Convex Optimization 2026, 09 - Log-Concave and Log-Convex Functions

A function f is logarithmically concave (log-concave) if f(x) > 0 for all x ∈ dom f and log f is concave. f is log-convex iff 1/f is log-concave. We set log(f = 0) to -∞ for convenience. If f is convex, then e^f is convex (3.5.1). If f is twice differentiable and dom f is convex, then

f is log-convex iff for all x ∈ dom f, f(x)∇²f(x) ⪰ ∇f(x)∇f(x)^T. For log-concavity, turn the relation around (3.5.2.1). Log-convexity and log-concavity are closed under multiplication and positive scaling (3.5.2.2). If f: ℝⁿ × ℝ^m → ℝ is log-concave, then g(x) = ∫ f(x, y) dy is log-concave in x on ℝⁿ (3.5.2.3).

A proper cone K with associated generalized inequality ⪯_K, f: ℝⁿ → ℝ is K-nondecreasing if x ⪯_K y ⇒ f(x) ≤ f(y), and K-increasing if x ⪯_K y, x ≠ y ⇒ f(x) < f(y). Analogously, K-nonincreasing and K-decreasing. A differentiable f with dom f convex is K-nondecreasing iff ∇f(x) ⪰_K* 0 ∀ x ∈ dom f (3.6.1). f is K-convex if ∀ x, y, 0 ≤ θ ≤ 1: f(θx + (1 - θ)y) ⪯_K θf(x) + (1 - θ)f(y). It's strictly K-convex for ≺_K. A function is K-convex iff for all w ⪰_K* the real-valued w^Tf is convex. It's strictly so if this function is strictly convex. A differentiable function is K-convex iff its domain is convex and ∀ x, y ∈ dom f: f(y) ⪰_K f(x) + Df(x)(y - x). Strictly so for ≻_K (3.6.2).