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Surrogate-assisted Bounding-Box approach for optimization problems with tunable objectives fidelity

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Abstract

In this work, we present a novel framework to perform multi-objective optimization when considering expensive objective functions computed with tunable fidelity. This case is typical in many engineering optimization problems, for example with simulators relying on Monte Carlo or on iterative solvers. The objectives can only be estimated, with an accuracy depending on the computational resources allocated by the user. We propose here a heuristic for allocating the resources efficiently to recover an accurate Pareto front at low computational cost. The approach is independent from the choice of the optimizer and overall very flexible for the user. The framework is based on the concept of Bounding-Box, where the estimation error can be regarded with the abstraction of an interval (in one-dimensional problems) or a product of intervals (in multi-dimensional problems) around the estimated value, naturally allowing the computation of an approximated Pareto front. This approach is then supplemented by the construction of a surrogate model on the estimated objective values. We first study the convergence of the approximated Pareto front toward the true continuous one under some hypotheses. Secondly, a numerical algorithm is proposed and tested on several numerical test-cases.

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Authors and Affiliations

  1. Team DeFI, Inria Saclay Île-de-France, 91120, Palaiseau, France

    M. Rivier & P. M. Congedo

  2. ArianeGroup, 33185, Le Haillan Cedex, France

    M. Rivier

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  1. M. Rivier
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  2. P. M. Congedo
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Appendices

Proof 1

Proof

The proof here is trivial. By definition,

$$\begin{aligned}&{\mathcal {X}}_{{\widetilde{{\mathcal {P}}}}}^{{\varvec{f}}}\Big (\big \{{\varvec{x}}_i\big \}_{i=1}^N\Big ) = \big \{{\varvec{x}}_i,\ i \in \llbracket 1,N \rrbracket \ |\ \text{ non-domination } \text{ condition }\big \} \\&\quad \subseteq \big \{{\varvec{x}}_i,\ i \in \llbracket 1,N \rrbracket \big \} \equiv \big \{{\varvec{x}}_i\big \}_{i=1}^N. \end{aligned}$$

The same can be said for the boxed Pareto optima. \(\square \)

Proof 2

Proof

For \({\varvec{y}}\notin {\mathcal {X}}_{{\widetilde{{\mathcal {P}}}}}^{{\varvec{f}}}\Big (\big \{{\varvec{x}}_i\big \}_{i=1}^N\Big )\), let us assume that \(\not \exists {\varvec{y}}' \in {\mathcal {X}}_{{\widetilde{{\mathcal {P}}}}}^{{\varvec{f}}}\Big (\big \{{\varvec{x}}_i\big \}_{i=1}^N\Big ),\ {\varvec{y}}' \succ {\varvec{y}}\), then \({\varvec{y}}\in {\mathcal {X}}_{{\widetilde{{\mathcal {P}}}}}^{{\varvec{f}}}\Big ({\mathcal {X}}_{{\widetilde{{\mathcal {P}}}}}^{{\varvec{f}}}\Big (\big \{{\varvec{x}}_i\big \}_{i=1}^N\Big ) \cup {\varvec{y}}\Big ) = {\mathcal {X}}_{{\widetilde{{\mathcal {P}}}}}^{{\varvec{f}}}\Big (\big \{{\varvec{x}}_i\big \}_{i=1}^N\Big )\) from Eq. 6, which proves the first implication by contradiction. The second implication is immediate from the definition of \({\mathcal {X}}_{{\widetilde{{\mathcal {P}}}}}^{{\varvec{f}}}\). \(\square \)

