minimize linear objective function with quadratic constraint

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As stated in Koenker (2005) "Quantile Regression" page 10 equation (1.20). Quantile regression problem has the form
may be reformulated as a linear program as :
where X now denotes the usual n × p matrix of regressors and y be the n × 1 vectors of outcomes and is a n × 1 vector of ones
or it can be written as:
In my case, I am trying to minimize the following quantile function
my objective function is linear with one quadratic constraint and the rest constraints are linears.
My quastion is how to solve my linear objective function subjecte to one quadratic constraint in matlab ?
Thanks in advance

Answers (2)

Alan Weiss
Alan Weiss on 1 Jul 2022
I'm not sure that I understand you correctly. If is an array, then perhaps you are looking for
z = zeros(size(y - yhat));
m1 = max(z,y - yhat);
m2 = max(z,-(y - yhat));
Alan Weiss
MATLAB mathematical toolbox documentation

Alan Weiss
Alan Weiss on 11 Jul 2022
Again, I am not sure that I understand you correctly. I do not know what your optimization variables are, and while you say that your constraints are linear and quadratic, the way you wrote the constraints makes them look more general. But if your and functions are linear in whatever your optimization variables are, then it is possible that the correct formulation of your problem would convert it to a coneprog form. See the coneprog reference page.
Alan Weiss
MATLAB mathematical toolbox documentation
  1 Comment
Az.Sa
Az.Sa on 11 Jul 2022
Edited: Az.Sa on 11 Jul 2022
my optimization variables are . are both linear functions.
to my understanding: minimizing same as minimizing - Norm which means it is maximizing and in this case my constraint will be , if it is minimizing without the negative sign then my constraint will be which means it is second order cone. please correct me if the way I wrote my inequality constraint is incorrect

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