## Cone projection algorithm

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## 1Definition of the Cone Projection Problem and Algorithm

We have the data set $(x_i,\phi _i),i=1,2,\ldots ,n$ which has emerged from a convex function f at least $C^{(2)}[x_1,x_n]$ by the process:
We want to find the vector y that has the smallest euclidean distance from $\phi$ subject to the requirement of convexity $A\,y\geq {0}$, thus we have to solve the next primal optimization problem:
There are two equivalent versions for the matrix A of the convexity inequalities constraints. The first one is is to observe that we have strict inequalities:
so starting from the definition of convexity we proceed to the inequalities:
By constructing now all the above inequalities for $i=1,2,\ldots ,n-2$ we have formulated the matrix $A^{(i)}$.

The second way is obtained if we have equal spaced $x_i$-data. Then it is easy to eliminate the same positive quantity $\Delta {x}=x_{j+1}-x_{j}$ from all inequalities:
again with $i=1,2,\ldots ,n-2$ and create the matrix $A^{(ii)}$.
Every algorithm that solves the Cone Projection Problem is called a Cone Projection Algorithm.
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