Computable error bounds for an optimization problem with parallelepiped constraint (Q1899951)

A Kuhn-Tucker point \(z^* = (x^{*T}, u^{*T})^T \in {\mathbb{R}}^{n + m}\) of the particular programming problem \[ \text{minimize } f(x) \quad (x \in \widehat{D} \subseteq D \subseteq {\mathbb{R}}^n) \quad \text{subject to } c_i(x) \geq 0 \quad (i = 1, \dots, m = 2n) \] \[ \text{with } c_i(x) = \begin{cases} x_i - \widehat {x}_{i,I} & (i = 1, \dots, n)\\ \widehat{x}_{i - n,S} - x_{i - n} & (i = n + 1, \dots, m)\end{cases} \] is defined by \(x^* \in \widehat {D}\), \(c_i(x^*) \geq 0\) for \(i = 1, \dots, m\), \(u^*_i c_i (x^*) = 0\) for \(i = 1, \dots, m\), \(u^*_i \geq 0\) for \(i = 1, \dots, m\), and \(\nabla_x L(x^*, u^*) = 0\) for the Lagrange function \(L(x,u) = f(x) - \sum^m_{i = 1} u_i c_i (x)\). Here, \(\widehat {\mathbf x} = ([ \widehat{x}_{i,I}, \widehat {x}_{i,S}]) \subseteq \widehat{D}\) is a given interval vector, \(\widehat {D}\) is an open set and \(f \in C^1(D)\) has its range in \(\mathbb{R}\). Assuming that \(\widehat {\mathbf x}\) contains the first block component \(x^*\) of a Kuhn-Tucker point \(z^*\) it is shown how to construct interval enclosures \(\widehat {\mathbf u}_i\), \(i = 1, \dots, m\), for the corresponding Lagrange multipliers \(u^*_i\) which are the components of the second block \(u^*\) of \(z^*\). For \(D \subseteq {\mathbb{R}}^2\) and for more general constraints as above, polar coordinates are introduced in order to transform the problem into the previous one, thus illustrating the applicability of the method.