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In mean-variance optimisation I have typically seen the below quadratic utility function where $𝐸[𝑅]$ is the expected return (or mean return) of a possible portfolio, $\sigma^{2}$ is the return variance of that portfolio and $A$ is a parameter that the determines the sensitivity to variance. $$𝑈=𝐸[𝑅]−\frac{1}{2}A\sigma^{2}$$ As an aside, maximising % ...

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Objective function of Lagrangian can be set up either with $+\lambda$ or $-\lambda$, depending on how you solve the budget constraint. Actually, for the solution it does not matter if $\lambda$ has negative or positive sign in the equation. You can clearly see it from the formula if you expand the second term:  \mathcal{L}(x,y, \lambda) = U(x,y) + \lambda(...

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