Optimization of a Water Tank: Convex Form of the GP

GP > Posynomials | Standard Forms > Example: Water Tank Optimization | Applications

The GP described here for the optimization of the shape of a water tank is

$min_{d>0,h>0} : C(d,h) := c_1 h^{-1}d^{-2} + c_2 d^2 + c_3 dh ~:~ begin{array}[t]{l} 0 < d_{rm max}^{-1} d le 1, ;; 0 < h_{rm max}^{-1} h le 1 kappa_{rm max}^{-1} hd^{-1} le 1. end{array}$

Let us define the new variables:

The objective function is transformed into

C(d,h) = exp( -2y_1 - y_2 + log c_1) + exp(2y_1 + log c_2) + exp(y_1+y_2+log c_3).

Taking logarithms, the new objective function is

$begin{array}{rcl} f_0(y) &:=& log left( exp( -2y_1 - y_2 + log c_1) + exp(2y_1 + log c_2) + exp(y_1+y_2+log c_3) right) &=& mbox{bf lse}(-2y_1 - y_2 + log c_1, 2y_1 + log c_2, y_1+y_2+log c_3), end{array}$

where mbox{bf lse} denotes the (convex) log-sum-exp function.

The constraints all involve monomials, so they will translate into linear or affine constraints. For example, the aspect ratio constraint on (d,h) :

$kappa_{rm max}^{-1} hd^{-1} le 1$

is equivalent to

$exp( -y_1 + y_2 + log kappa_{rm max} ) le 1,$

which, after taking logarithms, becomes the affine constraint:

$-y_1 + y_2 + log kappa_{rm max} le 0.$

Thus, the original GP takes the standard convex form:

$begin{array}[t]{ll} displaystylemin_{y} & log left( exp( -2y_1 - y_2 + log c_1) + exp(2y_1 + log c_2) + exp(y_1+y_2+log c_3) right) mbox{subject to} & y_1 le log d_{rm max}, ;; y_2 le log h_{rm max}, ;;-y_1 + y_2 + log kappa_{rm max} le 0. end{array}$

The water tank optimization problem in convex form, corresponding to the values

$begin{array}{l} V_{rm supp} = 800,000 L, ;; alpha_1 = 10 : $, ;; alpha_2 = 6 : $ / m^2, ;; alpha_3 = 2: $ / m^2, d_{rm max} = 20 m, ;; h_{rm max} = 30 m, ;; kappa_{rm max} = 3. end{array}$

The design constraints of the problem are pictured as a forbidden region, in white. We observe that the cost function in this formulation is convex, contrary to that of the original problem.