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recently, I have the following problem when designing the generalized benders decomposition.

Given the primal solution of a strict convex (nonlinear) optimization, is the dual variable computed from the KKT condition unique? Or is it possible that there may be multiple optimal solutions to the dual variables? We can assume the constraint qualifications hold.

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If the primal solution is unique, then the dual solution must be uniquely defined as well. It's possible to prove this using a generic strictly convex program, but for intuition, consider that such a strictly convex function can be cast as $f(\mathbf{x})=\mathbf{x}^TA\mathbf{x} + \mathbf{b}^T\mathbf{x}$, with $A\succ0$. Regardless of the feasible set, the the primal solution $\mathbf{x}^*$ will always involve an inverse of $A$, which is unique. Moreover, $A^{-1}$ is also PD, and so the dual program is unique and also strictly convex.

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