Proof 3

Proof

By using the explicit definition of the Pareto front as in Eq. 4, the proof is immediate as Assumption 1 gives \(\forall {\varvec{x}}\in {\mathcal {X}},\forall j \in \llbracket 1,N \rrbracket ,\ f_j({\varvec{x}}) \in \big [{\widetilde{f}}_j({\varvec{x}}) - {\overline{\varepsilon }}_j({\varvec{x}}),\ {\widetilde{f}}_j({\varvec{x}}) + {\overline{\varepsilon }}_j({\varvec{x}})\big ]\). Hence, \(\forall {\varvec{x}}_i \in {\mathcal {X}}_{{\widetilde{{\mathcal {P}}}}}^{{\varvec{f}}}\Big (\big \{{\varvec{x}}_i\big \}_{i=1}^N\Big )\):

$$\begin{aligned}&\forall k \in \llbracket 1,N \rrbracket , \exists j \in \llbracket 1,m \rrbracket ,\ \pm f_j({\varvec{x}}_i)< \pm f_j({\varvec{x}}_k), \\&\quad \pm {\widetilde{f}}_j({\varvec{x}}_i) - {\overline{\varepsilon }}_j({\varvec{x}}_i) < \pm {\widetilde{f}}_j({\varvec{x}}_k) + {\overline{\varepsilon }}_j({\varvec{x}}_k). \end{aligned}$$

Therefore, \({\varvec{x}}_i \in {\mathcal {X}}_{{\widetilde{{\mathcal {P}}}}_{{\mathcal {B}}}}^{{\varvec{f}}}\Big (\big \{\big ({\varvec{x}}_i,\overline{{\varvec{\varepsilon }}}({\varvec{x}}_i)\big )\big \}_{i=1}^N\Big )\), which ends the proof. \(\square \)

Proof 4

Proof

For proving this, mathematical induction can be used.

Let us assume that \(\exists l \in {\mathbb {N}}_+, \text{ so } \text{ that } {\mathcal {X}}_{{\widetilde{{\mathcal {P}}}}}^{{\varvec{f}}}\Big (\big \{{\varvec{x}}_i\big \}_{i=1}^N\Big ) \subseteq {\widetilde{{\mathcal {X}}}}_{{\widetilde{{\mathcal {P}}}}}^{{\varvec{f}},k}\). Then \(\forall {\varvec{x}}\in {\widetilde{{\mathcal {X}}}}_{{\widetilde{{\mathcal {P}}}}}^{{\varvec{f}},k}\):

  • if \({\varvec{x}}\in {\mathcal {X}}_{{\widetilde{{\mathcal {P}}}}}^{{\varvec{f}}}\Big (\big \{{\varvec{x}}_i\big \}_{i=1}^N\Big ),\ \not \exists {\varvec{y}}\in {\mathcal {X}}, \text{ so } \text{ that } {\varvec{y}}\succ {\varvec{x}}\). Hence, \(\not \exists {\varvec{y}}\in {\widetilde{{\mathcal {X}}}}_{{\widetilde{{\mathcal {P}}}}}^{{\varvec{f}},k} \subseteq {\mathcal {X}}, \text{ so } \text{ that } {\varvec{y}}\succ {\varvec{x}}\), and therefore \({\varvec{x}}\in {\mathcal {X}}_{{\widetilde{{\mathcal {P}}}}}^{{\varvec{f}}}\big ({\widetilde{{\mathcal {X}}}}_{{\widetilde{{\mathcal {P}}}}}^{{\varvec{f}},k}\big );\)

  • if \({\varvec{x}}\notin {\mathcal {X}}_{{\widetilde{{\mathcal {P}}}}}^{{\varvec{f}}}\Big (\big \{{\varvec{x}}_i\big \}_{i=1}^N\Big )\), with Proposition 2, \(\exists {\varvec{y}}\in {\mathcal {X}}_{{\widetilde{{\mathcal {P}}}}}^{{\varvec{f}}}\Big (\big \{{\varvec{x}}_i\big \}_{i=1}^N\Big ) \subseteq {\widetilde{{\mathcal {X}}}}_{{\widetilde{{\mathcal {P}}}}}^{{\varvec{f}},k}, \text{ so } \text{ that } {\varvec{y}}\succ {\varvec{x}}\), therefore, \({\varvec{x}}\notin {\mathcal {X}}_{{\widetilde{{\mathcal {P}}}}}^{{\varvec{f}}}\big ({\widetilde{{\mathcal {X}}}}_{{\widetilde{{\mathcal {P}}}}}^{{\varvec{f}},k}\big ).\)

