11

I suspect you want to minimize $\sum\limits_i \left[k_1\max(e_i,0)^2 + k_2\max(-e_i,0)^2\right]$. For instance, $a=0.5$ would correspond to $k_1 = 0.25$ and $k_2 = 2.25$). This can be formulated as a quadratic program by introducing new variables $u$ and $v$, using the objective $k_1 u^\top u + k_2 v^\top v$ with the constraints $u\geq 0, u \geq e, v\geq 0, ...


10

There is indeed a paper titled Loss Distributions that provides the limited expected value functions $L(x)$ for several probability distributions (on page 15). It is directly related to the first-order loss function $n(x)$ through $$n(x)=\Bbb E(X)-L(x)\tag1$$ and notice that the loss function can also be written as $$n(x)=\int_x^\infty yf(y)\,dy-x(1-F(x))\...


3

We want to measure the quality of a solution, that is given two solutions, $x, y \in A$, I want to tell if $x$ is a better solution than $y$. Real numbered as an ordered field enable us to compare two solutions conveniently. For example, for a minimization problem, we can say that $x$ is a better solution than $y$ if $f(x) < f(y)$. It can carry meaning of ...


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