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 = 
          algorithm: 'trust-region-reflective'
         iterations: 19
    constrviolation: 0
      firstorderopt: 8.3903e-06
       cgiterations: 1673
            message: 'Optimization terminated: relative function value changing by le...'

You can see that while convergence occurred in 20 iterations, the high number of CG iterations indicates that the cost of the linear system solve is high. In light of this cost, one strategy would be to limit the number of CG iterations per optimization iteration. The default number is the dimension of the problem divided by two, 200 for this problem. Suppose you limit it to 50 using the MaxPCGIter flag in options:

options = optimoptions(options,'MaxPCGIter',50);
[x,fval,exitflag,output] = ... 
        quadprog(H,f,[],[],[],[],lb,ub,xstart,options);

This time convergence still occurs and the total number of CG iterations (1547) has dropped:

exitflag,output

exitflag =
     3

output = 
          algorithm: 'trust-region-reflective'
         iterations: 36
    constrviolation: 0
      firstorderopt: 2.3821e-05
       cgiterations: 1547
            message: 'Optimization terminated: relative function value changing by le...'

A second strategy would be to use a direct solver at each iteration by setting the PrecondBandWidth option to inf:

options = optimoptions(options,'PrecondBandWidth',inf);
[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 = 
          algorithm: 'trust-region-reflective'
         iterations: 10
    constrviolation: 0
      firstorderopt: 2.8219e-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 = 
            message: 'Minimum found that satisfies the constraints.

Optimization com...'
          algorithm: 'interior-point-convex'
      firstorderopt: 1.4120e-06
    constrviolation: 0
         iterations: 8
       cgiterations: []
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