A study of phase transitions for convergence analysis of spin glasses: application to portfolio selection problems

To date, the spin glass paradigm has been gainfully used in solving a number of optimization problems by devising a mapping between our understanding of spin interactions within a natural spin glass and the given optimization problems. Among the determining factors in a natural spin glass, phase transition is a physical phenomenon that is controlled by temperature. Depending on the spin glass’s phase, spin glasses behave differently and may or may not reach the globally desired optimum. This study aims to determine this critical temperature below which convergence to a global optimum is more likely. Furthermore, we aim to determine the main parameters that characterize this critical temperature. Specifically, the critical temperature is studied as applied to the portfolio selection problem. It is shown that below the critical temperature, the glass consistently reaches the optimal states, whereas, convergence to optimum becomes increasingly unlikely if temperature exceeds this critical temperature. Application to five of the world’s major financial markets reveals that the critical temperature is directly proportional to covariance and the average return of assets and does not depend on the number of assets. In other words, all stock markets, that have the same asset covariance and average return, also have the same critical temperature. This is confirmed by several empirical tests such as correlation, entropy and hamming distance.