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Secondly, instead of simultaneously minimizing the mean and standard deviation of the objective(s), another type of formulation of robust optimization considers the standard deviation as the constraint: \begin{flalign}\label{eq:robust-optimization-2} \text{Type II:} \begin{cases} \text{find} &\vect{x} \\ \text{minimizing} &\mu\left\{f_{i}(\vect{x}, \vect{p})\right\} (i = 1, ...\, , n_{\text{objectives}}) \\ \text{subject to} &\sigma\left\{f_{i}(\vect{x}, \vect{p})\right\} \leq \sigma_{i}^{\text{crit}} (i = 1, ...\, , n_{\text{objectives}}) \\ &L_{j}(\vect{x}, \vect{p}) \leq 0 (j = 1, ...\, , n_{\text{constraints}}) \\ &\vect{x}^{-} \leq \vect{x} \leq \vect{x}^{+} \end{cases} \end{flalign} Thirdly, as previously discussed, in the robust design optimization problem as formulated by

How do I modify my code such that the following 2 requirements (brown and green color) are satisfied while other things remain as they are? enter image description here

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Just use an equation environment and nest aligned in cases:

\documentclass{article} \usepackage{empheq} \let\vect = \mathbf \begin{document} Secondly, instead of simultaneously minimizing the mean and standard deviation of the objective(s), another type of formulation of robust optimization considers the standard deviation as the constraint: \begin{equation}\label{eq:robust-optimization-2} \text{Type II:} \begin{cases}\begin{aligned} & \text{find} & &\vect{x} \\ & \text{minimizing} & &\mu\left\{f_{i}(\vect{x}, \vect{p})\right\} & & (i = 1, ...\, , n_{\text{objectives}}) \\ & \text{subject to} & & \sigma\left\{f_{i}(\vect{x}, \vect{p})\right\} \leq \sigma_{i}^{\text{crit}} & & (i = 1, ...\, , n_{\text{objectives}}) \\ & & &L_{j}(\vect{x}, \vect{p}) \leq 0 & & (j = 1, ...\, , n_{\text{constraints}}) \\ & & &\vect{x}^{-} \leq \vect{x} \leq \vect{x}^{+} \end{aligned}\end{cases} \end{equation} Thirdly, as previously discussed, in the robust design optimization problem as formulated by \end{document}

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With fleqn environment defined in the nccmath package:

\documentclass{article} \usepackage{bm} \usepackage{nccmath} \begin{document} \begin{fleqn} \begin{align}\label{eq:robust-optimization-2} \text{Type II:} \left\{\begin{aligned} \text{find} &&& \bm{x} \\ \text{minimizing} &&& \mu\left\{f_{i}(\bm{x}, \bm{p})\right\} && (i = 1, ...\, , n_{\text{objectives}}) \\ \text{subject to} &&& \sigma\left\{f_{i}(\bm{x}, \bm{p})\right\} \leq \sigma_{i}^{\text{crit}} && (i = 1, ...\, , n_{\text{objectives}}) \\ &&& L_{j}(\bm{x}, \bm{p}) \leq 0 && (j = 1, ...\, , n_{\text{constraints}}) \\ &&&\bm{x}^{-} \leq \bm{x} \leq \bm{x}^{+} && \end{aligned}\right. \end{align} \end{fleqn} \end{document}

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