Minimize Function with Opt Model

[harmonic]

Hello community,

I am starting using Proc Opt model, and I would like to minimize a function like this one:

min|y|=B1*type + B2*b + B3*c +  B4*d + B5*e + B6

Where values are taken from a dataset, here the values:

Y should be an integer number and this is the distribution

B1, B2, B3, B4, B5 and B6 are constants

'type' should be a string with values "white", "black"

Should I have different results for each string?

'b' should be with values 2 and 12

'c' should be in a float value in this distribution 

'd' should be in an interval 1 to 48 integer values

'e' should be with values 0, 2 or 12

Thank you in advance
@RobPratt @Ksharp 

[RobPratt]

I don't quite understand your problem.  Do you want to solve a regression problem where b, c, d, and e are regression coefficients?  If B1 is a constant and type is a string, what does B1*type mean?  Are you able to share example data and also an example (not necessarily optimal) solution?

[harmonic]

Yes,

For example, I would like to minimize y in this dataset. I was planning to set a lower limit of y to 1 because I don’t want it to be zero, and I also need to include constraints that are not explicitly visible in my data. For instance, the values in column 'a' should be within the integer range from 6 to 48.

Could you provide an example of how to minimize a regression function for y using this dataset or by applying the constraints I want, instead of using one of my columns?

Type is an example of a string column where I should perform minimization for each string value, but for now, we can ignore it to simplify the problem.

[RobPratt]

The following PROC OPTMODEL code solves an ordinary least squares regression:

proc optmodel;
   /* declare sets and input parameters */
   set OBS;
   num y {OBS};
   num a {OBS};
   num b {OBS};
   num c {OBS};
   num d {OBS};
   num e {OBS};

   /* read input data set */
   read data input into OBS=[_N_] y a b c d e;

   /* declare decision variables */
   var Beta {0..5};
   var Error {OBS};
   impvar Estimate {i in OBS} = Beta[0] + Beta[1]*a[i] + Beta[2]*b[i] + Beta[3]*c[i] + Beta[4]*d[i] + Beta[5]*e[i];

   /* declare objective to minimize sum of squared errors */
   min SSE = sum {i in OBS} Error[i]^2;

   /* declare constraints */
   con ErrorCon {i in OBS}:
      Error[i] = Estimate[i] - y[i];

   /* call quadratic programming (QP) solver */
   solve;

   /* print regression coefficients */
   print Beta;

   /* create output data sets */
   create data BetaData from [j] Beta;
   create data EstimateData from [i] a b c d e y Estimate Error;
quit;

The resulting regression coefficients are:

[1]	Beta
0	416.9191
1	19.3854
2	18.5124
3	-13.0827
4	-22.7077
5	-1.4499

Those are the same values that are returned by the following PROC REG call:

proc reg data=input;
   model y = a b c d e;
quit;

For the additional constraints, do you want to impose bounds on Beta[j]?

Or do you instead want to solve an optimization problem where a, b, c, d, and e are the (bounded) decision variables and you want to minimize the predicted value of y?  That is, with the fixed values of Beta[j] found earlier, do you want to minimize the linear function Beta[0] + Beta[1]*a + Beta[2]*b + Beta[3]*c + Beta[4]*d + Beta[5]*e subject to some constraints on a, b, c, d, and e?

[harmonic]

I don't want to find the betas, I already have the beta values. I would like to find the a, b, c , d , e values in an interval from my dataset or an interval that I will determine to minimize the y value. It's just an equation where my betas can be positive or negatives.

[Ksharp]

If I understood what you mean.

proc optmodel;
num a{1..2}=[2 12];
num b{1..3}=[2 12 24];
num d{i in 1..10}=i;
num e{i in 1..47}=i;
var var_a{1..2} binary;
var var_b{1..3} binary;
var var_c >=5 <=15;
var var_d{1..10} binary;
var var_e{1..47} binary;
impvar sum=-8*sum{i in 1..2} a[i]*var_a[i] +
           -2*sum{i in 1..3} b[i]*var_b[i] +
	       -2*var_c +
	       -4*sum{i in 1..10}d[i]*var_d[i] +
            2*sum{i in 1..47}e[i]*var_e[i] ;
min obj=sum;
con con_total:sum>=1;
con con_a:sum{i in 1..2} var_a[i]=1;
con con_b:sum{i in 1..3} var_b[i]=1;
con con_d:sum{i in 1..10}var_d[i]=1;
con con_e:sum{i in 1..47}var_e[i]=1;

solve ;

print a var_a ; 
print b var_b ;
print   var_c ;
print d var_d ;
print e var_e ;
quit;

[RobPratt]

Thanks for posting this reply.  +1

A slight improvement is that for variables d and e with consecutive integer domains, you can avoid num d, num e, con_d, and con_e by using integer variables:

   var var_d >= 1 <= 10 integer;
   var var_e >= 1 <= 47 integer;

[harmonic]

Thank you, I will try by today and tomorrow if it is work. If I would like to use steps of 4 within the interval? How can i change it?

