maximizing the negative norm is it convex problem? I have an objective function with two decision variables x1 and x2
\begin{equation} 
\begin{array}{cl}
\underset{\alpha_{1},\alpha_{2}}{ \max}  a_1x_1+ a_2x_2 -c\sqrt{h_{1}(x) ^2 + h_{2}(x)^2}  \\ 
\text { s.t. }  x_{1}+x_{2}=1 
\end{array}
\end{equation}
I tried to introduce new variables
$b_1=h_1(x) ,b_2=h_2(x) $ so my objective function will be
\begin{array}{cl}
\underset{\alpha_{1},\alpha_{2}}{ \max}  a_1x_1+ a_2x_2 -cu  \\ 
\text { s.t. }  x_{1}+x_{2}=1 ,\\
 h_1(x)=b_1, \\ 
h_2(x)=b_2 ,\\
 \sqrt{b_1^2 + b_2^2 } \leq u 
\end{array}
my questions:

*

*does the reformulation with the new constraints correct ?


*because I am maximizing the negative norm does that mean it is a convex function ?


*how can I solve my problem in MATLAB
Thanks in advance

Assuming $c$ is positive, it is a simple SOCP-representable convex problem. I would not be surprised if you can solve it analytically but otherwise solve it using an SOCP solver (or just use any nonlinear solver, it is easy enough)
Using YALMIP in MATLAB, you would just write
sdpvar x1 x2
h1 = ...
h2 = ...
Constraints= [x1+x2==1];
Objective = (a1*x1+a2*x2 - c*norm([h1;h2]);
optimize(Constraints,-Objective);

