Introduction

Abstract

Let ℝⁿ be the n-dimensional Euclidean space with the usual inner product 〈·, ·〉, 〈x, y〉 = x ^⊤ y, and norm $\left\| x \right\| = \sqrt {\left\langle {x,x} \right\rangle }$. Let M ⊂ ℝⁿ, M ≠ ∅ and f : M → ℝ a function. An $\bar x \in M$ M having the property: $f\left( {\bar x} \right)f(x)$ holds for all x ∈ M, is called a global minimum for f . If $f\left( {\bar x} \right) \leqslant f(x)$ holds for all x ∈ M ∩ O, O being a ℝⁿ-neighborhood of $\bar x$, then $\bar x$ is called a local minimum for f. Obviously, a global minimum is also a local minimum. Local (global) maxima of f are defined to be local (global) minima for the function -f.