linear objective function with linear constraints and one quadratic constraint

Question

I have an optimization problem in which the objective function and most constraints are linear, but I have one constraint is quadratic. I know my problem can not be reformulated as a convex , but if you have some advice on how to approach it in MATLAB ?

my problem as follows, $$\begin{array}{cl} \underset{w_{1},...,w_{6}}{ \min}& \sum_{t=1}^n ( - \mu_{t}(w)-u_t) \\s.t \sqrt{c_{1}^2+c_2^2} \geq u_{t} \\ c_{1}=h_{1t}(w) \\ c_{2}=h_{2t}(w)\\ w_1+...+w_6=1 \end{array} $$

please note, $ \mu_{t}(w) = a*w_1+...+a6*w_6 $ is a linear function. Also, $h_1t(w)= (b_1*w_1+...+b_6*w6 + d_1*w1+...+d_6*w_6)$ and $h_2t(w)= (f_1*w_1+...+f_6*w6 + g_1*w1+...+g_6*w_6)$. Finally, $a_1,...,a_6 , b_1,...,b_6 ,d_1,...,d_6 ,f_1,...,f_6$ and $g_1,...,g_6$ are constants.

EDIT: I add and remove some parts of my question.

Thanks in advance.

Answer 1

After eliminating one variable using your equality constraint, you are left with minimizing a linear objective function over the exterior of an ellipsoid. In general this will have no solution (you can make the objective function arbitrarily negative, while keeping $c_1^2 +c_2^2$ large enough to satisfy the inequality constraint, if you walk far enough in the direction of the negative objective gradient).

But in any case, in general when faced with a nonlinear optimization problem, you have several options:

1. Invoke Matlab's built-in black-box solvers for nonlinear problems; for instance fmincon implements an interior-point solver.
2. Try to use insights about the problem to simplify it yourself. For instance you can split your problem up into a regular linear program (in the case where the inequality constraint is inactive) and minimization of a linear function on a quadric surface (in the case where it's active). You can then apply some changes of variable to turn the quadric into a sphere or cylinder, etc.