We briefly review some elementary mathematical terminology and results on duality theory. All statements in this chapter are for sufficiently differentiable, real-valued functions, defined on real space.

Some reminders

Convex. A set X is convex if, and only if, all linear combinations of two arbitrary points belonging to the set X also belong to the set X. That is, if x ∈ X and ∈ X, then [λx + (1 − λ)] ∈ X for all scalars λ ∈ [0, 1].

Convex function. A function f defined on a convex set X is (strictly) convex if, and only if, the linear combination of the f -value of two (different) arbitrary points belonging to the set X (strictly) exceeds the f value of the linear combination itself. That is, if x ∈ X and ∈ X then λf(x) + (1 − λ)f ≥ f (λx + (1 − λ)) for all scalars λ ∈ [0, 1], with strict inequality for strictly convex functions. The function f is convex if, and only if, the Hessian matrix of second-order derivatives is positive semi-definite.

Concave function. A function f defined on a convex set X is (strictly) concave if, and only if, −f is (strictly) convex. Equivalently, a function f defined on a convex set X is (strictly) concave if, and only if, the linear combination of the f -value of two (different) arbitrary points belonging to the set X (strictly) falls short of the f -value of the linear combination itself.