Quadratic Minimization with Dense, Structured Hessian

The quadprog trust-region-reflective method can solve large problems where the Hessian is dense but structured. For these problems, quadprog does not compute H*Y with the Hessian H directly, as it does for trust-region-reflective problems with sparse H, because forming H would be memory-intensive. Instead, you must provide quadprog with a function that, given a matrix Y and information about H, computes W = H*Y.

In this example, the Hessian matrix H has the structure H = B + A*A' where B is a sparse 512-by-512 symmetric matrix, and A is a 512-by-10 sparse matrix composed of a number of dense columns. To avoid excessive memory usage that could happen by working with H directly because H is dense, the example provides a Hessian multiply function, qpbox4mult. This function, when passed a matrix Y, uses sparse matrices A and B to compute the Hessian matrix product W = H*Y = (B + A*A')*Y.

In the first part of this example, the matrices A and B need to be provided to the Hessian multiply function qpbox4mult. You can pass one matrix as the first argument to quadprog, which is passed to the Hessian multiply function. You can use a nested function to provide the value of the second matrix.

The second part of the example shows how to tighten the TolPCG tolerance to compensate for an approximate preconditioner instead of an exact H matrix.

Step 1: Decide what part of H to pass to quadprog as the first argument.

Either A or B can be passed as the first argument to quadprog. The example chooses to pass B as the first argument because this results in a better preconditioner (see Preconditioning).

Step 2: Write a function to compute Hessian-matrix products for H.

Now, define a function runqpbox4 that:

• Contains a nested function qpbox4mult that uses A and B to compute the Hessian matrix product W, where W = H*Y = (B + A*A')*Y. The nested function must have the following form, where the first two arguments Hinfo and Y are required:

W = qpbox4mult(Hinfo,Y,...)

• Loads the problem parameters from qpbox4.mat.

• Uses optimoptions to set the HessianMultiplyFcn option to a function handle that points to qpbox4mult.

• Calls quadprog with B as the first argument.

The first argument to the nested function qpbox4mult must be the same as the first argument passed to quadprog, which in this case is the matrix B.

The second argument to qpbox4mult is the matrix Y (of W = H*Y). Because quadprog expects Y to be used to form the Hessian matrix product, Y is always a matrix with n rows, where n is the number of dimensions in the problem. The number of columns in Y can vary. The function qpbox4mult is nested so that the value of the matrix A comes from the outer function. The runqpbox4.m file appears at the end of this example.

Step 3: Call a quadratic minimization routine with a starting point.

To call the quadratic minimizing routine contained in runqpbox4, enter

[fval,exitflag,output] = runqpbox4;

Local minimum possible.

quadprog stopped because the relative change in function value is less than the sqrt of the function tolerance, the rate of change in the function value is slow, and no negative curvature was detected.

Then display the values for fval, exitflag, output.iterations, and output.cgiterations.

fval,exitflag,output.iterations, output.cgiterations

fval = -1.0538e+03

exitflag = 3

ans = 18

ans = 30

After 18 iterations with a total of 30 PCG iterations, the function value is reduced to

fval

fval = -1.0538e+03

and the first-order optimality is

output.firstorderopt

ans = 0.0043

Preconditioning

Sometimes quadprog cannot use H to compute a preconditioner because H only exists implicitly. Instead, quadprog uses B, the argument passed in instead of H, to compute a preconditioner. B is a good choice because it is the same size as H and approximates H to some degree. If B were not the same size as H, quadprog would compute a preconditioner based on some diagonal scaling matrices determined from the algorithm. Typically, this would not perform as well.

Because the preconditioner is more approximate than when H is available explicitly, adjusting the TolPCG parameter to a somewhat smaller value might be required. This example is the same as the previous one, but reduces TolPCG from the default 0.1 to 0.01. The runqpbox4prec function is listed at the end of this example.

[fval,exitflag,output] = runqpbox4prec;

Local minimum possible.

quadprog stopped because the relative change in function value is less than the sqrt of the function tolerance, the rate of change in the function value is slow, and no negative curvature was detected.

After 18 iterations and 50 PCG iterations, the function value has the same value to five significant digits

fval

fval = -1.0538e+03

and the first-order optimality is essentially the same:

output.firstorderopt

ans = 0.0028

Note: Decreasing TolPCG too much can substantially increase the number of PCG iterations.

Helper Functions

This code creates the runqpbox4 helper function.

function [fval, exitflag, output, x] = runqpbox4
%RUNQPBOX4 demonstrates 'HessianMultiplyFcn' option for QUADPROG with bounds.

problem = load('qpbox4'); % Get xstart, u, l, B, A, f
xstart = problem.xstart; u = problem.u; l = problem.l;
B = problem.B; A = problem.A; f = problem.f;
mtxmpy = @qpbox4mult; % function handle to qpbox4mult nested function

% Choose algorithm and the HessianMultiplyFcn option
options = optimoptions(@quadprog,'Algorithm','trust-region-reflective','HessianMultiplyFcn',mtxmpy);

% Pass B to qpbox4mult via the H argument. Also, B will be used in
% computing a preconditioner for PCG.
[x, fval, exitflag, output] = quadprog(B,f,[],[],[],[],l,u,xstart,options);

    function W = qpbox4mult(B,Y)
        %QPBOX4MULT Hessian matrix product with dense structured Hessian.
        %   W = qpbox4mult(B,Y) computes W = (B + A*A')*Y where
        %   INPUT:
        %       B - sparse square matrix (512 by 512)
        %       Y - vector (or matrix) to be multiplied by B + A'*A.
        %   VARIABLES from outer function runqpbox4:
        %       A - sparse matrix with 512 rows and 10 columns.
        %
        %   OUTPUT:
        %       W - The product (B + A*A')*Y.
        %

        % Order multiplies to avoid forming A*A',
        %   which is large and dense
        W = B*Y + A*(A'*Y);
    end

end

This code creates the runqpbox4prec helper function.

function [fval, exitflag, output, x] = runqpbox4prec
%RUNQPBOX4PREC demonstrates 'HessianMultiplyFcn' option for QUADPROG with bounds.

problem = load('qpbox4'); % Get xstart, u, l, B, A, f
xstart = problem.xstart; u = problem.u; l = problem.l;
B = problem.B; A = problem.A; f = problem.f;
mtxmpy = @qpbox4mult; % function handle to qpbox4mult nested function

% Choose algorithm, the HessianMultiplyFcn option, and override the TolPCG option
options = optimoptions(@quadprog,'Algorithm','trust-region-reflective',...
    'HessianMultiplyFcn',mtxmpy,'TolPCG',0.01);

% Pass B to qpbox4mult via the H argument. Also, B will be used in
% computing a preconditioner for PCG.
% A is passed as an additional argument after 'options'
[x, fval, exitflag, output] = quadprog(B,f,[],[],[],[],l,u,xstart,options);

    function W = qpbox4mult(B,Y)
        %QPBOX4MULT Hessian matrix product with dense structured Hessian.
        %   W = qpbox4mult(B,Y) computes W = (B + A*A')*Y where
        %   INPUT:
        %       B - sparse square matrix (512 by 512)
        %       Y - vector (or matrix) to be multiplied by B + A'*A.
        %   VARIABLES from outer function runqpbox4prec:
        %       A - sparse matrix with 512 rows and 10 columns.
        %
        %   OUTPUT:
        %       W - The product (B + A*A')*Y.
        %

        % Order multiplies to avoid forming A*A',
        %   which is large and dense
        W = B*Y + A*(A'*Y);
    end

end