\$title Linear Regression with Various Criteria (LINEAR,SEQ=23) \$onText This example solves linear models with differing objective functions. Absolute deviations cannot be solved in a reliable manner with most NLP systems and one has to resort to a formulation with negative and positive deviations (models ending with the letter a). Bracken, J, and McCormick, G P, Chapter 8.2. In Selected Applications of Nonlinear Programming. John Wiley and Sons, New York, 1968, pp. 86-88. Keywords: linear programming, nonlinear programming, discontinuous derivatives, linear regression, econometrics \$offText Set i 'observation number' / 1*20 / n 'index of independent variables' / a, b, c, d /; Table dat(i,*) y a b c d 1 99 1 85 76 44 2 93 1 82 78 42 3 99 1 75 73 42 4 97 1 74 72 44 5 90 1 76 73 43 6 96 1 74 69 46 7 93 1 73 69 46 8 130 1 96 80 36 9 118 1 93 78 36 10 88 1 70 73 37 11 89 1 82 71 46 12 93 1 80 72 45 13 94 1 77 76 42 14 75 1 67 76 50 15 84 1 82 70 48 16 91 1 76 76 41 17 100 1 74 78 31 18 98 1 71 80 29 19 101 1 70 83 39 20 80 1 64 79 38; Variable obj 'objective value' dev(i) 'total deviation' devp(i) 'positive deviation' devn(i) 'negative deviation' b(n) 'estimates'; Positive Variable devp, devn; Equation ddev 'definition of deviations using total deviations' ddeva 'definition of deviations using positive and negative deviations' ls1 ls1a ls2 ls3 ls4 ls5 ls5a ls6 ls7 ls8; ddev(i).. dev(i) =e= dat(i,"y") - sum(n, b(n)*dat(i,n)); ddeva(i).. devp(i) - devn(i) =e= dat(i,"y") - sum(n, b(n)*dat(i,n)); ls1.. obj =e= sum(i, abs(dev(i))); ls1a.. obj =e= sum(i, devp(i)+devn(i)); ls2.. obj =e= sum(i, sqr(dev(i))); ls3.. obj =e= sum(i, power(abs(dev(i)),3)); ls4.. obj =e= sum(i, power(dev(i),4)); ls5.. obj =e= sum(i, abs(dev(i)/dat(i,"y"))); ls5a.. obj =e= sum(i, (devp(i)+devn(i))/dat(i,"y")); ls6.. obj =e= sum(i, sqr(dev(i)/dat(i,"y"))); ls7.. obj =e= sum(i, power(abs(dev(i)/dat(i,"y")),3)); ls8.. obj =e= sum(i, power(dev(i)/dat(i,"y"),4)); Model mod1 / ddev, ls1 / mod1a / ddeva,ls1a / mod2 / ddev, ls2 / mod3 / ddev, ls3 / mod4 / ddev, ls4 / mod5 / ddev, ls5 / mod5a / ddeva,ls5a / mod6 / ddev, ls6 / mod7 / ddev, ls7 / mod8 / ddev, ls8 /; Parameter result 'summary table'; b.l(n) = 1; dev.l(i) = dat(i,"y") - sum(n, b.l(n)*dat(i,n)); dev.up(i) = 100; dev.lo(i) = -100; devp.up(i) = 100; devn.up(i) = 100; option limRow = 0, limCol = 0; solve mod1 min obj using dnlp; result("mod1" ,n) = b.l(n); result("mod1" ,"obj") = obj.l; solve mod1a min obj using lp; result("mod1a",n) = b.l(n); result("mod1a","obj") = obj.l; solve mod2 min obj using nlp; result("mod2" ,n) = b.l(n); result("mod2" ,"obj") = obj.l; solve mod3 min obj using dnlp; result("mod3" ,n) = b.l(n); result("mod3" ,"obj") = obj.l; solve mod4 min obj using nlp; result("mod4" ,n) = b.l(n); result("mod4" ,"obj") = obj.l; solve mod5 min obj using dnlp; result("mod5" ,n) = b.l(n); result("mod5" ,"obj") = obj.l; solve mod5a min obj using lp; result("mod5a",n) = b.l(n); result("mod5a","obj") = obj.l; solve mod6 min obj using nlp; result("mod6" ,n) = b.l(n); result("mod6" ,"obj") = obj.l; solve mod7 min obj using dnlp; result("mod7" ,n) = b.l(n); result("mod7" ,"obj") = obj.l; solve mod8 min obj using nlp; result("mod8" ,n) = b.l(n); result("mod8" ,"obj") = obj.l; display result;