which means that:

$$\begin{aligned} {\mathcal {X}}_{{\widetilde{{\mathcal {P}}}}}^{{\varvec{f}}}\Big (\big \{{\varvec{x}}_i\big \}_{i=1}^N\Big ) = {\mathcal {X}}_{{\widetilde{{\mathcal {P}}}}}^{{\varvec{f}}}\big ({\widetilde{{\mathcal {X}}}}_{{\widetilde{{\mathcal {P}}}}}^{{\varvec{f}},k}\big ). \end{aligned}$$
(13)

Finally, Lemma 1 gives \({\mathcal {X}}_{{\widetilde{{\mathcal {P}}}}}^{{\varvec{f}}}\big ({\widetilde{{\mathcal {X}}}}_{{\widetilde{{\mathcal {P}}}}}^{{\varvec{f}},k}\big ) \subseteq {\mathcal {X}}_{{\widetilde{{\mathcal {P}}}}_{{\mathcal {B}}}}^{{\varvec{f}}}\Big (\Big \{\Big ({\widetilde{{\mathcal {X}}}}_{{\widetilde{{\mathcal {P}}}}_i}^{{\varvec{f}},k},\overline{{\varvec{\varepsilon }}}^{k}\Big ({\widetilde{{\mathcal {X}}}}_{{\widetilde{{\mathcal {P}}}}_i}^{{\varvec{f}},k}\Big )\Big )\Big \}_{i=1}^N \Big ) = {\widetilde{{\mathcal {X}}}}_{{\widetilde{{\mathcal {P}}}}}^{{\varvec{f}},k+1}\), which ends the inductive step of the proof, yielding \({\mathcal {X}}_{{\widetilde{{\mathcal {P}}}}}^{{\varvec{f}}}\Big (\big \{{\varvec{x}}_i\big \}_{i=1}^N\Big ) \subseteq {\widetilde{{\mathcal {X}}}}_{{\widetilde{{\mathcal {P}}}}}^{{\varvec{f}},k+1}\).

Of course, \({\mathcal {X}}_{{\widetilde{{\mathcal {P}}}}}^{{\varvec{f}}}\Big (\big \{{\varvec{x}}_i\big \}_{i=1}^N\Big ) \subseteq \big \{{\varvec{x}}_i\big \}_{i=1}^N = {\widetilde{{\mathcal {X}}}}_{{\widetilde{{\mathcal {P}}}}}^{{\varvec{f}},0}\), therefore, the mathematical induction proves that the robustness inclusion is verified. \(\square \)

Proof 5

Proof

The triangle inequality gives :

$$\begin{aligned}&d_H\Big ({\widetilde{{\mathcal {P}}}}_c\big ({\widetilde{{\varvec{f}}}}^{l_k}\big ({\widetilde{{\mathcal {X}}}}_{{\widetilde{{\mathcal {P}}}}}^{{\varvec{f}},k}\big )\big ),\ {\mathcal {P}}_c\Big ) \le d_H\Big ({\widetilde{{\mathcal {P}}}}_c\big ({\widetilde{{\varvec{f}}}}^{l_k}\big ({\widetilde{{\mathcal {X}}}}_{{\widetilde{{\mathcal {P}}}}}^{{\varvec{f}},k}\big )\big ),{\widetilde{{\mathcal {P}}}}_c\big ({\varvec{f}}\big ({\widetilde{{\mathcal {X}}}}_{{\widetilde{{\mathcal {P}}}}}^{{\varvec{f}},k}\big )\big )\Big )\nonumber \\&\quad + d_H\Big ({\widetilde{{\mathcal {P}}}}_c\big ({\varvec{f}}\big ({\widetilde{{\mathcal {X}}}}_{{\widetilde{{\mathcal {P}}}}}^{{\varvec{f}},k}\big )\big ),\ {\mathcal {P}}_c\Big ). \end{aligned}$$
(14)

