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Assume $x\in \operatorname{cone}\{a_1, a_2,\ldots, a_m\}$. Is there a systematic way to find out the coefficients of $x$ with respect to $a_i$'s?

When $a_i$'s are independent, it should easy. What about otherwise?

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up vote 0 down vote accepted

I presume by $\operatorname{cone}$ you mean the convex cone generated by $a_i$.

The problem is equivalent to solving $A \mu = x$, $\mu \geq 0$, with $\mu \in \mathbb{R}^n$, and the inequality interpreted component-wise.

You could solve the quadratic program: $\min_{\mu} \{\frac{1}{2} || A \mu -x||^2 \; | \; \mu \geq 0 \}$.

Or you could solve the linear program: $\min_{(\mu, \alpha)} \{\alpha \; | \; A \mu = x, \; -\mu_i-\alpha \leq 0, \; \forall i \}$.

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