Help framing optimization problem


(Rahul) #1

Hi,

I am new to this website and convex optimization. I am seeking help to model and code following optimization problem at hand:

a_i, b_i and c_i are three series which are indexed from i = 1 to 700, such that,

\sum_{i=1}^{700}{a_i} = 1,
\sum_{i=1}^{700}{b_i} = 1,
0<a_i, b_i<1,
c_i=1, and hence \sum_{i=1}^{700}{c_i} = 700.

Now i is divided into 12 sets by making 11 cuts at i = x_p, p = 0, 1, 2, …, 12. Let x_0=0 and x_{12}=700.

Sets \lambda_{x_p} are defined such that ,
\lambda_{x_p} = [a_{x_p}, b_{x_p}, c_{x_p}] where,

a_{x_p} = \sum_{x_p}({a_i}) = \sum_{i=x_{p-1}}^{i=x_{p}}({a_i}),

b_{x_p} = \sum_{x_p}({b_i}) = \sum_{i=x_{p-1}}^{i=x_{p}}({b_i}),

c_{x_p} = \sum_{x_p}({c_i}) = \sum_{i=x_{p-1}}^{i=x_{p}}({c_i}),

c_{x_p} \geq 15

Also, \sum_{p=1}^{12}{y_p} = 1,
0<y_p<1, where 0<p \leq12
y_0 = 0

My cost function is as follows:
$$f(x_p, y_p) = \sum_{p=1}^{12}\frac{{(b_{x_p})}^{\beta}{(y_{p})}^{1+\beta}}{({a_{x_p})}^{\beta}}$$
where, 0.4 \leq \beta \leq0.65 (I am flexible with \beta as long as it is possible to get a convex approximation)


So, ultimately find the values of following 23 variables,
x_1, x_2, ..., x_{11} and y_1, y_2, ..., y_{12} that corresponds to minimum value of f(x_p, y_p)

Thanks,
Rahul


(Michael C. Grant) #2

This is definitely the wrong forum. Your Math.SE post is a better fit.


(Michael C. Grant) #3