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I have a convex optimization problem and I am trying to solve it using lagrange duality method. For one of the solutions of my problem, negative values are obtained for some the lagrange multipliers. My question is: this solution is not optimal or not feasible? Indeed I want to know what does a negative lagrange multiplier imply? What if we find a solution to the original problem which yields some negative lagrange multipliers?

Answer: I think I have found the answer to my question. If a Lagrange Multiplier is computed as negative, its associated constraint is redundant, i.e., the optimal solution with and without that constraint is the same.

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closed as unclear what you're asking by Brian Borchers, Chris Custer, JonMark Perry, Claude Leibovici, Riccardo.Alestra Apr 27 '18 at 9:34

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    $\begingroup$ You'll need to provide more information about your problem. For equality constraints, the Lagrange multiplier can have either sign, while for inequality constraints there will be a sign restriction. $\endgroup$ – Brian Borchers Apr 27 '18 at 0:36
  • $\begingroup$ All of the constraints are inequality constraints. However, one of the feasible solutions of the original problem results in negative lagrange multipliers. Does this mean that the solution is not optimal or Am I missing something? $\endgroup$ – Hasti Apr 27 '18 at 0:50
  • $\begingroup$ Are the constraints $\geq$ or $\leq$? Is the objective minimized or maximized? $\endgroup$ – Brian Borchers Apr 27 '18 at 0:59
  • $\begingroup$ The constraint are $\leqslant$ and it is a maximization problem. $\endgroup$ – Hasti Apr 27 '18 at 1:09