Convex Optimization 2026, 16 - Geometric Interpretation

A simple geometric interpretation of the dual function emerges from the description as the set 𝔾 = {(fi(x), ..., fm(x), h1(x), ..., hp(x), f0(x)) ∈ ℝ^m^ × ℝ^p^ × ℝ | x ∈ 𝔻} (Eq. 5.36), where p* = inf{t | (u, v, t) ∈ 𝔾, u ⪯ 0, v = 0} and (λ, ν, 1)^T^(u, v, t) = Σ^m^i=1 λiui + Σ^p^i=1νivi + t. Given g(λ, ν) = inf{(λ, ν, 1)^T^(u, v, t) | (u, v, t) ∈ 𝔾}, if the infimum is finite, then (λ, ν, 1)^T^(u, v, t) ≥ g(λ, ν) describes a supporting hyperplane to 𝔾. If A ⊆ ℝ^m^ × ℝ^p^ × ℝ, then A = 𝔾 + (ℝ^m^+ × {0} × ℝ+). A then describes an epigraph form of 𝔾, so that p* = inf{t | (0, 0, t) ∈ A} (5.3.1). Slater's constraint guarantees strong duality (5.3.2). The Lagrange duality for a problem without equality constraints connects to the scalarization method for the unconstrained multicriterion problem. In scalarization, there needs to be a positive scaling vector λ', to minimize λ'^T^F(x). These minimizers are Pareto-optimal. Without loss of generality, λ' = (λ, 1) can be assumed. In scalarization, λ'^T^F(x) = f0(x) + ∑^m^i=1 λifi(x). All Pareto-optimal points of convex multicriterion problem minimize λ'^T^F(x) for λ' nonnegative.