I am an electrical engineer who is currently working with some optimization problem. From this "MM Algorithms for Geometric and Signomial Programming" paper, it seems that although signomial programming is hard and it does not guarantee global solution, local method still exists. Furthermore, it seems that many problem can be expressed as signomial problem.
I was a bit quite fascinated by this prospect, so I am having a quite of a stupid question about the expressive power of signomial programming. Therefore, my question is
Is it possible to reduce/transform/ or present a linear programming problem to a signomial programming problem ?
To be honest, my question is just academic curiosity and it is a little bit stupid because in practice nobody would even make this move.
For example: how to transform this LP problem into a signomial optimization problem
$\begin{array}{*{20}{c}} {\min }&{{x_1} + {x_2} - {x_3}}\\ {}&{{x_1} + {x_2} \ge 1}\\ {}&{{x_2} - {x_3} \le 4} \end{array}$
Would you kindly help me with this ?
Thank you for your enthusiasm !
Yes it is possible.
Each linear constraint can be transformed to a signomial constraint involving a single monomial experssion.
Each equality linear constraint (the same can be done for an inequality constraint) $$a_1x_1 +\dots+a_n x_n =b$$ can be transformed to the following:
$$\alpha \times y_1^{a_1} \dots y_n^{a_n} =1$$
with $\alpha=e^{-b}$ and $y_i>0$ using the transformation:
$$ y_i=e^{x_i}, i=1,\dots,n. $$
Similarly, an objective
$$\min \quad c_1x_1 +\dots+x_n x_n$$
can be given as
$$\min \quad y_1^{c_1}\dots y_n^{c_n}.$$
For the example, first write it as follows:
$\begin{array}{*{20}{c}} {\min }&{{x_1} + {x_2} - {x_3}}\\ {}&{-{x_1} - {x_2} +1 \le 0}\\ {}&{{x_2} - {x_3} -4\le 0} \end{array}.$
Then apply $e^x$ anywhere:
$\begin{array}{*{20}{c}} {\min}&{ e^{{x_1} + {x_2} - {x_3}} }\\ {}&{e^{{-x_1} -{x_2} +1} \le e^0}\\ {}&{e^{{x_2} - {x_3} -4}\le e^0} \end{array}.$
Now use $y_1,y_2,y_3>0$
$\begin{array}{*{20}{c}} {\min}&{y_1 y_2 y_3^{-1} }\\ {}&{ e \,y_1^{-1} y_2^{-1}\le 1}\\ {}&{e^{-4} \,y_2 y_3^{-1}\le 1} \end{array}.$