Optimization Toolbox

Equality Constrained Example

For routines that permit equality constraints, nonlinear equality constraints must be computed in the M-file with the nonlinear inequality constraints. For linear equalities, the coefficients of the equalities are passed in through the matrix Aeq and the right-hand-side vector beq.

For example, if you have the nonlinear equality constraint and the nonlinear inequality constraint , rewrite them as

and then solve the problem using the following steps.

Step 1: Write an M-file objfun.m.

• function f = objfun(x)
  f = exp(x(1))*(4*x(1)^2+2*x(2)^2+4*x(1)*x(2)+2*x(2)+1);
  

Step 2: Write an M-file confuneq.m for the nonlinear constraints.

• function [c, ceq] = confuneq(x)
  % Nonlinear inequality constraints
  c = -x(1)*x(2) - 10;
  % Nonlinear equality constraints
  ceq = x(1)^2 + x(2) - 1;
  

Step 3: Invoke constrained optimization routine.

• x0 = [-1,1];            % Make a starting guess at the solution
  options = optimset('LargeScale','off');
  [x,fval] = fmincon(@objfun,x0,[],[],[],[],[],[],... 
     @confuneq,options)
  [c,ceq] = confuneq(x) % Check the constraint values at x
  

After 21 function evaluations, the solution produced is

• x =
     -0.7529    0.4332
  fval =
      1.5093
  c =
      -9.6739
  ceq =
    4.0684e-010
  

Note that ceq is equal to 0 within the default tolerance on the constraints of 1.0e-006 and that c is less than or equal to zero as desired.

Gradient Check: Analytic Versus Numeric

Maximization