The following PROC TRANSPOSE statements and DATA step create an input data set for the CLP procedure in SAS/OR software:

proc transpose data=a_data out=trans(drop=_name_) prefix=x;
run;

data condata_feas(drop=j);
   length _type_ $8;
   array x[&n];
   set trans;
   _type_ = 'le';
   _rhs_ = &b;
   output;
   do j = 1 to &n;
      x[j] = 1;
   end;
   _type_ = 'binary';
   _rhs_ = .;
   output;
run;

The following PROC CLP statements find all feasible solutions and save them in the out_feas data set:

proc clp data=condata_feas out=out_feas allsolns usecondatavars=1;
run;

The following DATA step creates another input data set for PROC CLP, for the purpose of finding all binary solutions that violate the original constraint:

data condata_infeas;
   set condata_feas;
   if _N_ = 1 then do;
      _type_ = 'ge';
      _rhs_ = _rhs_ + 1;
   end;
run;

The following PROC CLP statements find all such solutions and save them in the out_infeas data set:

proc clp data=condata_infeas out=out_infeas allsolns usecondatavars=1;
run;

The first several PROC OPTMODEL statements declare sets and parameters and read the original constraint data:

proc optmodel;
   set VARS;
   set VARS0 = VARS union {0};
   num a {VARS0};
   read data a_data into VARS=[_N_] a;
   a[0] = &b;

The following statements declare the FEAS_POINTS set and the x_feas parameter and populate them by reading the out_feas data set:

   set FEAS_POINTS;
   num x_feas {FEAS_POINTS, VARS};
   read data out_feas into FEAS_POINTS=[_N_]
      {j in VARS} <x_feas[_N_,j]=col('x'||j)>;

The following statements declare the INFEAS_POINTS set and the x_infeas parameter and populate them by reading the out_infeas data set:

   set INFEAS_POINTS;
   num x_infeas {INFEAS_POINTS, VARS};
   read data out_infeas into INFEAS_POINTS=[_N_]
      {j in VARS} <x_infeas[_N_,j]=col('x'||j)>;

The following statements declare the variables and constraints:

   var Scale {VARS0} >= 0;
   impvar Alpha {j in VARS0} = a[j] * Scale[j];

   con Feas_con {point in FEAS_POINTS}:
      sum {j in VARS} Alpha[j] * x_feas[point,j] <= Alpha[0];
   con Infeas_con {point in INFEAS_POINTS}:
      sum {j in VARS} Alpha[j] * x_infeas[point,j] >= Alpha[0] + 1;

The following statements solve the problem by using the first objective and then print the solution:

   min Objective1 = abs(a[0]) * Scale[0];
   solve;
   print a Scale Alpha;

Figure 18.1 shows the output from the linear programming solver for the first objective.

The following statements solve the problem by using the second objective and then print the solution:

   min Objective2 = sum {j in VARS} abs(a[j]) * Scale[j];
   solve;
   print a Scale Alpha;
quit;

Figure 18.2 shows the output from the linear programming solver for the second objective.

For this test instance, it turns out that the optimal solutions for the two different objectives agree.