## 3 Variable-Thickness Domain
### Variable-Thickness Domain  
The surrogate model domain defined using an appropriate value of the thickness parameter \(T\) (cf. Figure 3) ensures incorporation of the majority of the optimum design manifold \(U_{F}(F)\). At the same time, maintaining fixed \(T\) for the entire objective space is not really a necessity. In particular, as \(U_{F}(F)\) coincides with \(s_{I}(F)\) (the image of \(F\) through the first-level surrogate) at all reference designs \(\mathbf{x}^{(i)}\), at these points, the required domain thickness is zero. Based on these observations, this section introduces a variable thickness domain as a mean to reduce the number of training data samples without compromising the model predictive power. The prerequisites are as follows:  
The thickness parameter should be a function of the objective vector \(\mathbf{f}\), i.e., \(T=T(\mathbf{f})\);  
\(T(\mathbf{f}^{(i)})=0\) for all \(j=1,\dots,p\), i.e., the domain thickness is zero for all reference designs \(\mathbf{x}^{(i)}=s_{I}(\mathbf{f}^{(i)})\);  
\(T(\mathbf{f})=T_{\max}\) (the \(T\)-value ensuring \(U_{F}(F)\subset X_{S}\)) for \(\mathbf{f}\) corresponding to objective space locations where the expected first-level surrogate inaccuracy (w.r.t. \(U_{F}(F)\)) is the highest;  
\(T(\mathbf{f})\) changes monotonically for intermediate locations. The first task is to determine vectors \(\mathbf{f}\) for which \(\|s_{I}(\mathbf{f})-U_{F}(\mathbf{f})\|\) is maximized. One option would be to identify locations maximizing the mean square error (MSE) of the first-level model, which can be determined directly from the formulation of the kriging interpolation surrogate [52]. Another approach is to make a reasonable assumption that the error is the highest near the geometrical centers of the simplexes \(S^{(i)}\), \(j=1,\dots,N_{S}\), obtained by triangulating the reference designs, cf. Figure 4. Here, Delaunay triangulation is employed [53] to avoid degenerated simplexes. The simplex centers in the objective space will be denoted as \(\mathbf{f}_{T}^{(i)}\); their parameter space counterparts are \(\mathbf{x}_{T}^{(i)}=s_{I}(\mathbf{f}^{(i)})\). Figure 3: The meaning of the domain thickness parameter \(T\), here, explained for two-dimensional parameter space. Increasing \(T\) enlarges the model domain which allows for encapsulating the optimum design manifold \(U_{F}(F)\). At the same time, larger domain requires more training samples to render reliable surrogate. The desired trade-off is to find the minimum value of \(T\) enabling incorporation of the majority of \(U_{F}(F)\) into \(X_{S}\). In order to develop the analytical form of the thickness function \(T(f)\), the location of any \(f\in F\) with respect to the simplexes \(S^{(j)}\) has to be established. Let \(x^{(j,k)}\), \(k=1,\ldots,N+1\), be the simplex vertices, where \(x^{(j,k)}\in\{x^{(1)},\ldots,x^{(p)}\}\) (the set of reference designs) for \(j=1,\ldots,N_{S}\), and \(k=1,\ldots,N+1\). For any \(j=1,\ldots,N_{S}\), the following expansion holds  
\[f=\sum_{k=1}^{N+1}a_{j,k}(f)x^{(j,k)} \tag{6}\]  
where \(a_{j,1}(f)+\ldots+a_{j,N+1}(f)=1\). Let \(J(f)\) be any \(j\) for which \(\min\{ka_{j,k}(f)\geq 0\}\). It follows that \(f\in H_{conv}(S^{(j)}(f))\), where \(H_{conv}(S)\) is the convex hull of \(S\). Note that \(a_{j,k}(f_{T}{}^{(i)})=(N+1)^{-1}\) for \(j=J(f_{T}{}^{(i)})\) and all \(i\) = 1, \(\ldots,N_{S}\) (recall that \(f_{T}{}^{(i)}\) were the simplex centers). At the same time, \(a_{j,k}(f^{(i)})=1\) for a certain \(k=1\), \(\ldots,N+1\), and zero for the remaining values of \(k\). Furthermore, for \(N>1\), the expansion coefficients assume \"intermediate\" distributions for \(f\) being at the boundaries between the simplexes. For example, if \(N=2\), then, for \(f\) allocated at the center of the edge between two adjacent simplexes, we have \(a_{j,k}=1/N\) for two out of three values of \(k\) and zero for the remaining one.