of matrix X iff dom g is a convex set and the sublevel set corresponding to each and every S ∈ S M L S g = {X ∈ domg | g(X) ≼ S } ⊆ R p×k (567) is convex. Vectors are compared with respect to the nonnegative orthant R M + while matrices are with respect to the positive semidefinite cone S M + . Convexity of the superlevel set corresponding to each and every S ∈ S M , likewise L S g = {X ∈ domg | g(X) ≽ S } ⊆ R p×k (579) is necessary and sufficient for quasiconcavity. △ 3.3.0.0.3 Exercise. Nonconvexity of matrix product. Consider real function f on a positive definite domain f(X) = tr(X 1 X 2 ) , X ∆ = [ X1 X 2 ] ∈ domf ∆ = [ rel int S N + rel int S N + ] (580) with superlevel sets L s f = {X ∈ domf | 〈X 1 , X 2 〉 ≥ s } (581) Prove: f(X) is not quasiconcave except when N = 1, nor is it quasiconvex unless X 1 = X 2 . 3.3.1 bilinear Bilinear function 3.20 x T y of vectors x and y is quasiconcave (monotonic) on the entirety of the nonnegative orthants only when vectors are of dimension 1. 3.20 Convex envelope of bilinear functions is well known. [2]
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240 CHAPTER 3. GEOMETRY OF CONVEX FUNCTIONS<br />
Figure 69: Iconic unimodal differentiable quasiconvex function of two<br />
variables graphed in R 2 × R on some open disc in R 2 . Note reversal of<br />
curvature in direction of gradient.
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