minimize linear objective function with quadratic constraint

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

Answer 1

[Alan Weiss]

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

[Az.Sa]

I edited my question to be clear

Answer 2

[Alan Weiss]

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

[Az.Sa]

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