Convex Functions |

"The link between convex sets and convex functions is via the epigraph: A function is convex if and only if its epigraph is a convex set." Any convex real function f(X) has unique minimum value over any convex subset of its domain. Yet solution to some convex optimization problem is, in general, not unique; It is customary to consider only a real function for the objective of a convex optimization problem because vector- or matrix-valued functions can introduce ambiguity into the optimal value of the objective. Quadratic real functions x^T P x + q^T x + r characterized by a symmetric positive definite matrix P are strictly convex. The vector 2-norm squared |x|^2 (Euclidean norm squared) and Frobenius norm squared |X|_F^2, for example, are strictly convex functions of their respective argument (each absolute norm is convex but not strictly convex). |