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- - **Optimization problem**
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Optimization problemI have the following optimization problem: $\displaystyle \textrm{Minimize }f(x) + g(y)$ $\displaystyle x, y$ $\displaystyle \textrm{Subject to } h(x,y) = x + y - k = 0$ where the cost functions are quadratic: $\displaystyle f(x) = a_1 + b_1x + \frac{c_1}{2} x^2$ $\displaystyle g(y) = a_2 + b_2y + \frac{c_2}{2} y^2$ The Lagrangian is defined as $\displaystyle \mathcal{L}(x, y, \lambda) = f(x) + g(y) - \lambda h(x,y)$ Therefore, the Lagrangian of this problem is $\displaystyle \mathcal{L}(x, y, \lambda) = a_1 + b_1x + \frac{c_1}{2} x^2 + a_2 + b_2y + \frac{c_2}{2} y^2- \lambda(x + y - k)$ The book that I have says that the Karush-Kuhn-Tacker (KKT) optimality conditions for this problem are $\displaystyle \frac{\partial \mathcal{L}}{\partial x} = c_1 x - \lambda = 0$ $\displaystyle \frac{\partial \mathcal{L}}{\partial y} = c_2 y - \lambda = 0$ $\displaystyle \frac{\partial \mathcal{L}}{\partial \lambda} = -(x + y - k) = 0$ but if these are just partial derivatives of the Lagrangian, then what happened to $\displaystyle b_1$ and $\displaystyle b_2$? Should they instead be $\displaystyle \frac{\partial \mathcal{L}}{\partial x} = b_1 + c_1 x - \lambda = 0$ $\displaystyle \frac{\partial \mathcal{L}}{\partial y} = b_2 + c_2 y - \lambda = 0$ $\displaystyle \frac{\partial \mathcal{L}}{\partial \lambda} = -(x + y - k) = 0$ The book states that the solutions are: $\displaystyle x = \frac{c_2}{c_1 + c_2} k$ $\displaystyle y = \frac{c_1}{c_1 + c_2} k$ $\displaystyle \lambda = \frac{c_1 c_2}{c_1 + c_2} k$ If I include $\displaystyle b_1$ and $\displaystyle b_2$, I get the following solutions instead: $\displaystyle x = \frac{b_2 - b_1 + c_2 k}{c_1 + c_2}$ $\displaystyle y = \frac{b_1 - b_2 + c_1 k}{c_1 + c_2}$ $\displaystyle \lambda = \frac{b_1 c_c + b_2 c_2 + c_1 c_2 k}{c_1 + c_2} $ |

This is truly a superficial response, but the book's answer (if correct) seems to imply that the b parameters are known (somehow) to equal zero (or to be negligible in practice). Since this model seems to come from economics, implicit assumptions may be hidden almost anywhere. |

You are correct and the book is wrong. |

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