# Surrogate metric for Linear Program problem

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 $$b$$.

• 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. Commented Jun 4 at 15:24