Table 6.

The dose mimicking models presented in vector and matrix notation with their dual models. Terms that follow colons indicate the dual variables for that constraint.

	Dose mimicking model	Dual mode
MeanAbs	$$\begin{gathered} \underset{x,\sigma ,\delta }{\mathrm{minimize}}{\mathbf{\text{e}}}^{\prime }\mathbf{\sigma }+ϵ{\mathbf{\text{e}}}^{\prime }\mathbf{\delta } \\ \text{subject to}\mathbf{\text{Cx}}=\mathbf{\text{C}}\widehat{\mathbf{x}}+\mathbf{\sigma }+\mathbf{\delta }:\alpha \\ \mathbf{\text{Ax}}=\mathbf{\text{b}}:\mathbf{\text{p}} \\ \mathbf{x}⩾\mathbf{0} \\ \mathbf{\sigma }⩾\mathbf{0} \\ \mathbf{\delta }⩽\mathbf{0} \end{gathered}$$	$$\begin{gathered} \underset{\alpha ,\mathbf{p}}{\mathrm{minimize}}{\mathit{\text{α}}}^{\prime }\mathbf{C}\widehat{\mathbf{x}}-{\mathbf{\text{b}}}^{\prime }\mathbf{\text{p}} \\ \text{subject to}{\mathbf{C}}^{\prime }\mathit{\alpha }⩾{\mathbf{A}}^{\prime }\mathbf{p}:\mathbf{x} \\ \mathit{\alpha }⩽\mathbf{e}:\mathbf{\sigma } \\ \mathit{\alpha }⩾ϵ\mathbf{e}:\mathit{\delta } \end{gathered}$$
MaxAbs	$$\begin{gathered} \underset{\mathbf{x},\zeta }{\mathrm{minimize}}\zeta \\ \text{subject to}\mathbf{\text{Cx}}⩽\mathbf{\text{C}}\widehat{\mathbf{x}}+\zeta \mathbf{\text{e}}:\mathit{\alpha } \\ \mathbf{\text{Ax}}=\mathbf{\text{b}}:\mathbf{\text{p}} \\ \mathbf{x}⩾\mathbf{0} \end{gathered}$$	$$\begin{gathered} \underset{\alpha ,\mathbf{p}}{\mathrm{minimize}}{\mathit{\text{α}}}^{\prime }\mathbf{C}\widehat{\mathbf{x}}-{\mathbf{\text{b}}}^{\prime }\mathbf{\text{p}} \\ \text{subject to}{\mathbf{C}}^{\prime }\mathit{\alpha }⩾{\mathbf{A}}^{\prime }\mathbf{p}:\mathbf{x} \\ {\mathit{\alpha }}^{\prime }=1:\zeta \\ \mathit{\alpha }⩾0 \end{gathered}$$
MeanRel	$$\begin{gathered} \underset{x,\sigma ,\delta }{\mathrm{minimize}}{\mathbf{\text{e}}}^{\prime }\mathbf{\sigma }+ϵ{\mathbf{\text{e}}}^{\prime }\mathbf{\delta } \\ \text{subject to}\mathbf{\text{Cx}}=\mathbf{\text{C}}\widehat{\mathbf{x}}\odot (\mathbf{\text{e}}+\mathbf{\sigma }+\mathbf{\delta }):\alpha \\ \mathbf{\text{Ax}}=\mathbf{\text{b}}:\mathbf{\text{p}} \\ \mathbf{x}⩾\mathbf{0} \\ \mathbf{\sigma }⩾\mathbf{0} \\ \mathbf{\delta }⩽0 \end{gathered}$$	$$\begin{gathered} \underset{\alpha ,\mathbf{p}}{\mathrm{minimize}}{\mathit{\text{α}}}^{\prime }\mathbf{C}\widehat{\mathbf{x}}-{\mathbf{\text{b}}}^{\prime }\mathbf{\text{p}} \\ \text{subject to}{\mathbf{C}}^{\prime }\mathit{\alpha }⩾{\mathbf{A}}^{\prime }\mathbf{p}:\mathbf{x} \\ \mathit{\alpha }\odot \mathbf{\text{C}}\widehat{\mathbf{x}}⩽\mathbf{\text{e}}:\mathbf{\sigma } \\ \mathit{\alpha }\odot \mathbf{\text{C}}\widehat{\mathbf{x}}⩾ϵ\mathbf{\text{e}}:\mathit{\delta } \end{gathered}$$
MaxRel	$$\begin{gathered} \underset{\mathbf{x},\zeta }{\mathrm{minimize}}\zeta \\ \text{subject to}\mathbf{\text{Cx}}⩽\mathbf{\text{C}}\widehat{\mathbf{x}}\odot (\mathbf{\text{e}}+\zeta \mathbf{\text{e}}):\mathit{\alpha } \\ \mathbf{\text{Ax}}=\mathbf{\text{b}}:\mathbf{\text{p}} \\ \mathbf{x}⩾\mathbf{0} \end{gathered}$$	$$\begin{gathered} \underset{\alpha ,\mathbf{p}}{\mathrm{minimize}}{\mathit{\text{α}}}^{\prime }\mathbf{C}\widehat{\mathbf{x}}-{\mathbf{\text{b}}}^{\prime }\mathbf{\text{p}} \\ \text{subject to}{\mathbf{C}}^{\prime }\mathit{\alpha }⩾{\mathbf{A}}^{\prime }\mathbf{p}:\mathbf{x} \\ {\mathit{\text{α}}}^{\prime }\mathbf{C}\widehat{\mathbf{x}}=1:\zeta \\ \mathit{\alpha }⩾0 \end{gathered}$$