Convex Optimization · Problem 9.1

Unconstrained minimization — Boyd & Vandenberghe

Mathematics — Optimization · unconstrained-minimization-quadratic

Problem

Consider the general convex quadratic minimization problem minimize(1/2)xTPx+qTx+r,\text{minimize} \quad (1/2)x^T P x + q^T x + r, where P∈S+nP \in \mathbf{S}^n_+, q∈Rnq \in \mathbf{R}^n, and r∈Rr \in \mathbf{R}. Show that if Px⋆=−qPx^\star = -q does not have a solution, then the problem is unbounded below.

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