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Quadratic Minimization with Bound Constraints

To minimize a large-scale quadratic with upper and lower bounds, you can use the quadprog function with the 'trust-region-reflective' algorithm.

The problem stored in the MAT-file qpbox1.mat is a positive definite quadratic, and the Hessian matrix H is tridiagonal, subject to upper (ub) and lower (lb) bounds.

Step 1: Load the Hessian and define f, lb, and ub.

load qpbox1   % Get H
lb = zeros(400,1); lb(400) = -inf;
ub = 0.9*ones(400,1); ub(400) = inf;
f = zeros(400,1); f([1 400]) = -2;

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

xstart = 0.5*ones(400,1);
options = optimoptions('quadprog','Algorithm','trust-region-reflective');
[x,fval,exitflag,output] = ... 
        quadprog(H,f,[],[],[],[],lb,ub,xstart,options);

Looking at the resulting values of exitflag and output,

exitflag,output

exitflag =
     3

output =
  struct with fields:
          algorithm: 'trust-region-reflective'
         iterations: 19
    constrviolation: 0
      firstorderopt: 9.5437e-06
       cgiterations: 1638
            message: 'Optimization terminated: relative function value changing by le…'

You can see that while convergence occurred in 19 iterations, the high number of CG iterations indicates that the cost of solving the linear system is high. In light of this cost, try using a direct solver at each iteration by setting the SubproblemAlgorithm option to 'factorization':

options = optimoptions(options,'SubproblemAlgorithm','factorization');
[x,fval,exitflag,output] = ... 
        quadprog(H,f,[],[],[],[],lb,ub,xstart,options);

Now the number of iterations has dropped to 10:

exitflag,output

exitflag =
     3

output =
  struct with fields:
          algorithm: 'trust-region-reflective'
         iterations: 10
    constrviolation: 0
      firstorderopt: 1.2656e-06
       cgiterations: 0
            message: 'Optimization terminated: relative function value changing by le…'

Using a direct solver at each iteration usually causes the number of iterations to decrease, but often takes more time per iteration. For this problem, the tradeoff is beneficial, as the time for quadprog to solve the problem decreases by a factor of 10.

You can also use the default 'interior-point-convex' algorithm to solve this convex problem:

options = optimoptions('quadprog','Algorithm','interior-point-convex');
[x,fval,exitflag,output] = ... 
        quadprog(H,f,[],[],[],[],lb,ub,[],options);

Check the exit flag and output structure:

exitflag,output

exitflag =
     1

output =
  struct with fields:
            message: 'Minimum found that satisfies the constraints.…'
          algorithm: 'interior-point-convex'
      firstorderopt: 1.4120e-06
    constrviolation: 0
         iterations: 8
       cgiterations: []
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