# Is the optimization of the following composite function even possible, and if so, how would I go about solving it? Hi Guys,

So when I formulate a problem I am trying to solve for work, the above (please see attached figure) optimization framework results. I am not too familiar with optimization techniques (apart from linear and simple convex techniques). Can anybody tell me if this is problem has a solution, and if so, how I would go about acheiving it? Any references or leads would be appreciated as well.

As an engineer, I am interested in a numeric solution (I understand that this problem can have multiple solutions), but if this can be reduced to some analytical form to make it resemble some other standard optimization problem, I would be interested in that too.

Thanks a lot in advance for taking the time to read through this!

• What do you mean by $X_t+E_t$? – copper.hat May 31 '14 at 22:03
• The set resulting from sum of all elements in X_i (which, in turn, is from set X) and all elements in E_i (which in turn is from set E). So, if X_i = {x_1, x_2 .. x_n} , E_i ={e_1, e_2 ... e_n} then X_i + E_i = {x_1 + e_1, x_2 + e_2, ...x_n + e_n} Thanks! – F. Riberi May 31 '14 at 22:41

Assuming the subsets $X_i,E_i$ are given and their indices are denoted as subsets $S_i$, I guess you just need an integer programming formulation in which you can use a binary variable $f_i$ in place of the function:

$\max \sum_t f_i$

$k - \sum_{i \in S_i} (x_i+e_i) \leq M_i f_i \quad i=1,\ldots,m$

$\sum_{i \in S_i} (x_i+e_i) -k \leq M_i (1-f_i) \quad i=1,\ldots,m$

$lb\leq e_i \leq ub \quad i=1,\ldots,n$

$\sum_{i=1}^n e_i x_i= 0$

$f_i\in \{0,1\} \quad i=1,\ldots,m$

where $M$ is an upper bound to $|k - \sum_{i \in S_i} x_i+e_i |$, the so-called big$-M$. The tighter $M_i$, the better. Please check I got the right logical implication. But the overall idea should be OK.

• Thanks for taking the time to read through the problem and respond. But I do not think this is correct (or perhaps I don't understand this formulation). What I am trying to search for is the values of $e_i$ 's themselves. So, when you say $E_i$ is given, that is incorrect. I am trying to find the values for $e_i$ s that will maximize the objective function. – F. Riberi Jun 2 '14 at 3:12
• I guess I have not been clear: I meant the structure of the subsets, i.e. the definition of $E_i$, is given. Of course all $e_i$'s must be determined. – AndreaCassioli Jun 2 '14 at 18:32