Summary: A risk measure in a portfolio selection problem is linear programming (LP) solvable, if it has a linear formulation when the asset returns are represented by discrete random variables, i.e., they are defined by their realizations under specified scenarios. The efficient frontier corresponding to an LP solvable model is a piecewise linear curve. In this paper we describe a method which realizes and produces a tangency portfolio as a by-product during the procedure of tracing out of the efficient frontier of risky assets in an LP solvable model, when our portfolio contains some risky assets and a riskless asset, using nonsmooth optimization methods. We show that the test of finding the tangency portfolio can be limited only for two portfolios. Also, we describe that how this method can be employed to trace out the efficient frontier corresponding to a portfolio selection problem in the presence of a riskless asset.