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It's a matter of choice how one writes the Lagrangian in the context of Lagrange/KKT. Depending on how it's written, the gradients of the objective and constraint functions are either parallel or anti-parallel at a (suitable) optimum, and the Lagrange multiplier is neither negative or positive. At the end of the day, it is the same (subset of) optima that ...


1

The two ways of properly setting up Lagrangian are completely equivalent (although you have mistakes above) so it does not matter which way you set it up. You actually have mistake there resulting in the two ways being different: In the first equation you forgot minus sign since: $ \sum_i^l p_i w_i = \sum_i^l p_i x_i \implies 0 = \sum_i^l p_i w_i - \sum_i^l ...


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