Abstract: This paper is part one of an homage to the seminal paper of Artzner, Delbaen, Eber, and Heath (1997) [1], who proposed a set of axioms that must be satisfied by risk measures in order to be “coherent”.

This paper does not aim to add to the knowledge of coherent risk measures, but it aims to prove that coherence matters not only for the mathematician, but also for the investment manager and his clients by constructing simple and transparent examples that show the dangers of working with incoherent risk measures. This way the author hopes to improve communication between the academic communities on one side and on the others side policy-makers and operational decision makers at financial institutions, their regulators and law-makers, who fifteen year after that paper still underestimate the importance of “thinking coherently”.

Keywords: Portfolio Selection, Personal Financial Decision Making, Coherent Risk Measures, Strategic Asset Allocation, Suitability of Investments, Value at Risk, Expected Shortfall, Global Exposure, Risk Classification, risk and reward indicator, variance, UCITS IV, FINRA 2011.

Abstract: Graph theory has been a powerful tool in solving difficult and complex problems arising in all disciplines. In particular, graph matching is a classical problem in pattern analysis with enormous applications. Many graph problems have been formulated as a mathematical program then solved using exact, heuristic and/or approximated-guaranteed procedures. On the other hand, graph theory has been a powerful tool in visualizing and understanding of complex mathematical programming problems, especially integer programs. Formulating a graph problem as a natural integer program (IP) is often a challenging task. However, an IP formulation of the problem has many advantages. Several researchers have noted the need for natural IP formulation of graph theoretic problems. The aim of the present study is to provide a unified framework for IP formulation of graph matching problems. Although there are many surveys on graph matching problems, however, none is concerned with IP formulation. This paper is the first to provide a comprehensive IP formulation for such problems. The framework includes variety of graph optimization problems in the literature. While these problems have been studied by different research communities, however, the framework presented here helps to bring efforts from different disciplines to tackle such diverse and complex problems. We hope the present study can significantly help to simplify some of difficult problems arising in practice, especially in pattern analysis.

Keywords: Combinatorial optimization, graph matching, integer programming, quadratic assignment problem.