I would like to minimize linear pseudo-boolean function

$$obj = \sum_i c_i sel_i$$

subject to $$\sum_i c_i sel_i \geq Value (1)$$ where

$c_1 ... c_5, Value$ - some constants;

$sel_1 ... sel_5$ - sort of selector variables, $ 0 \leq sel_i \leq 1$

Everything is pretty straightforward, but I need to check constraint (1) if and only if some of $sel_i = 1$. In the other words, if $obj = 0$, that is OK with me, but if some or all of $sel_i$ are picked up, I need to check $obj \geq Value$. Currently I'm using SAT to solve similar problem, but I prefer to study something else. Maybe I need consider different framework for this task? Any directions will be very useful. Thanks in advance.

I would like to minimize linear pseudo-boolean function

$$\mathrm{obj} = \sum_i c_i \mathrm{sel}_i$$

subject to $$\sum_i c_i sel_i \geq \mathrm{Value} \qquad\qquad(1)$$ where

$c_1,\dots c_5, \mathrm{Value}$ are some constants;

$\mathrm{sel}_1,\dots,\mathrm{sel}_5$ are sort of selector variables, $ 0 \leq \mathrm{sel}_i \leq 1$

Everything is pretty straightforward, but I need to check constraint (1) if and only if some of $\mathrm{sel}_i = 1$. In the other words, if $\mathrm{obj} = 0$, that is OK with me, but if some or all of $\mathrm{sel}_i$ are picked up, I need to check $\mathrm{obj} \geq \mathrm{Value}$. Currently I'm using SAT to solve similar problem, but I prefer to study something else. Maybe I need consider different framework for this task? Any directions will be very useful. Thanks in advance.