Convex Optimization · Problem 2.10

Convex functions — Boyd & Vandenberghe

Mathematics — Optimization · quadratic-inequality-convexity

Problem

Solution set of a quadratic inequality. Let C⊆RnC \subseteq \mathbf{R}^n be the solution set of a quadratic inequality, C={x∈Rn∣xTAx+bTx+c≤0},C = \{x \in \mathbf{R}^n \mid x^T A x + b^T x + c \leq 0\}, with A∈SnA \in \mathbf{S}^n, b∈Rnb \in \mathbf{R}^n, and c∈Rc \in \mathbf{R}. ((a)) Show that CC is convex if A⪰0A \succeq 0. ((b)) Show that the intersection of CC and the hyperplane defined by gTx+h=0g^T x + h = 0 (where g≠0g \neq 0) is convex if A+λggT⪰0A + \lambda g g^T \succeq 0 for some λ∈R\lambda \in \mathbf{R}. Are the converses of these statements true?

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