Minimization with Dense Structured Hessian, Linear Equalities
Hessian Multiply Function for Lower Memory
The fmincon interior-point and trust-region-reflective algorithms,
and the fminunc trust-region algorithm
can solve problems where the Hessian is dense but structured. For
these problems, fmincon and fminunc do
not compute H*Y with the Hessian H directly,
because forming H would be memory-intensive. Instead,
you must provide fmincon or fminunc with
a function that, given a matrix Y and information
about H, computes W = H*Y.
In this example, the objective function is nonlinear and linear
equalities exist so fmincon is used. The description
applies to the trust-region reflective algorithm; the fminunc trust-region algorithm
is similar. For the interior-point algorithm, see the 'HessMult' option
in Hessian Multiply Function. The
objective function has the structure
where V is a 1000-by-2 matrix. The Hessian
of f is dense, but the Hessian of is sparse. If the Hessian of
is , then H,
the Hessian of f, is
To avoid excessive memory usage that could happen by working
with H directly, the example provides a Hessian
multiply function, hmfleq1. This function, when
passed a matrix Y, uses sparse matrices Hinfo,
which corresponds to
, and V to compute the Hessian
matrix product
W = H*Y = (Hinfo - V*V')*Y
In this example, the Hessian multiply function needs
and V to
compute the Hessian matrix product. V is a constant,
so you can capture V in a function handle to an
anonymous function.
However,
is not a constant and must be computed at the
current x. You can do this by computing
in the objective
function and returning
as Hinfo in the third output
argument. By using optimoptions to set the 'Hessian' options
to 'on', fmincon knows to get
the Hinfo value from the objective function and
pass it to the Hessian multiply function hmfleq1.
Step 1: Write a file brownvv.m that computes the objective function, the gradient, and the sparse part of the Hessian.
The example passes brownvv to fmincon as
the objective function. The brownvv.mbrownvv.m file
is long and is not included here. You can view the code with the command
type brownvv
Because brownvv computes the gradient and
part of the Hessian as well as the objective function, the example
(Step
3) uses optimoptions to
set the GradObj and Hessian options
to 'on'.
Step 2: Write a function to compute Hessian-matrix products for H given a matrix Y.
Now, define a function hmfleq1 that uses Hinfo,
which is computed in brownvv, and V,
which you can capture in a function handle to an anonymous function,
to compute the Hessian matrix product W where W = H*Y = (Hinfo - V*V')*Y. This function must
have the form
W = hmfleq1(Hinfo,Y)
The first argument must be the same as the third argument returned
by the objective function brownvv. The second argument
to the Hessian multiply function is the matrix Y (of W
= H*Y).
Because fmincon expects the second argument 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. Finally, you can use a function handle to an anonymous function
to capture V, so V can be the third argument to 'hmfleqq'.
function W = hmfleq1(Hinfo,Y,V); %HMFLEQ1 Hessian-matrix product function for BROWNVV objective. % W = hmfleq1(Hinfo,Y,V) computes W = (Hinfo-V*V')*Y % where Hinfo is a sparse matrix computed by BROWNVV % and V is a 2 column matrix. W = Hinfo*Y - V*(V'*Y);
Note
The function |
Step 3: Call a nonlinear minimization routine with a starting point and linear equality constraints.
Load the problem parameter, V, and the sparse
equality constraint matrices, Aeq and beq,
from fleq1.mat, which is available in the optimdemos folder.
Use optimoptions to set the GradObj and Hessian options
to 'on' and to set the HessMult option
to a function handle that points to hmfleq1. Call fmincon with
objective function brownvv and with V as
an additional parameter:
function [fval, exitflag, output, x] = runfleq1
% RUNFLEQ1 demonstrates 'HessMult' option for FMINCON with linear
% equalities.
problem = load('fleq1'); % Get V, Aeq, beq
V = problem.V; Aeq = problem.Aeq; beq = problem.beq;
n = 1000; % problem dimension
xstart = -ones(n,1); xstart(2:2:n,1) = ones(length(2:2:n),1); % starting point
options = optimoptions(@fmincon,'Algorithm','trust-region-reflective','GradObj','on', ...
'Hessian','user-supplied','HessMult',@(Hinfo,Y)hmfleq1(Hinfo,Y,V),'Display','iter', ...
'TolFun',1e-9);
[x,fval,exitflag,output] = fmincon(@(x)brownvv(x,V),xstart,[],[],Aeq,beq,[],[], ...
[],options);
To run the preceding code, enter
[fval,exitflag,output,x] = runfleq1;
Because the iterative display was set using optimoptions,
this command generates the following iterative display:
Norm of First-order
Iteration f(x) step optimality CG-iterations
0 2297.63 1.41e+03
1 1084.59 6.3903 578 1
2 1084.59 100 578 3
3 1084.59 25 578 0
4 1084.59 6.25 578 0
5 1047.61 1.5625 240 0
6 761.592 3.125 62.4 2
7 761.592 6.25 62.4 4
8 746.478 1.5625 163 0
9 546.578 3.125 84.1 2
10 274.311 6.25 26.9 2
11 55.6193 11.6597 40 2
12 55.6193 25 40 3
13 22.2964 6.25 26.3 0
14 -49.516 6.25 78 1
15 -93.2772 1.5625 68 1
16 -207.204 3.125 86.5 1
17 -434.162 6.25 70.7 1
18 -681.359 6.25 43.7 2
19 -681.359 6.25 43.7 4
20 -698.041 1.5625 191 0
21 -723.959 3.125 256 7
22 -751.33 0.78125 154 3
23 -793.974 1.5625 24.4 3
24 -820.831 2.51937 6.11 3
25 -823.069 0.562132 2.87 3
26 -823.237 0.196753 0.486 3
27 -823.245 0.0621202 0.386 3
28 -823.246 0.0199951 0.11 6
29 -823.246 0.00731333 0.0404 7
30 -823.246 0.00505883 0.0185 8
31 -823.246 0.00126471 0.00268 9
32 -823.246 0.00149326 0.00521 9
33 -823.246 0.000373314 0.00091 9
Local minimum possible.
fmincon stopped because the final change in function value relative to
its initial value is less than the selected value of the function tolerance.Convergence is rapid for a problem of this size with the PCG iteration cost increasing modestly as the optimization progresses. Feasibility of the equality constraints is maintained at the solution.
problem = load('fleq1'); % Get V, Aeq, beq
V = problem.V; Aeq = problem.Aeq; beq = problem.beq;
norm(Aeq*x-beq,inf)
ans =
2.3093e-14Preconditioning
In this example, fmincon cannot use H to
compute a preconditioner because H only exists
implicitly. Instead of H, fmincon uses Hinfo,
the third argument returned by brownvv, to compute
a preconditioner. Hinfo is a good choice because
it is the same size as H and approximates H to
some degree. If Hinfo were not the same size as H, fmincon would
compute a preconditioner based on some diagonal scaling matrices determined
from the algorithm. Typically, this would not perform as well.