A problem with a linear objective function and having linear as well as second order cone constraints is called a second order cone programming (SOCP) problem.
A simple type of closed convex pointed cone that captures many optimization problems of interest is the second order cone. Mathematically, this can be represented as:
x(3)< 10- r,
where r = x(1)^2 + x(2)^2
A second order cone (SOC) constraint of dimension n specifies that the vector formed by a set of n decision variables must belong to this cone.
SOCP problem can be solved in MATLAB using the interior-point algorithm implemented in the fmincon function. The SOC constraint can be defined as the non-linear constraint.
A similar problem (here the objective function is non-linear) is mentioned in our documentation :