Mechanics of Continua and Structures
Theorem Let \(f\) be defined on the set \(\mathcal{D}\), and if \((x^{i;*})\) is a local minimizer that lies in the interior of the domain \(\mathcal{D}\) and \(f\) is continuously differentiable in an open neighborhood of \((x^{i;*})\), then
\[\begin{align} \frac{\partial f} {\partial x^{j}}[(x^{i;*})] & =0, \quad j=1,\ldots, n. \end{align}\]The above theorem is sometimes referred to as Fermat’s theorem.
Theorem (Th 2.3 Nocedal and Wright, p. 15) If \(\boldsymbol{x}^*\) is a local minimizer of \(f\) and \(\nabla^2 f\) exits and is continuous in an open neighborhood of \(\boldsymbol{x}^*\), then \(\nabla f[\boldsymbol{x}^{*}]=\boldsymbol{0}\) and \(\nabla^2 f[\boldsymbol{x}^{*}]\) is positive semi-definite.
Theorem (p. 115 FMEA) Let \((x^{i;*})\) be an interior stationary point in \(\mathcal{D}\). Assume that \(f: \mathcal{D}\subseteq \mathbb{R}^n \to \mathbb{R}\) is \(C^2\) on an open ball centered at \((x^{i;*})\). Let \(\Delta_{k}\) denote a generic principal minor of order \(k\) of the Hessain matrix. If \((x^{i,*})\) is a local minimizer only if \(\Delta_{k}[(x^{i;*})]\ge 0\) for all principal minors of order \(k=1,\ldots,n\).
Example For the case \(n=2\), the first order principal minors are $A$ and $C$. The second order principal minors are \(AC-B^2\). So, for a interior stationary point \((x^{1;*},x^{2;*})\) to be local minimizer, it is necessary that
\[\begin{align} A&\ge 0\\ C&\ge 0 \\ AC-B^2&\ge 0 \end{align}\]Theorem (Th 2.4 Nocedal and Wright, p. 16) Suppose that \(D_{ij} f[(x^{i})]\) is continuous in an open neighborhood of \((x^{i;*})\) and that \(D_i f[(x^{i;*})]=0\) and \((D_{ij} f[(x^{k,*})])\) is positive definite. Then \((x^{i,*})\) is a strict local minimizer of \(f\).
Theorem (FMEA, p. 112) Let \((x^{i,*})\) be an interior point of \(\mathcal{D}\) and a stationary point of \(f:\mathcal{D}\subseteq \mathbb{R}^2\to \mathbb{R}\).