Assumption 1 implies that \(\forall {\varvec{x}}\in {\widetilde{{\mathcal {X}}}}_{{\widetilde{{\mathcal {P}}}}}^{{\varvec{f}},k},\ d_{\infty }\big ({\varvec{f}}({\varvec{x}}),{\widetilde{{\varvec{f}}}}^{l_k}({\varvec{x}})\big ) \le \underset{j}{\max }\ \overline{{\varvec{\varepsilon }}}_j^{l_k}({\varvec{x}})\).

Now, let us suppose that \(\exists {\varvec{a}}\in {\widetilde{{\mathcal {P}}}}_c\big ({\varvec{f}}\big ({\widetilde{{\mathcal {X}}}}_{{\widetilde{{\mathcal {P}}}}}^{{\varvec{f}},k}\big )\big )\), so that \(d_{\infty }\Big ({\varvec{a}}, {\widetilde{{\mathcal {P}}}}_c\big ({\widetilde{{\varvec{f}}}}^{l_k}\big ({\widetilde{{\mathcal {X}}}}_{{\widetilde{{\mathcal {P}}}}}^{{\varvec{f}},k}\big )\big )\Big ) > \underset{(i,j)}{\max }\ \overline{{\varvec{\varepsilon }}}_j^{l_k}\big ({\widetilde{{\mathcal {X}}}}_{{\widetilde{{\mathcal {P}}}}_i}^{{\varvec{f}},k}\big )\).

This means that \({\widetilde{{\mathcal {P}}}}_c\big ({\widetilde{{\varvec{f}}}}^{l_k}\big ({\widetilde{{\mathcal {X}}}}_{{\widetilde{{\mathcal {P}}}}}^{{\varvec{f}},k}\big )\big )\cap \big ({\varvec{a}},\overline{{\varvec{\varepsilon }}}_{max}\big ) = \emptyset \) with \(\overline{{\varvec{\varepsilon }}}_{max}\) being the m-dimensional vector where each component is equal to \(\underset{(i,j)}{\max }\ \overline{{\varvec{\varepsilon }}}_j^{l_k}\big ({\widetilde{{\mathcal {X}}}}_{{\widetilde{{\mathcal {P}}}}_i}^{{\varvec{f}},k}\big )\).

Therefore, from Definition 8, (i) either \(\exists i \in \llbracket 1,N \rrbracket \), then \(\big ({\widetilde{{\varvec{f}}}}^{l_k}\big ({\widetilde{{\mathcal {X}}}}_{{\widetilde{{\mathcal {P}}}}_i}^{{\varvec{f}},k}\big ),0\big ) \underset{{\mathcal {B}}}{\succ \succ } \big ({\varvec{a}},\overline{{\varvec{\varepsilon }}}_{max}\big )\) or (ii) \(\forall {\varvec{a}}' \in \big ({\varvec{a}},\overline{{\varvec{\varepsilon }}}_{max}\big ), \not \exists i \in \llbracket 1,N \rrbracket \), so that \({\widetilde{{\varvec{f}}}}^{l_k}\big ({\widetilde{{\mathcal {X}}}}_{{\widetilde{{\mathcal {P}}}}_i}^{{\varvec{f}},k}\big ) \succ {\varvec{a}}'\).

