I am trying to decide the best way to solve this:
Minimise $Y = | K + ax^2 + bxy + cy^2 |$,
with decision variables $x$ and $y$ and $a > 0$. I also have a number of convex and affine constraints I won't bother listing to do with the ratio of $x$ to $y$.
My problem is the absolute value operation in the objective function. If it wasn't there, this would (to me) appear to be a simple quadratic program, as the objective function would be convex. But no matter how I carve it up, (e.g., I've thought about substitute variables, doing it in two stages, etc.), but I always end up with a concave part of the problem that (I presume) standard QP software won't like.
(can't seem to add a comment so have to place edit here): Thanks for advice but is Minimise in this situation p-time efficient? I don't want to use a non-linear optimiser - I'd much prefer to map it to a nice convex (hopefully quadratic-convex) problem if possible. My real problem has x1 to xn with non-sparse A matrix. I have been looking into second-order cone but haven't got very far on that track.
(second edit). Thanks again guys for additional comments. Rahul I guess that is fair enough; I think given that insight my question basically becomes "what is the most elegant way of solving this" rather than "help me force this to be convex".
Sjoerd thankyou - if I read you right you are proposing to minimise Y^2 = px^4 + qx^3y + rx^2y^2 + sxy^3 + ty^4 + c. Can you elaborate on how the minimisation is better behaved? I seem to recall polynomial techniques are available but not sure where to start; I presume the mission is - assuming p is positive - to evaluate Y^2 at all the places where the first deriv of Y^2 is zero? And that means the problem then reduces to locating all of those points (not straight forward I imagine in n dimensions?).