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$$\min_{x\geq 0}\sum_{i=1}^n (a_i-x b_i)^2 [a_i-x b_i\leq 0],\quad a_i,b_i\in\mathbb R,n\in\mathbb N$$

where $[p]$ is an Iverson bracket.

The objective function seemed easy (convex).

1.Is there any the concrete name for this kind of optimization or specific method to solve?

2.Any reference material?

3.To find the solution, is there suitable package in R?

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    Well, it's a constrained optimization (programming) problem... the piecewise nature of your function would certainly trip up most of the routines, since they assume nice behavior of the derivatives...2011-07-24
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    Unless the rest of you guys have better ideas, you might want to check out any of the stochastic methods (e.g. "differential evolution" or "simulated annealing"); they take a fair bit of computational effort, but at least they don't assume that the objective function has neat derivatives...2011-07-24
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    Thanks for your kindness.I haven't thought so looking-easy problem is difficult. Maybe I should quit. But I will go to browse materials as you mentioned to try.2011-07-24
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    The derivative of your function is piecewise linear, with corners at $x_j = \frac{a_j}{b_j}$. At each such corner, the slope of $f'(x)$ increases by $2b_i^2$. A few application of the cumsum function plus a linear interpolation should allow you to find the answer.2011-07-24

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