  • In the first case i), the dominance can be formulated as follows: \(\exists j \in \llbracket 1,N \rrbracket \), so that \(\big ({\widetilde{{\varvec{f}}}}^{l_k}\big ({\widetilde{{\mathcal {X}}}}_{{\widetilde{{\mathcal {P}}}}_j}^{{\varvec{f}},k}\big ),\overline{{\varvec{\varepsilon }}}_{max}\big ) \underset{{\mathcal {B}}}{\succ \succ } \big ({\varvec{a}},0\big )\) and as \({\varvec{a}}\in {\widetilde{{\mathcal {P}}}}_c\big ({\varvec{f}}\big ({\widetilde{{\mathcal {X}}}}_{{\widetilde{{\mathcal {P}}}}}^{{\varvec{f}},k}\big )\big ), \not \exists i \in \llbracket 1,N \rrbracket , {\varvec{f}}\big ({\widetilde{{\mathcal {X}}}}_{{\widetilde{{\mathcal {P}}}}_i}^{{\varvec{f}},k}\big ) \succ \succ {\varvec{a}}\). Therefore, it can be inferred that \(\not \exists i \in \llbracket 1,N \rrbracket , {\varvec{f}}\big ({\widetilde{{\mathcal {X}}}}_{{\widetilde{{\mathcal {P}}}}_i}^{{\varvec{f}},k}\big ) \in \big ({\widetilde{{\varvec{f}}}}^{l_k}\big ({\widetilde{{\mathcal {X}}}}_{{\widetilde{{\mathcal {P}}}}_j}^{{\varvec{f}},k}\big ),\overline{{\varvec{\varepsilon }}}_{max}\big )\). However, this would mean \(\exists {\varvec{x}}\in {\widetilde{{\mathcal {X}}}}_{{\widetilde{{\mathcal {P}}}}}^{{\varvec{f}},k},\ d_{\infty }\big ({\varvec{f}}({\varvec{x}}),{\widetilde{{\varvec{f}}}}^{l_k}({\varvec{x}})\big )> \overline{{\varvec{\varepsilon }}}_{max_i} > \underset{j}{\max }\ \overline{{\varvec{\varepsilon }}}_j^{l_k}({\varvec{x}})\), which is contradictory with Assumption 1.

  • The second case ii) implies that \(\exists {\varvec{b}}\in {\widetilde{{\mathcal {P}}}}_c\big ({\widetilde{{\varvec{f}}}}^{l_k}\big ({\widetilde{{\mathcal {X}}}}_{{\widetilde{{\mathcal {P}}}}}^{{\varvec{f}},k}\big )\big )\), so that \(\big ({\varvec{a}},\overline{{\varvec{\varepsilon }}}_{max}\big ) \underset{{\mathcal {B}}}{\succ \succ } \big ({\varvec{b}},0\big )\). However, \(\exists j \in \llbracket 1,N \rrbracket \), so that \({\varvec{f}}\big ({\widetilde{{\mathcal {X}}}}_{{\widetilde{{\mathcal {P}}}}_j}^{{\varvec{f}},k}\big ) \succ {\varvec{a}}\). Therefore, it follows \(\big ({\varvec{f}}\big ({\widetilde{{\mathcal {X}}}}_{{\widetilde{{\mathcal {P}}}}_j}^{{\varvec{f}},k}\big ),\overline{{\varvec{\varepsilon }}}_{max}\big ) \underset{{\mathcal {B}}}{\succ \succ } \big ({\varvec{b}},0\big )\). As \({\varvec{b}}\in {\widetilde{{\mathcal {P}}}}_c\big ({\widetilde{{\varvec{f}}}}^{l_k}\big ({\widetilde{{\mathcal {X}}}}_{{\widetilde{{\mathcal {P}}}}}^{{\varvec{f}},k}\big )\big )\), from Definition 8, it follows that \(\not \exists i \in \llbracket 1,N \rrbracket \), so that \({\widetilde{{\varvec{f}}}}^{l_k}\big ({\widetilde{{\mathcal {X}}}}_{{\widetilde{{\mathcal {P}}}}_i}^{{\varvec{f}},k}\big ) \succ \succ {\varvec{b}}\), hence, \(\not \exists i \in \llbracket 1,N \rrbracket , {\widetilde{{\varvec{f}}}}^{l_k}\big ({\widetilde{{\mathcal {X}}}}_{{\widetilde{{\mathcal {P}}}}_i}^{{\varvec{f}},k}\big ) \in \big ({\varvec{f}}\big ({\widetilde{{\mathcal {X}}}}_{{\widetilde{{\mathcal {P}}}}_j}^{{\varvec{f}},k}\big ),\overline{{\varvec{\varepsilon }}}_{max}\big )\), which contradicts again Assumption 1.

