PROC OPTMODEL Statements and Output

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 19.1 shows the output from the linear programming solver for the first objective.

Figure 19.1: Output from Linear Programming Solver, First Objective

The OPTMODEL Procedure

Problem Summary
Objective Sense Minimization
Objective Function Objective1
Objective Type Linear
   
Number of Variables 9
Bounded Above 0
Bounded Below 9
Bounded Below and Above 0
Free 0
Fixed 0
   
Number of Constraints 256
Linear LE (<=) 152
Linear EQ (=) 0
Linear GE (>=) 104
Linear Range 0
   
Constraint Coefficients 1280

Performance Information
Execution Mode Single-Machine
Number of Threads 1

Solution Summary
Solver LP
Algorithm Dual Simplex
Objective Function Objective1
Solution Status Optimal
Objective Value 25
   
Primal Infeasibility 4.218847E-15
Dual Infeasibility 0
Bound Infeasibility 0
   
Iterations 24
Presolve Time 0.02
Solution Time 0.02

[1] a Scale Alpha
0 37 0.67568 25
1 9 0.66667 6
2 13 0.69231 9
3 -14 0.71429 -10
4 17 0.70588 12
5 13 0.69231 9
6 -19 0.68421 -13
7 23 0.69565 16
8 21 0.66667 14


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 19.2 shows the output from the linear programming solver for the second objective.

Figure 19.2: Output from Linear Programming Solver, Second Objective

Problem Summary
Objective Sense Minimization
Objective Function Objective2
Objective Type Linear
   
Number of Variables 9
Bounded Above 0
Bounded Below 9
Bounded Below and Above 0
Free 0
Fixed 0
   
Number of Constraints 256
Linear LE (<=) 152
Linear EQ (=) 0
Linear GE (>=) 104
Linear Range 0
   
Constraint Coefficients 1280

Performance Information
Execution Mode Single-Machine
Number of Threads 1

Solution Summary
Solver LP
Algorithm Dual Simplex
Objective Function Objective2
Solution Status Optimal
Objective Value 89
   
Primal Infeasibility 4.218847E-15
Dual Infeasibility 0
Bound Infeasibility 0
   
Iterations 21
Presolve Time 0.02
Solution Time 0.02

[1] a Scale Alpha
0 37 0.67568 25
1 9 0.66667 6
2 13 0.69231 9
3 -14 0.71429 -10
4 17 0.70588 12
5 13 0.69231 9
6 -19 0.68421 -13
7 23 0.69565 16
8 21 0.66667 14


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