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mathopt5.gms : MathOptimizer Example 5

Eight univariate test models with simple box constraints. Compares
the performance on eight models.

More information at http://www.wolfram.com/products/applications/mathoptimizer/
References:
Small Model of Type: DNLP
$title MathOptimizer Example 5 (MATHOPT5,SEQ=259) * Eight univariate test models with simple box constraints. Compares * the performance on eight models. * * More information at http://www.wolfram.com/products/applications/mathoptimizer/ * * * Mathematica, MathOptimizer - An Advanced Modeling and Optimization System * for Mathematica Users, http://www.wolfram.com/products/applications/mathoptimizer/ * * Janos D Pinter, Global Optimization in Action, Kluwer Academic Publishers, * Dordrecht/Boston/London, 1996. * * Janos D Pinter, Computational Global Optimization in Nonlinear Systems, * Lionheart Publishing, Inc., Atlanta, GA, 2001 * $eolcom // sets ff / f1*f8 /, f(ff) dynamic version of ff k / k0*k6 /, k5(k) / k1*k5 / parameters c4(k) / k4 2, k3 -13, k2 18, k1 -10, k0 3 / c7(k) / k5 0.2, k4 -1.6995, k3 0.998266, k2 -0.0218343, k1 -0.000089248 / c8(k) / k2 2, k4 -1.05, k6 0.1666667, k1 -1 /; * Instead of defining eight different models we combine the * the models into one single objective function. Note the use * of the dynamic set f and the function sameas() which allows * us to select any component at runtime. The equation objdef is * declared using the domain ff, but defined with the dynamic set f. * This will permit convenient testing of all eight models. variables x, obj; equation objdef(ff); objdef(f).. obj =e= ( Exp[-x] - power(Sin[x],3) )$sameas('f1',f)+( sqr(x) - Cos[18*x] )$sameas('f2',f)+( Sin[x]*sqr(Cos[x] - Sin[x]) )$sameas('f3',f)+( sqr(sum(k, c4(k)*power(x,ord(k)-1))) )$sameas('f4',f)+( Sum[k5, ord(k5)*Sin[(ord(k5)+1)*x+ord(k5)]] )$sameas('f5',f)+( 0.1*x + Sqrt[Abs[x]]*sqr(Sin[x]) )$sameas('f6',f)+( 0.01*sum(k, c7(k)*power(x,ord(k)-1)) )$sameas('f7',f)+( sum(k, c8(k)*power(x,ord(k)-1)) )$sameas('f8',f); model m / objdef /; * You could use the convert option to expand all equation to allow * better verification with the original source. * * f(ff) = yes; option dnlp=convert; solve m using dnlp min obj; table bounds(ff,*) bounds and starting values used by J Pinter lo up l f1 -5 10 1 f2 -5 5 2 f3 0 10 3 f4 -1 4 3 f5 -10 10 5 f6 -10 5 6 f7 0 8 2 f8 -2 2.5 1 set col / x_initial, f_solver, f_global, f_reldiff, x_solver, x_global, status /; acronym global,local,failed,capability; scalar stat; * note that in some cases we have multiple global optima. For * example f5 has three global optima spaced 2*pi apart. * the x_global is just used a possible starting point for * local optimizers. table data(ff,col) solution summary x_global f_global f1 7.85411102 -0.99961182 f2 0 -1 f3 5.22405862 -1.61642493 f4 1 f5 -7.39728499 -14.83795000 f6 -9.44104654 -0.94329150 f7 6.32565486 -4.43672834 f8 1.75767181 -0.68607228 ; * Solve for each component of the objective function and collect * the result to report performance. f(ff) = no; loop(ff, x.lo = bounds(ff,'lo'); x.up = bounds(ff,'up'); x.l = bounds(ff,'l'); * try different starting points * x.l = uniform(x.lo,x.up); * x.l = bounds(ff,'l'); * x.l = data(ff,'x_global'); * x.l = 0; data(ff,'x_initial') = x.l; f(ff) = yes; solve m us dnlp min obj; f(ff) = no; if(m.modelstat=1 or m.modelstat = 2, // good return data(ff,'f_reldiff') = abs(data(ff,'f_global') - obj.l)/(1+abs(data(ff,'f_global'))); if(data(ff,'f_reldiff') < 1e-6, stat = global else stat = local ); data(ff,'x_solver') = x.l; data(ff,'f_solver') = obj.l else data(ff,'f_reldiff') = na; if(m.solvestat = 6, stat = capability else stat = failed); data(ff,'x_solver') = na; data(ff,'f_solver') = na ); data(ff,'status') = stat ); options decimals=7; display data;