Hence, we prove by contradiction that \(\forall {\varvec{a}}\in {\widetilde{{\mathcal {P}}}}_c\big ({\varvec{f}}\big ({\widetilde{{\mathcal {X}}}}_{{\widetilde{{\mathcal {P}}}}}^{{\varvec{f}},k}\big )\big )\), it follows that \(d_{\infty }\Big ({\varvec{a}}, {\widetilde{{\mathcal {P}}}}_c\big ({\widetilde{{\varvec{f}}}}^{l_k}\big ({\widetilde{{\mathcal {X}}}}_{{\widetilde{{\mathcal {P}}}}}^{{\varvec{f}},k}\big )\big )\Big ) \le \underset{(i,j)}{\max }\ \overline{{\varvec{\varepsilon }}}_j^{l_k}\big ({\widetilde{{\mathcal {X}}}}_{{\widetilde{{\mathcal {P}}}}_i}^{{\varvec{f}},k}\big )\). This statement holds also when inverting the Pareto front continuous sets (real and approximated), and this can proved in the same way. As a consequence, the Hausdorff distance can be written as follows:

$$\begin{aligned} d_H\Big ({\widetilde{{\mathcal {P}}}}_c\big ({\widetilde{{\varvec{f}}}}^{l_k}\big ({\widetilde{{\mathcal {X}}}}_{{\widetilde{{\mathcal {P}}}}}^{{\varvec{f}},k}\big )\big ),{\widetilde{{\mathcal {P}}}}_c\big ({\varvec{f}}\big ({\widetilde{{\mathcal {X}}}}_{{\widetilde{{\mathcal {P}}}}}^{{\varvec{f}},k}\big )\big )\Big ) \le \underset{(i,j)}{\max }\ \overline{{\varvec{\varepsilon }}}_j^{l_k}\big ({\widetilde{{\mathcal {X}}}}_{{\widetilde{{\mathcal {P}}}}_i}^{{\varvec{f}},k}\big ). \end{aligned}$$

Hence, Assumption 2 implies that:

$$\begin{aligned} \lim _{k \rightarrow \infty } d_H\Big ({\widetilde{{\mathcal {P}}}}_c\big ({\widetilde{{\varvec{f}}}}^{l_k}\big ({\widetilde{{\mathcal {X}}}}_{{\widetilde{{\mathcal {P}}}}}^{{\varvec{f}},k}\big )\big ),{\widetilde{{\mathcal {P}}}}_c\big ({\varvec{f}}\big ({\widetilde{{\mathcal {X}}}}_{{\widetilde{{\mathcal {P}}}}}^{{\varvec{f}},k}\big )\big )\Big ) = 0. \end{aligned}$$
(15)

Let us focus now on the second part of the sum in Eq. 14.

