===== Optimization set =====
==== Set ====
| @#55CCEE: context     | @#55CCEE: $ B $ |
| @#55CCEE: context     | @#55CCEE: $ \langle Y, \le \rangle $ ... Non-strict partially ordered set |
| @#55CCEE: context     | @#55CCEE: $ r:B\to Y $ |
| @#FF9944: definition  | @#FF9944: $ O_r := \{\beta\in B\mid \forall(b\in B).\,r(\beta)\le{r(b)}\}$ |

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>todo
>#tag 
>If p are parameters and c_p(x) curves with x_min(c_p)=f(p) known, try to find x_min(c') by fitting c_p to c'. Now what's p here. Is there a scheme so that we can extend the list p to have guaranteed that there are parameters so that eventually c_p=c'?

If ${\mathrm{min}(r)}\subseteq Y$ denote the minimum values of $r$, then 

$O_r = r^{-1}({\mathrm{min}(r)})$

with $r^{-1}:{\mathcal P}Y\to{\mathcal P}B$.

Compare with [[Solution set]].

=== Parametrized regression ===
Consider a test set

$\langle x_0,y_0\rangle \in X\times Y$,

where $x_0$ somehow depends on $y_0$.

Use $B$-family of fit functions

$f:B\to(X\to Y)$

(the indexed subspace of $X\to Y$ is called hypotheses space)

and find from this set find the optimal fit (given by optimal $\beta\in B$) w.r.t. loss function 

$V:Y\times Y\to Y$ 

by optimizing

$r(\beta):=V(f(\beta,x),y)$

As a remark, given a function $f$ (resp. a $\beta$), the value $V(f(\beta,x_0),y_0)$ (or a multiple thereof) is called "empirical risk" in Statistical learning theory.

== Linear regression w.r.t. least square ==
$f(\beta,x):=\beta_0+\sum_{i=1}^N\beta_ix_i$

with loss function

$V({\hat y},y)=({\hat y}-y)\cdot({\hat y}-y)$

In practice, $x_i$ may be vectors and then $V$ is taken to be an inner product.

=== Reference ===
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=== Context ===
[[Non-strict partial order]]
=== Related ===
[[Solution set]]