Geometric compromise programming: application in portfolio selection

Compromise programming (CP) aims to find solutions by minimising distances to an ideal point with maximum achievement which is usually infeasible. A common assumption in CP is that it is highly unlikely that the optimum decision will lie out of the bounds of the compromise set given by metrics p=1$p=1$ and p=infinity$p=\infty$ of the Minkowski distance function. This assumption excludes the use of multiplicative functions as a measure of achievement. We propose geometric CP (GCP) to provide alternative solutions based on multiplicative functions to overcome this limitation. This methodology is an extension of CP that allows to incorporate the principle of limited compensability. An additional interesting feature of GCP is that, under reasonable assumptions, characterises extreme seekers' behaviour with non-concave utility functions (expressing no preference for any of the extremes). We discuss the practical implications of our approach and present three numerical illustrations in the context portfolio selection.