Of course, \(\forall {\varvec{a}}\in {\widetilde{{\mathcal {P}}}}_c\big ({\varvec{f}}\big ({\widetilde{{\mathcal {X}}}}_{{\widetilde{{\mathcal {P}}}}}^{{\varvec{f}},k}\big )\big ), \exists {\varvec{a}}' \in {\mathcal {P}}_c\), so that \({\varvec{a}}' \succ {\varvec{a}}\) (or \({\varvec{a}}' = {\varvec{a}}\)). Moreover, \(\forall {\varvec{b}}\in {\mathbb {R}}^m\) such that \(\exists {\varvec{a}}' \in {\mathcal {P}},\ {\varvec{a}}' \succ \succ {\varvec{b}}\), Assumption 3 with Theorem 1 provides evidence that the recursive discrete efficient set converges toward the continuous real one and that this efficient set is included in \({\widetilde{{\mathcal {X}}}}_{{\widetilde{{\mathcal {P}}}}}^{{\varvec{f}},k}\). In other words, \(\forall k \in {\mathbb {N}}, \exists M \in {\mathbb {N}}^*, \exists i \in \llbracket 1,M \rrbracket , s.t. \forall j \in \llbracket 1,m \rrbracket , \big |a'_j - f_j\big ({\widetilde{{\mathcal {X}}}}_{{\widetilde{{\mathcal {P}}}}_i}^{{\varvec{f}},k}\big )\big | < \big |a'_j - b_j\big |\). Thus, \(\forall k \in {\mathbb {N}}, \exists M \in {\mathbb {N}}^*, \exists i \in \llbracket 1,M \rrbracket , {\varvec{f}}\big ({\widetilde{{\mathcal {X}}}}_{{\widetilde{{\mathcal {P}}}}_i}^{{\varvec{f}},k}\big ) \succ \succ {\varvec{b}}\). Hence, \({\widetilde{{\mathcal {P}}}}_c\) is always dominated by \({\mathcal {P}}_c\) and any element dominated by \({\mathcal {P}}_c\) is dominated by \({\widetilde{{\mathcal {P}}}}_c\) with a sufficient number of points. From Definition 8, it can be deduced that:

$$\begin{aligned} \lim _{N \rightarrow \infty } d_H\Big ({\widetilde{{\mathcal {P}}}}_c\big ({\varvec{f}}\big ({\widetilde{{\mathcal {X}}}}_{{\widetilde{{\mathcal {P}}}}}^{{\varvec{f}},k}\big )\big ),\ {\mathcal {P}}_c\Big ) = 0. \end{aligned}$$
(16)

Finally, by combining Eqs. 14, 15 and 16, it comes:

$$\begin{aligned} \lim _{(N,l) \rightarrow (+\infty ,+\infty )}d_H\Big ({\widetilde{{\mathcal {P}}}}_c\big ({\widetilde{{\varvec{f}}}}^{l_k}\big ({\widetilde{{\mathcal {X}}}}_{{\widetilde{{\mathcal {P}}}}}^{{\varvec{f}},k}\big )\big ),\ {\mathcal {P}}_c\Big ) = 0, \end{aligned}$$

which ends the proof. \(\square \)

Proof 6

Proof

The proof is straightforward and comes from the following inequalities:

If \(\overline{{\varvec{\varepsilon }}}_{SA}^t({\varvec{x}}_i) > {\varvec{s}}_1\),

$$\begin{aligned} \big |{\varvec{f}}_{opt}({\varvec{x}}_i) - {\varvec{f}}({\varvec{x}}_i)\big | = \big |{\widetilde{{\varvec{f}}}}^l({\varvec{x}}_i) - {\varvec{f}}({\varvec{x}}_i)\big | \le \overline{{\varvec{\varepsilon }}}^l({\varvec{x}}_i). \end{aligned}$$

Else,

$$\begin{aligned} \big |{\varvec{f}}_{opt}({\varvec{x}}_i) - {\varvec{f}}({\varvec{x}}_i)\big | = \big |{\varvec{f}}^t_{SA}({\varvec{x}}_i) - {\varvec{f}}({\varvec{x}}_i)\big | \le \overline{{\varvec{\varepsilon }}}_{SA}^t({\varvec{x}}_i) \end{aligned}$$

which comes from Eq. 3. \(\square \)

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Rivier, M., Congedo, P.M. Surrogate-assisted Bounding-Box approach for optimization problems with tunable objectives fidelity. J Glob Optim 75, 1079–1109 (2019). https://doi.org/10.1007/s10898-019-00823-9

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