Type of constraints

Transformation

Deterministic counter part

$\tilde{a}x\le \tilde{b}$

$D{F}_1$

$D{F}_1\left(\tilde{a}x\le \tilde{b}\right)=\left\{x\mathrm{<mo>\; |\; </mo>}{\tilde{a}}_{\alpha }x\le {\tilde{b}}_{\alpha }\mathrm{<mo>\; ;\; </mo>}\forall \alpha \in \left(\mathrm{0,1}\right]\right\}$

$\tilde{a}x\le \tilde{b}$

$D{F}_2$

$D{F}_2\left(\tilde{a}x\le \tilde{b}\right)=\left\{x\mathrm{<mo>\; |\; </mo>}{\tilde{a}}_{{\alpha }_i}x\le {\tilde{b}}_{{\alpha }_i}\mathrm{<mo>\; ;\; </mo>}{\alpha }_i\in \left(\mathrm{0,1}\right]\mathrm{<mo>\; ;\; </mo>}i=\mathrm{1,}\cdots \mathrm{,}p\right\}$

$\tilde{a}x\le \tilde{b}$

$D{F}_3$

$D{F}_3\left(\tilde{a}x\le \tilde{b}\right)=\left\{x\mathrm{<mo>\; |\; </mo>}T\left(\tilde{a}x\le \tilde{b}\right)\ge \beta \right\}$

where $T\in \left\{Poss\mathrm{,}Nec\right\}$ and $\beta$ is a prescribed threshold.

Poss and Nec stand for possibility and necessity respectively.

$\tilde{a}x\le \tilde{b}$

$D{F}_4$

$D{F}_4\left(\tilde{a}x\le \tilde{b}\right)=\left\{x\mathrm{<mo>\; |\; </mo>}Poss\left(\tilde{a}x\le \tilde{b}\right)\ge \alpha \text{and}Nec\left(\tilde{a}x\le \tilde{b}\right)\ge \delta \right\}$

$\alpha$ and stand for prescribed thresholds.

$\overline{a}x\le \overline{b}$

$D{R}_1$

$D{R}_1\left(\overline{a}x\le \overline{b}\right)=\left\{x\mathrm{<mo>\; |\; </mo>}\left(E\overline{a}\right)x\le E\left(\overline{b}\right)\right\}$

where E stands for the expectation operator.

$\overline{a}x\le \overline{b}$

$D{R}_2$

$D{R}_2\left(\overline{a}x\le \overline{b}\right)=\left\{x\mathrm{<mo>\; |\; </mo>}Prob\left(\overline{a}x\le \overline{b}\right)\ge \beta \right\}$

where Prob is the probability measure and $\beta$ a fixed threshold.