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I would like to solve the following quadratically constrained linear programming problem.

I have written a Matlab code (R2091b) that solve the problem using Gurobi. Now, I would like to rewrite the code using fmincon instead of Gurobi. This is because the optimisation problem will have to be solve thousands of times and Gurobi's academic license does not allow to parallelise via array jobs in a cluster. However, I'm encountering a huge problem: Gurobi takes 0.23 second to give a solution, fmincon takes 13 sec. I suspect this should be due to my mistakes/inefficiences in providing gradient, hessian, etc. Could you kindly help me to improve below?

Also, Gurobi gives me 0.2 as solution, fmincon gives 0.089. Can the accuracy of fmincon be improved without trying other starting points?

This is my code with Gurobi

clear

rng default

load matrices

%1) GUROBI

model.A=[Aineq; Aeq];

model.obj=f;

model.modelsense='max';

model.sense=[repmat('<', size(Aineq,1),1); repmat('=', size(Aeq,1),1)];

model.rhs=[bineq; beq];

model.ub=ub;

model.lb=lb;

model.quadcon(1).Qc=Q;

model.quadcon(1).q=q;

model.quadcon(1).rhs=d;

params.outputflag = 0;

result=gurobi(model, params);

max_problem_Gurobi=result.objval;

This is my code with fmincon

%2) FMINCON

options = optimoptions(@fmincon,'Algorithm','interior-point',...

'SpecifyObjectiveGradient',true,...

'SpecifyConstraintGradient',true,...

'HessianFcn',@(z,lambda)quadhess(z,lambda,Q));

fun = @(z)quadobj(z,f.');

nonlconstr = @(z)quadconstr(z,Q,d);

[~,fval] = fmincon(fun,z0,Aineq,bineq,Aeq,beq,lb,ub,nonlconstr,options);

max_problem_fmincon=-fval;

function [y,yeq,grady,gradyeq] = quadconstr(z,Q,d)

y= z'*Q*z -d;

yeq = []; %no quadratic inequalities

if nargout > 2

grady = 2*Q*z;

end

gradyeq = []; %no quadratic inequalities

end

function hess = quadhess(z,lambda,Q) %#ok<INUSL>

hess = 2*lambda.ineqnonlin*Q;

end

function [y,grady] = quadobj(z,f)

y = -f'*z;

if nargout > 1

grady = -f;

end

Further, if I run the code with fmincon with as starting point the optimal point given by Gurobi, I still get the solution 0.089 (instead of 0.2 as in Gurobi). Why?

%3) FMINCON

z0=result_Gurobi.x;

options = optimoptions(@fmincon,'Algorithm','interior-point',...

'SpecifyObjectiveGradient',true,...

'SpecifyConstraintGradient',true,...

'HessianFcn',@(z,lambda)quadhess(z,lambda,Q));

fun = @(z)quadobj(z,f.');

nonlconstr = @(z)quadconstr(z,Q,d);

[~,fval] = fmincon(fun,z0,Aineq,bineq,Aeq,beq,lb,ub,nonlconstr,options);

max_problem_fmincon=-fval;

function [y,yeq,grady,gradyeq] = quadconstr(z,Q,d)

y= z'*Q*z -d;

yeq = []; %no quadratic inequalities

if nargout > 2

grady = 2*Q*z;

end

gradyeq = []; %no quadratic inequalities

end

function hess = quadhess(z,lambda,Q) %#ok<INUSL>

hess = 2*lambda.ineqnonlin*Q;

end

function [y,grady] = quadobj(z,f)

y = -f'*z;

if nargout > 1

grady = -f;

end

end

Matt J
on 30 Mar 2020

Well, it would be interesting to know what algorithm Gurobi uses, but the issue of the objective function difference appears to be a matter of the tolerances

options = optimoptions(@fmincon,'Algorithm','interior-point',...

'SpecifyObjectiveGradient',true,...

'SpecifyConstraintGradient',true,...

'HessianFcn',@(z,lambda)quadhess(z,lambda,Q),...

'StepTolerance',1e-30,'OptimalityTolerance',1e-10);

fun = @(z)quadobj(z,f);

nonlconstr = @(z)quadconstr(z,Q,d);

tic;

[~,fval] = fmincon(fun,z0(:),Aineq,bineq,Aeq,beq,lb,ub,nonlconstr,options);

toc

max_problem_fmincon=-fval

max_problem_fmincon =

0.2000

Matt J
on 31 Mar 2020

Matt J
on 31 Mar 2020

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