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The portfolio optimization problem is modeled as a mean-risk bicriteria optimization problem where the expected return is maximized and some (scalar) risk measure is minimized. In the original Markowitz model the risk is measured by the variance while several polyhedral risk measures have been introduced leading to Linear Programming (LP) computable portfolio optimization models in the case of discrete random variables represented by their realizations under specified scenarios. Among them, the second order quantile risk measures, recently, become popular in finance and banking. The simplest such measure, now commonly called the Conditional Value at Risk (CVaR) or Tail VaR, represents the mean shortfall at a specified confidence level. Recently, the second order quantile risk measures have been introduced and become popular in finance and banking. The corresponding portfolio optimization …