Quadprog 'interior-point-convex' failure
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Dear Matlab Gurus,
I have a problem with Matlab quadprog function.
A few years ago I implemented a code using quadprog with the 'active-set' algorithm, and it worked pretty well. The problem is characterised by a structurally dense hessian and inequality constraint matrix (see data attached).
Now this algorithm is no more available and only 'interior-point-convex' and 'trust-region-reflective' can be used. However, the 'trust-region-reflective' cannot be used with inequality constraints. So the choice is reduced to 'interior-point-convex' that is not able to solve the problem:
clear variables, close all
% load problem
load('testData.mat');
% quadprog setup (R2018b)
opts = optimoptions(@quadprog,...
'Algorithm','interior-point-convex');
tic
[x,fval,exitflag,output,lambda] = quadprog(H,f,A,b,Aeq,beq,[],[],x0,opts);
toc
The output is
The interior-point-convex algorithm does not accept an initial point.
Ignoring X0.
No feasible solution found.
quadprog stopped because it was unable to find a point that satisfies
the constraints within the default value of the constraint tolerance.
<stopping criteria details>
The funny is that I already have a feasible solution x0, but the 'interior-point-convex' algorithm does not accept an initial point:
% feasibility check
any(A*x0-b > 0)
ans =
0
When using the 'active-set' algorithm with a Matlab version < 2016 everything is fine and the algorithm ends correctly.
clear variables, close all
% load problem
load('testData.mat');
% quadprog setup (R2015b)
opts = optimoptions(@quadprog,...
'Algorithm','active-set');
tic
[x,fval,exitflag,output,lambda] = quadprog(H,f,A,b,Aeq,beq,[],[],x0,opts);
toc
with output
Warning: The 'active-set' algorithm will be removed in a future release. To avoid this warning or a future error, choose a
different algorithm: 'interior-point-convex' or 'trust-region-reflective'.
> In quadprog (line 427)
In test_quadprog_solvers (line 10)
Optimization terminated.
Elapsed time is 2.239872 seconds.
I have seen that fmincon still have the 'active-set' algorithm and I am wondering if with a proper setup of options I can trick the optimizer to solve a purely quadratic programming. I tried:
clear variables, close all
% load problem
load('testData.mat');
% fmincon setup (R2018b)
opts = optimoptions(@fmincon,...
'Algorithm','active-set',...
'SpecifyObjectiveGradient',true);
tic
[xred,fval,exitflag,output,lambda] = fmincon(@(x)quadFcn(x,H,f),x0,A,b,Aeq,beq,[],[],[],opts);
toc
With
function [fval,gval] = quadFcn(x,H,f)
fval = .5*x'*H*x+f'*x;
if nargout > 1 % gradient required
gval = H*x+f;
end
end
And the result is fine
Local minimum possible. Constraints satisfied.
fmincon stopped because the predicted change in the objective function
is less than the default value of the function tolerance and constraints
are satisfied to within the default value of the constraint tolerance.
<stopping criteria details>
Elapsed time is 3.966116 seconds.
But the calculation time is almost doubled with respect to the quadprog function with 'active-set' algorithm, and things are worse for more complex problems than this simple benchmark.
Any idea to restore the original behavior or performance?
Thank you in advance
Accepted Answer
Matt J
on 16 Jun 2019
Edited: Matt J
on 17 Jun 2019
Simply l2-normalizing the rows of A also seems to help,
load('testData-Matt.mat');
opts = optimoptions(@quadprog,'Algorithm','interior-point-convex');
anorm=vecnorm(A,2,2);
AAA=A./anorm;
bbb=b./anorm;
tic;
[y,fval,exitflag,output,lambda] = quadprog(H,f,AAA,bbb,[],[],[],[],[],opts);
toc%Elapsed time is 1.533334 seconds.
>> exitflag
exitflag =
1
4 Comments
Bruno Luong
on 17 Jun 2019
"It is still puzzling me why quadprog does not make scaling by itself."
Because to do it in the "right" way, user can do more than simple centering and scaling, such as linear transformation (or even non-linear in case of fmincon) of the input variables. Mathematically it's call right preconditioning of the problem.
Just keep in mind that when you do numerical minimization, precondingtioning is an important aspect that can break or make your problem resolution.
More Answers (2)
Bruno Luong
on 16 Jun 2019
You can shift the input variable, it seems to work
[y,fval,exitflag,output,lambda] = quadprog(H,f,A,b-A*x0,[],[],[],[],[],opts);
x=x0+y
4 Comments
Matt J
on 16 Jun 2019
Edited: Matt J
on 16 Jun 2019
@Fabio,
In addition to what Bruno mentions, you should rescale your constraints so that all b(i)=1. This noticeably speeds things up. Note that I've chosen a random objective vector f here because the one in testData.mat is just zero.
f=rand(size(f));
opts = optimoptions(@quadprog,'Algorithm','interior-point-convex');
bb=b-A*x0;
AA=A./bb;
bb=ones(size(bb));
tic
[y,fval,exitflag,output,lambda] = quadprog(H,f,A,b-A*x0,[],[],[],[],[],opts);
toc%Elapsed time is 15.233883 seconds - and still non-convergent
tic;
[y,fval,exitflag,output,lambda] = quadprog(H,f,AA,bb,[],[],[],[],[],opts);
toc %Elapsed time is 0.604501 seconds.
Matt J
on 16 Jun 2019
Edited: Matt J
on 16 Jun 2019
I don't have a pre-R2016a Matlab release readily at hand, and so I cannot compare performance. However, if the problem is simply that you would like to be able to incorporate the initial x0 into quadprog's interior-point-convex algorithm, you could try adding one more inequality constraint,
f.'*x <= f.'*x0
With this constraint, quadprog should restrict its search to points with objective better than x0.
7 Comments
Bruno Luong
on 16 Jun 2019
You might convince me if you show if there is any saving in the CPU time. Not sure adding this constraint help QUADPROG to better start or it make it even worse by dealing with more constraints.
Matt J
on 16 Jun 2019
Edited: Matt J
on 16 Jun 2019
Here is a case where it helped CPU time, though perhaps it is difficult to guarantee.
load('testData-Matt.mat');
anorm=vecnorm(A,2,2);
AAA=A./anorm;
bbb=b./anorm;
tic;
[y,fval,exitflag,output,lambda] = quadprog(H,f,AAA,bbb,[],[],[],[],[],opts);
toc %Elapsed time is 1.527962 seconds.
fff=f./norm(f);
bbbf=fff.'*x0;
AAAA=[AAA;fff.'];
bbbb=[bbb;bbbf];
tic;
[y,fval,exitflag,output,lambda] = quadprog(H,f,AAAA,bbbb,[],[],[],[],[],opts);
toc%Elapsed time is 1.068352 seconds.
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