Capital budgeting problem

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In its elementary form there is a fixed amount of capital, say LaTeX: C, that can be allocated to any of LaTeX: n investments. Each investment has a minimum level, say LaTeX: L, and a maximum level, say LaTeX: U. The expected return on investment is a function, LaTeX: v_j(x_j), where LaTeX: x_j is the level of the LaTeX: j-th investment opportunity (LaTeX: L_j \le x_j \le U_j). Risk is measured by a standard deviation from the expected return, say LaTeX: s_j(x_j). The problem is to maximize total expected return, subject to a budget constraint: LaTeX: \sum_j x_j \le C, and a risk constraint: LaTeX: \sum_j v_j(x_j) + a_j s_j(x_j)} \le b, where LaTeX: a_j and LaTeX: b are parameters. The returns on the investments could be correlated. Then, if LaTeX: Q is the variance-covariance matrix, the risk constraint is quadratic: LaTeX: v^Tx + x^T Q x \le b. (Also see the portfolio selection problem.)

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