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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?).

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Is this question related to the software package Mathematica? –  Sjoerd C. de Vries Nov 18 '13 at 23:12
    
If constant a, b, c, k are given you can find minimum with Minimize, e.g. Minimize[{Abs[3 + 2 x^2 - x y + y^2]}, {x, y}] yields {3, {x -> 0, y -> 0}}. Your question seems to be a duplicate of How do I determine the maximum value for a polynomial, given a range of x values? –  Artes Nov 18 '13 at 23:21
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Instead of minimizing Abs[...] you could try to minimize Power[...,2]. Should yield the same result, but is better behaved. –  Sjoerd C. de Vries Nov 18 '13 at 23:52
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I guess we can't help having no idea what exactly your function is. I mean you should provide a minimal example of Y (matrix A etc.) to get a grasp what your real problem is. So far it seems to be too general. –  Artes Nov 19 '13 at 0:06
    
You can't possibly map it to a convex problem because it can have multiple local minima. For example, let $Y = \lvert-100+x^2+y^2\rvert$ with $x,y\in[-1,1]$: the minimum is attained at $(x,y)=(\pm1,\pm 1)$. –  Rahul Nov 19 '13 at 0:28

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