Formulating a continuous NLP problem with a class variable

Question

In this minimization problem we have $N$ items, $j= 1, 2, \dots, N$ and a decision variable $x_j$ which are continuous values. For every item, we have a nonlinear objective function $f$ in function of the decision variables $x_j$ that we want to minimize. We also have a variable $d_j$ that is different for every item and a variable $a_j$ that contains values from the set $\lbrace 0, 1, 2\rbrace$. Look at this $a$-variable as a classification that puts all items in one of three classes. I want to formulate the problem where we minimize $\sum_{j=1}^Nf(x_{j})$ but $x_j$ must be fixed for each class, so there are at most $3$ different values for $x_j$. Apart from that, we have the constraints $0.00 \le x_j < 1.00$ and $$\frac{\sum_{j=1}^N d_jx_{j}}{\sum_{j=1}^N d_j} = \beta,$$ where $0.00 \le \beta < 1.00$.

To make it a bit more clear, you can imagine the following table as an example of a valid (but not necessarily optimal) solution:

item    a    x     d     f(x)

1       0   0.98  198    212.5 
2       1   0.95  50     1245.2  
3       0   0.98  110    100.2     
4       2   0.92  20     120.8
5       1   0.95  80     521.2
6       1   0.95  36     8232.1
7       0   0.98  109    3245.7
8       2   0.92  15     58.2
9       0   0.98  140    5123.2
10      2   0.92  10     4128

In this valid solution, $\beta = 0.97$ and the result of the objective function is $22987.1$.

How can I formulate this NLP problem by enforcing the constraints mentioned?

Answer 1

Given, $a_j\in\{0,1,2\}$, $d_j$, and $\beta$, your problem is to find $x_0,x_1,x_2\in [0,1]$ to minimize $\sum_{j=1}^N f(x_{a_j})$ subject to $$\frac{\sum_{j=1}^N d_j x_{a_j}}{\sum_{j=1}^N d_j} = \beta.$$

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For example, here's what it would look like in SAS, where I have used $f(x)=(x-1/2)^4$:

data indata;
   input item a d;
   datalines;
 1 0 198
 2 1  50
 3 0 110
 4 2  20
 5 1  80
 6 1  36
 7 0 109
 8 2  15
 9 0 140
10 2  10
;

proc optmodel;
   set ITEMS;
   num a {ITEMS};
   num d {ITEMS};
   num beta = 0.97;
   set LABELS = setof {j in ITEMS} a[j];

   read data indata into ITEMS=[item] a d;

   var X {LABELS} >= 0 <= 1;
   min Z = sum {j in ITEMS} (X[a[j]] - 1/2)^4;
   con Ratio:
      (sum {j in ITEMS} d[j] * X[a[j]]) / (sum {j in ITEMS} d[j]) = beta;

   solve;
   print a d {j in ITEMS} X[a[j]];
quit;