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I am working on a constrained optimization problem modeled as a Mixed-Integer Linear Program,

$\textrm{argmax}_x c \cdot x \hspace{3mm}$ such that $\hspace{3mm} Ax \leq b \hspace{5mm}$ (1) .

Here $c \in \mathbb{R}^n$ is a vector of forecasted prices. This is available before some auction. $x \in \mathbb{R}^n$ denotes a vector of positions taken (the portfolio vector).

After the auction, the actual prices $\widehat{c}$ are known. Having taken positions $x$, the profit generated is

profit $= \widehat{c} \cdot x \hspace{5mm}$ (2) .

Eq. (2) would be maximized with a perfect portfolio $\widehat{x}$ obtained using a perfect forecast $\widehat{c}$ in eq. (1). But in reality my forecasts are trained to minimize Ordinary Least Squares (OLS),

$\mathrm{dist}(c, \widehat{c}) = |c - \widehat{c}|^2 \hspace{5mm}$ (3)

and have some nonzero distance from $\widehat{c}$. In practice, OLS does not provide an ideal loss function, as moving away from $\widehat{c}$ in some directions leads to a faster decrease in profit than in other directions. My question is now if anyone has a suggestion for a better metric than in eq. (3)? I.e. what form of dist ensures $\mathrm{dist}(c, \widehat{c}) \approx 0$ implies $c \cdot x \approx \widehat{c} \cdot \widehat{x}$?

Of course, the perfect metric dist will depend on the constraint matrix $A$ and vector $b$. More specifically, my question is if there is some proxy metric which is independent of or easily determined from $A$ and $b$.

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    $\begingroup$ Smart predict-then-optimize might be interesting for you. The idea is that the loss is not determined over the forecast error, but the decision error. Check out this. $\endgroup$
    – PeterD
    Commented Jun 4 at 15:24

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