Bias, exploitation and proxies in scenario-based risk minimization

When minimizing a risk measure over a set of scenarios, solutions are often optimistic in the sense that the in-sample, or perceived, risk is much less than the out-of-sample risk. Optimism, which can be attributed to the bias of the risk estimator and the exploitation of the scenarios' idiosyncracies by the optimization, increases with the amount of sampling error inherent in the scenarios and the flexibility afforded by the problem's formulation. Minimizing a proxy, namely an estimator of a risk measure different from the one that is actually of interest, can reduce optimism and improve out-of-sample performance. The effectiveness of a proxy depends on the sizes of its perceived risk, bias and exploitation in relation to those of the estimator for the risk measure of interest.