## 3 Multiobjective Optimization Approach
### Two-Slope Parameterized ASFs
Then there exists a feasible solution \(\mathbf{x}^{\prime}\in X\) such that \(\mathbf{f}(\mathbf{x}^{\prime})<\mathbf{f}(\mathbf{x}^{*})\). For any \(\mathbf{x}\in X\), denote \(I_{\mathbf{x}}=\{i\in I^{q}\,|\,f_{i}^{R}\leq f_{i}(\mathbf{x})\}\), \(J_{\mathbf{x}}=\{i\in I^{q}\,|\,f_{i}^{R}>f_{i}(\mathbf{x})\}\) and \(s_{R}^{q}(\mathbf{f}(\mathbf{x}),\mathbf{\lambda}^{U},\mathbf{\lambda}^{A})\) as the objective of the scalarized problem (32). Now  
\[s_{R}^{q}(\mathbf{f}(\mathbf{x}^{\prime}),\mathbf{\lambda}^{U},\mathbf{\lambda}^ {A}) =\max_{ \begin{subarray}{c}I^{q}\subseteq I\\ |I^{q}|=q\end{subarray} }\Bigg{\{}\sum_{i\in I_{\mathbf{x}^{\prime}}}\lambda_{i}^{U }(f_{i}(\mathbf{x}^{\prime})-f_{i}^{R})+\sum_{i\in I_{\mathbf{x}^{\prime}}}\lambda_{i} ^{A}(f_{i}(\mathbf{x}^{\prime})-f_{i}^{R})\Bigg{\}}\] \[<\max_{ \begin{subarray}{c}I^{q}\subseteq I\\ |I^{q}|=q\end{subarray} }\Bigg{\{}\sum_{i\in I_{\mathbf{x}^{\prime}}}\lambda_{i}^{U }(f_{i}(\mathbf{x}^{*})-f_{i}^{R})+\sum_{i\in I_{\mathbf{x}^{\prime}}}\lambda_{i}^{A}( f_{i}(\mathbf{x}^{*})-f_{i}^{R})\Bigg{\}}\] \[\leq\max_{ \begin{subarray}{c}I^{q}\subseteq I\\ |I^{q}|=q\end{subarray} }\Bigg{\{}\sum_{i\in I_{\mathbf{x}^{\prime}}}\lambda_{i}^{U }(f_{i}(\mathbf{x}^{*})-f_{i}^{R})+\sum_{i\in I_{\mathbf{x}^{\prime}}}\lambda_{i}^{A} (f_{i}(\mathbf{x}^{*})-f_{i}^{R})\Bigg{\}}\] \[=s_{R}^{q}(\mathbf{f}(\mathbf{x}^{*}),\mathbf{\lambda}^{U},\mathbf{\lambda}^{A})\]  
yielding to a contradiction. (ii) Assume that \(\mathbf{x}^{*}\) is an optimal solution of the problem (32) but not a Pareto optimal solution of the problem (31). Therefore, there exists \(\mathbf{x}^{\prime}\in X\) such that \(\mathbf{f}(\mathbf{x}^{\prime})\leq\mathbf{f}(\mathbf{x}^{*})\) and at least one index \(j\in I\) such that \(f_{j}(\mathbf{x}^{\prime})<f_{j}(\mathbf{x}^{*})\). Similarly to (i), we can deduce that \(s_{R}^{q}(\mathbf{f}(\mathbf{x}^{\prime}),\mathbf{\lambda}^{U},\mathbf{\lambda}^{A})\leq s_{R} ^{q}(\mathbf{f}(\mathbf{x}^{*}),\mathbf{\lambda}^{U},\mathbf{\lambda}^{A})\). If the equality holds, \(\mathbf{x}^{\prime}\) is an optimal solution for the problem (32) and Pareto optimal for the problem (31). In the case of strict inequality, this yields to a contradiction with an assumption that \(\mathbf{x}^{*}\) is an optimal solution for the problem (32). (iii) First, we observe that \(s_{R}^{q}\) is strictly increasing (i.e., \(s_{R}^{q}(\mathbf{f}(\mathbf{x}_{1}),\mathbf{\lambda}^{U},\mathbf{\lambda}^{A})<s_{R}^{q}(\mathbf{f }(\mathbf{x}_{2}),\mathbf{\lambda}^{U},\mathbf{\lambda}^{A})\) for any \(\mathbf{f}(\mathbf{x}_{1}),\mathbf{f}(\mathbf{x}_{2})\) having \(\mathbf{f}(\mathbf{x}_{1})<\mathbf{f}(\mathbf{x}_{2})\) and \(\mathbf{x}_{1},\mathbf{x}_{2}\in X\)). Indeed, by taking \(\mathbf{x}_{1},\mathbf{x}_{2}\in X\) such that \(\mathbf{f}(\mathbf{x}_{1})<\bmThus, we know that every Pareto optimal solution can be obtained and the solution of the scalarized problem (32) is weakly Pareto optimal for the original multiobjective problem. In order to guarantee the Pareto optimality of solutions, a so-called augmentation term [32]  
\[\rho\sum_{i\in I}\lambda_{i}(f_{i}(\mathbf{x})-f_{i}^{R}),\quad\rho>0 \tag{33}\]  
may be added to the objective of the scalarized problem (32) [25]. Note that similarly to Theorem 8 in [25] it can be proven that if the set \(X\) and the objectives \(f_{i}\), \(i\in I\) are convex, then \(s_{R}^{q}(f(\mathbf{x}),\mathbf{\lambda}^{U},\mathbf{\lambda}^{A})\) preserves the convexity.