Matlab optimization - variables to satisfy L1 norm

Hi,

I am trying to set up an optimization problem, the variables of my problem are stored in vector x, which is of length 4.

How do I constrain my optimization such that the first three elements in my vector x satisfy the L1 norm.

In my current setup i am using fmincon with the following code:

    x0=[A_init'; th_init];
    opt=optimoptions('fmincon','MaxFunctionEvaluations',1e5,'Display','iter','MaxIterations',1e3,...
        'Algorithm','sqp','FiniteDifferenceStepSize',1e-6,'DiffMinChange',0.00051,'DiffMaxChange',0.2);
    [x,fval,exitflag,output]=fmincon(@(x)costFunction_es_sa(x,a,N,epRm,epR_target),x0,[],[],[],[],[0.1 0.1 0.1 0.01]',[1 1 1 0.61]',[],opt)
    

Answer 1

[Manikanta Aditya]

To constrain the first three elements of your vector ( x ) to satisfy the ( L1 ) norm in an optimization problem using fmincon in MATLAB, you can add a nonlinear constraint function.

% Initial guess
x0 = [A_init'; th_init];

% Optimization options
opt = optimoptions('fmincon', 'MaxFunctionEvaluations', 1e5, 'Display', 'iter', ...
    'MaxIterations', 1e3, 'Algorithm', 'sqp', 'FiniteDifferenceStepSize', 1e-6, ...
    'DiffMinChange', 0.00051, 'DiffMaxChange', 0.2);

% Nonlinear constraint function
nonlcon = @(x) l1NormConstraint(x);

% Call fmincon with the nonlinear constraint
[x, fval, exitflag, output] = fmincon(@(x) costFunction_es_sa(x, a, N, epRm, epR_target), ...
    x0, [], [], [], [], [0.1 0.1 0.1 0.01]', [1 1 1 0.61]', nonlcon, opt);

% Nonlinear constraint function definition
function [c, ceq] = l1NormConstraint(x)
    % L1 norm constraint for the first three elements of x
    c = sum(abs(x(1:3))) - c_bound; % c_bound is your desired bound for the L1 norm
    ceq = []; % No equality constraints
end

[Simon Philipp Hehenberger]

Thank you for your answer.

In fact I want the l1 norm of the first three values to be exactly 1.

So I have set up the nonlinear constraint function like this:

function [c,ceq] = l1NormConstraint(x)
    % L1 norm constraint for the first three elements of x
    c=[];
    ceq = sum(abs(x(1:3)))-1; 
end

But I am getting the error: "Failure in initial nonlinear constraint function evaluation. FMINCON cannot continue." What am I missing

[Aquatris]

Probably your x0 does not satisfy the L1 norm condition. You need to provide more info for us to be helpful.

[Manikanta Aditya]

+1 to what @Aquatris mentioned.

Also refer this MATLAB Answer to know more about the error:

• https://www.mathworks.com/matlabcentral/answers/694040-failure-in-initial-nonlinear-constraint-function-evaluation-fmincon-cannot-continue

[Simon Philipp Hehenberger]

Initial condition not satisfying the nonlinear constraint was indeed the issue.

Thanks alot for your help @Aquatris and @Manikanta Aditya

[Manikanta Aditya]

Thanks for informing @Simon Philipp Hehenberger

Answer 2

[Aquatris]

You create a nonlinear constraint functio and give it as an argument to the fmincon.

[x,fval,exitflag,output]=fmincon(@(x)costFunction_es_sa(x,a,N,epRm,epR_target),x0,...
    [],[],[],[],[0.1 0.1 0.1 0.01]',[1 1 1 0.61]',@normL1,opt)

function y = normL1(x); %fmincon will try to satisfy myFun(x)<=0
    y = sum(abs(x(1:3))) - 5; % want it to be less than 5 for instance
end