Convex Optimization 2026, 06 - Dual Cones and Generalized Inequalities

A cone K defines a set K* = {y | x^Ty ≥ 0 ∀ x ∈ K}, which is the dual cone of K. It's always convex, regardless of K's convexity. y ∈ K* iff -y is the normal hyperplane supporting K at the origin. It's also always closed. K₁ ⊂ K₂ implies the inverse for their dual cones. If K has nonempty interior, K* is pointed. If the closure of K is pointed, then K* has nonempty interior. Define K** as the closure of the convex hull of K (2.6.1).

Assume K is convex and proper, then it induces a generalized inequality ⪯_K. K* is then also proper, and induces a generalized inequality ⪯_K*. x ⪯_K y iff λ^Tx ≤ λ^Ty ∀ λ ⪰_K* 0, and x ≺_K y iff λ^Tx < λ^Ty ∀ λ ⪰_K* 0, λ ≠ 0 (2.6.2).

The minimum element x of S w.r.t. ⪯_K is equivalent to x being the unique minimizer of λ^Tz for z in S, which implies that the hyperplane {z | λ^T(z - x) = 0} is a strict supporting hyperplane (2.6.3, p.54).