Optimization Toolbox

Nonlinear Minimization with Equality Constraints

The large-scale method for fmincon can handle equality constraints if no other constraints exist. Suppose you want to minimize the same objective as in Eq. 2-7, which is coded in the function brownfgh.m, where n = 1000, such that for Aeq that has 100 equations (so Aeq is a 100-by-1000 matrix).

Step 1: Write an M-file brownfgh.m that computes the objective function, the gradient of the objective, and the sparse tridiagonal Hessian matrix.

As before, this file is rather long and is not included here. You can view the code with the command

• type brownfgh
  

Because brownfgh computes the gradient and Hessian values as well as the objective function, you need to use optimset to indicate that this information is available in brownfgh, using the GradObj and Hessian parameters.

The sparse matrix Aeq and vector beq are available in the file browneq.mat:

• load browneq
  

The linear constraint system is 100-by-1000, has unstructured sparsity (use spy(Aeq) to view the sparsity structure), and is not too badly ill-conditioned:

• condest(Aeq*Aeq')
  ans =
    2.9310e+006
  

Step 2: Call a nonlinear minimization routine with a starting point xstart.

• fun = @brownfgh;
  load browneq          % Get Aeq and beq, the linear equalities
  n = 1000;
  xstart = -ones(n,1); xstart(2:2:n) = 1;
  options = optimset('GradObj','on','Hessian','on', ...
                     'PrecondBandWidth', inf); 
  [x,fval,exitflag,output] = ...
     fmincon(fun,xstart,[],[],Aeq,beq,[],[],[],options); 
  

Setting the parameter PrecondBandWidth to inf causes a sparse direct solver to be used instead of preconditioned conjugate gradients.

The exitflag indicates convergence with the final function value fval after 16 iterations:

• exitflag =
       1
  fval =
    205.9313
  output = 
         iterations: 16
          funcCount: 16
       cgiterations: 14
      firstorderopt: 2.1434e-004
          algorithm: 'large-scale: projected trust-region Newton'
  

The linear equalities are satisfied at x.

• norm(Aeq*x-beq)
  ans =
    1.1913e-012
  

Nonlinear Minimization with Bound Constraints and Banded Preconditioner

Nonlinear Minimization with a Dense but Structured Hessian and Equality Constraints