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I have to minimize the function: $$F(x) = \sum_{i=1}^{M}\left\|x_{i+1} - x_i - K\left(\frac{x_{i+1} + x_i}{2}\right)\right\|^2 + \|x_1-c_1\|^2 + \|x_N-c_2\|^2,$$ where $x$ is a vector of $N$ scalars, $c$ are constants and the latter two terms are my positional constraints.

Now, the $K(s)$ function (scalar in, scalar out) is just a $1 \rm D$ linear (but non-convex) data interpolator
I have a feeling that this could be simplified further (to a linear problem? quadratic?)

For example I was thinking to express the $K(s)$ function as a sum of tent functions (one at each data point), but I'm not sure if I'm getting anywhere better with that.

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You need to explain with $K$ is in more detail, as linearity obviously hinges of $K$. Also, what do you mean by optimizing the function $F(x) = 0$??? – copper.hat May 15 '12 at 17:49
Is $K$ convex, linear,... (This depends on the data you have, of course.) – copper.hat May 15 '12 at 18:05
I understand that, but if your data is 'convex' then the function $K$ will be too. – copper.hat May 15 '12 at 18:10
Well, in this case, your $F_i$ is convex iff $K$ is concave. Linear terms do not affect convexity (or concavity). – copper.hat May 15 '12 at 18:20
I've reformulated the problem based on your comments (and deleted my comments as well). – Babis May 15 '12 at 18:25

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