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If I am trying to solve an optimization problem of a $f(x,y)$ subject to the following constraints:

$$0\le x \le y$$ $$ 0\le y\le 1$$

Then do I setup the Lagrangian as follows?

$$L(x,y,\lambda)=f(x,y)-\lambda_1(x-y)-\lambda_2(-x)-\lambda_3(-y)-\lambda_4(y-1)$$

I am trying to figure out how to transform this constrained optimization into an unconstrained one but having issues on how to account for $0\le y,x$

nvm
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  • It seems you have already transformed the problem to an unconstrained optimization and you have already accounted for the $0\leq y$ and $0\leq x$ constraints. It may help to note whether you are trying to maximize or minimize $f(x,y)$, and state whether or not $f$ is convex. This may affect the signs of your Lagrange multipliers. The usual convention is to minimize convex functions, and that multipliers for inequality constraints should be nonnegative. Your convention seems different to me. – Michael Jan 23 '20 at 04:42
  • @Michael I just wanted to make sure that the way I wrote my lagrangian was correct. – nvm Jan 23 '20 at 04:49
  • I see: You want us to tell you if you are doing things correctly, but you do not want to reveal whether you are trying to maximize or minimize $f$, and you do not want to reveal any convexity properties of $f$. – Michael Jan 23 '20 at 04:54
  • @michael I am trying to determine the general form for the lagrangian not dependent on the form of f. The function I am working with is f(x,y)=-x+y – nvm Jan 23 '20 at 04:59

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