input-algo-han-powell

有限元分析源代码

/*  
 *  ================================================================
 *  Example Problem 1:                              
 *  
 *    minmize f(x) = (x1 -x2)^2 + (x2+x3-2)^2 + (x4-1)^2 + (x5 -1)^2
 *  
 *  Subject to constraints:                                   
 *  
 *         x1+3*x2            -4 = 0                   
 *                 x3+x4-2*x5    = 0                   
 *              x2        -x5    = 0                   
 *  
 *  Notation -- f = f(x), G = g(x), Df = grad f(x), Dg = grad g(x).
 *  
 *  Written By: X.G. Chen                                  June 1994
 *  ================================================================
 */  

SetUnitsOff;

/* [a] : Initialize and print variables and matrices */

   Epsilon = 1E-6;
   beta    = 0.5;
   alpha   = 0.1;
   IMAX    = 20; /* maximum of iterations */
   n       = 5;  /* number of variables   */
   M       = 3;  /* number of constriants */

   print "\n ****** Initial Guess Value ******* \n";

   x = [20; 0.1; 10; -1; -10];   /* not feasible */
   PrintMatrix(x);

   x_old      = x;
   lambda     = Zero([M, 1]);
   lambda_old = Zero([M, 1]);
   temp1 = (x[1][1] -x[2][1])*(x[1][1] -x[2][1]);
   temp2 = (x[2][1] +x[3][1] -2)*(x[2][1] +x[3][1] - 2);
   temp3 = (x[4][1] -1)*(x[4][1] -1);
   temp4 = (x[5][1] -1)*(x[5][1] -1);

   f = temp1 + temp2 + temp3 + temp4;
   A = Zero([n+M, n+M]);
   C = Zero([n+M, 1]);
   B  = Zero([n,n]);
   Df = Zero([n,1]);
   DF = Zero([n,1]);
   b  = Zero([M,1]);
   d  = Zero([n,1]);

   Df[1][1] =  2*(x[1][1]- x[2][1]);
   Df[2][1] = -2*(x[1][1]- x[2][1]) + 2*(x[2][1]+ x[3][1] -2);
   Df[3][1] =  2*(x[2][1]+ x[3][1]-2);
   Df[4][1] =  2*(x[4][1]- 1);
   Df[5][1] =  2*(x[5][1]- 1);

   B = Diag([n,1]); /* initialize B as identity matrix */

   b[1][1] = -4;
   b[2][1] =  0;
   b[3][1] =  0;

   k = 1;
   Trans_Dg = [1,3,0,0,0; 0,0,1,1,-2; 0, 1, 0, 0,-1];
   Dg       = Trans(Trans_Dg);
   G       = Trans_Dg*x+b;

/* [b] : Initialize matrices A and C */

   for ( i = 1; i <= n+M; i = i + 1) {
       for(j = 1; j <= n + M; j = j + 1) {
          if( i <= n) then {
             C[i][1] = -Df[i][1];
             if(j <= n) then {
                A[i][j] = B[i][j];
             } else {
                k = j - n;
                A[i][j] = Dg[i][k];
             }
          } else {
            k = i - n;
            C[i][1] = -G[k][1];
            if(j <= n) {
               A[i][j] = Trans_Dg[k][j];
            }
          }
       }
   }

   q = Zero([n,1]);

/* [c] : Main Loop */

   for ( ii = 1; ii <= IMAX; ii = ii + 1 ) {

   /* [c.1] : Solve QP for direction d */

      y = Solve(A,C);
        
      for ( i = 1; i <= n; i = i + 1) {
            d[i][1] = y[i][1];
      }

      for ( i = n+1; i <= n+M; i = i + 1) {
            j = i-n;
            lambda[j][1] = y[i][1];
      }

      if(L2Norm(d) <= Epsilon) {
         break;
      }

   /* [c.2] : Line Search with Armijo's Rule */

      t       = 1;
      x       = x_old + t*d;
      temp1   = (x[1][1] -x[2][1])*(x[1][1] -x[2][1]);
      temp2   = (x[2][1] +x[3][1] -2)*(x[2][1] +x[3][1] - 2);
      temp3   = (x[4][1] -1)*(x[4][1] -1);
      temp4   = (x[5][1] -1)*(x[5][1] -1);
      f_new   = temp1 + temp2 + temp3 + temp4;

      temp    = alpha*QuanCast(Trans(Df)*d);
      counter = 0;
      while (f_new > f + t*temp) { /* t is too large */
         counter = counter + 1;
         t       = t*beta;
         x       = x_old + t*d;
         temp1   = (x[1][1] -x[2][1])*(x[1][1] -x[2][1]);
         temp2   = (x[2][1] +x[3][1] -2)*(x[2][1] +x[3][1] - 2);
         temp3   = (x[4][1] -1)*(x[4][1] -1);
         temp4   = (x[5][1] -1)*(x[5][1] -1);
         f_new   = temp1 + temp2 + temp3 + temp4;
         if(counter > 5) {
            print " Too much iterations for line search \n";
            print " counter =", counter," \n";
            break;
         }
      }

   /* [c.3] : Modified BFGS matrix update */

      DF[1][1] =  2*(x[1][1]- x[2][1]);
      DF[2][1] = -2*(x[1][1]- x[2][1]) + 2*(x[2][1]+ x[3][1] -2);
      DF[3][1] =  2*(x[2][1]+ x[3][1]-2);
      DF[4][1] =  2*(x[4][1]- 1);
      DF[5][1] =  2*(x[5][1]- 1);

      q     = DF + Dg*lambda - Df - Dg*lambda_old;
      A1    = QuanCast(Trans(d)*B*d)*t*t;
      A2    = QuanCast(Trans(d)*q)*t;

      if(A2 >= (0.2*A1)) then {
         theta = 1.0;
      } else {
         theta = 0.8*A1/(A1-A2);
      } 

      r          = theta*q + (1-theta)*t*B*d;
      A3         = QuanCast(Trans(d)*r)*t;
      B          = B - B*d*Trans(d)*B/A1 + r*Trans(r)/A3;
      Df         = DF;
      x_old      = x;
      lambda_old = lambda;
      G          = Trans_Dg*x+b;

      for ( i = 1; i <= n; i = i + 1) {
          C[i][1] = -Df[i][1];
          for(j = 1; j <= n; j = j + 1) {
              A[i][j] = B[i][j];
          }
      }

      for ( i = 1; i <= M; i = i + 1) {
            k = i + n;
            C[k][1] = -G[i][1];
      }
   }

/* [d] : Print results and terminate program execution */

   print" Results: \n --------------\n";
   print" Iteration Number =", ii-1, "\n\n";
   PrintMatrix(x);
   quit;