[RobPratt]

Here's one way to enforce that x is in {2,6,10,...,42}:

   var x;
   set MYSET = 2..42 by 4;
   var y {MYSET} binary;
   con Mycon1: sum {i in MYSET} y[i] = 1;
   con Mycon2: sum {i in MYSET} i*y[i] = x;

Here's another way that uses fewer variables and constraints:

   var x >= 2 <= 42;
   var y integer;
   con Mycon: x = 2 + 4*y;

See also https://or.stackexchange.com/questions/6545/how-to-linearize-membership-in-a-finite-set

[harmonic]

How can I integrate that in the code of the proc optmodel above? @RobPratt 

[RobPratt]

Sorry, this thread is getting hard to follow. What do you want to integrate into which PROC OPTMODEL code?

[harmonic]

How can I insert these con

   var x >= 2 <= 42;
   var y integer;
   con Mycon: x = 2 + 4*y;

to this proc opt model

proc optmodel;
num a{1..2}=[2 12];
num b{1..3}=[2 12 24];
num d{i in 1..10}=i;
num e{i in 1..47}=i;
var var_a{1..2} binary;
var var_b{1..3} binary;
var var_c >=5 <=15;
var var_d{1..10} binary;
var var_e{1..47} binary;
impvar sum=-8*sum{i in 1..2} a[i]*var_a[i] +
           -2*sum{i in 1..3} b[i]*var_b[i] +
	       -2*var_c +
	       -4*sum{i in 1..10}d[i]*var_d[i] +
            2*sum{i in 1..47}e[i]*var_e[i] ;
min obj=sum;
con con_total:sum>=1;
con con_a:sum{i in 1..2} var_a[i]=1;
con con_b:sum{i in 1..3} var_b[i]=1;
con con_d:sum{i in 1..10}var_d[i]=1;
con con_e:sum{i in 1..47}var_e[i]=1;

solve ;

print a var_a ; 
print b var_b ;
print   var_c ;
print d var_d ;
print e var_e ;
quit;

@RobPratt 

[Ksharp]

proc optmodel;
num a{1..2}=[2 12];
num b{1..3}=[2 12 24];
num d{i in 1..10}=i;
num e{i in 1..47}=i;
var var_a{1..2} binary;
var var_b{1..3} binary;
var var_c >=2 <=42 ;
var var_d integer;
impvar sum= 4*sum{i in 1..2} a[i]*var_a[i] +
           -2*sum{i in 1..3} b[i]*var_b[i] +
	       -2*var_c +
	        4*var_d  ;
min obj=sum;
con con_total:sum>=1;
con con_a:sum{i in 1..2} var_a[i]=1;
con con_b:sum{i in 1..3} var_b[i]=1;
con con_d:var_c=2+4*var_d;

solve ;

print a var_a ; 
print b var_b ;
print   var_c ;
print   var_d ;
quit;

[harmonic]

If I would like to read the data from my dataset?

[Ksharp]

I don't understand. If you want to import these interval of variable from XLSX, you could make a macro variable (assuming the levels of these variable was not too many).

proc import datafile="C:\Users\xiakeshan\Downloads\input.xlsx" out=have dbms=xlsx replace;
run;
proc sql noprint;
select distinct a into :a separated by ' ' from have;
select count(distinct a) into :_a separated by ' ' from have;

select distinct b into :b separated by ' ' from have;
select count(distinct b) into :_b separated by ' ' from have;

select floor(min(c)) into :min_c from have;
select ceil(max(c)) into :max_c from have;

select distinct d into :d separated by ' ' from have;
select count(distinct d) into :_d separated by ' ' from have;

select distinct e into :e separated by ' ' from have;
select count(distinct e) into :_e separated by ' ' from have;

quit;

proc optmodel;
num a{1..&_a.}=[&a.];
num b{1..&_b.}=[&b.];
num d{1..&_d.}=[&d.];
num e{1..&_e.}=[&e.];
var var_a{1..&_a.} binary;
var var_b{1..&_b.} binary;
var var_c >=&min_c.  <=&max_c.;
var var_d{1..&_d.} binary;
var var_e{1..&_e.} binary;
impvar sum=-8*sum{i in 1..&_a.} a[i]*var_a[i] +
           -2*sum{i in 1..&_b.} b[i]*var_b[i] +
	       -2*var_c +
	       -4*sum{i in 1..&_d.}d[i]*var_d[i] +
            2*sum{i in 1..&_e.}e[i]*var_e[i] ;
min obj=sum;
con con_total:sum>=1;
con con_a:sum{i in 1..&_a.} var_a[i]=1;
con con_b:sum{i in 1..&_b.} var_b[i]=1;
con con_d:sum{i in 1..&_d.} var_d[i]=1;
con con_e:sum{i in 1..&_e.} var_e[i]=1;

solve ;

print a var_a ; 
print b var_b ;
print   var_c ;
print d var_d ;
print e var_e ;
quit;