2017年11月10日 · i.e. $\mathrm{strong ~ duality} \implies \mathrm{KKT ~ is ~ necessary ~ condition ~ for ~ optimal ~ solution}$ and in pp. 244, (When the primal problem is convex) if $\tilde{x}, \tilde{\lambda}, \tilde{\mu}$ are any points that satisfy the KKT conditions, then $\tilde{x}$ and $(\tilde{\lambda}, \tilde{\mu})$ are primal and dual optimal, with ...

In general, the KKT conditions DO NOT HAVE TO BE SATISFIED at an optimal solution. Nonetheless, Fritz John's conditions must always be satisfied. In fact, Fritz John's condition is a generalization of the KKT conditions. Fritz John's conditions reduce to the KKT conditions if some regularity conditions are satisfied at that point.

2020年5月18日 · In Hastie, Tibshirani and Wainwright "Statistical Learning with Sparsity" page 99 equation (5.11) in section 5.2.2 they use the generalized KKT-conditions in the special case of the Lasso-estimator. I am going to refer to this in my assignment, even though it isn't proved in …

2021年4月14日 · I am currently reading the book An introduction to optimization by Chang and Zak. When reading about the Karush Kuhn Tucker (KKT) conditions, I came across this geometrical explanation of the KKT t...

(a): yes, the KKT conditions will by construction "miss" solutions that aren't regular. If you want to find those, you must use other means. Note that there are methods tailored to specific constraint structures, which are able to identify local solutions where the KKT conditions fail to apply. This is more of a case-by-case game. (b): yes ...

2020年4月9日 · The discussion indicates for non-convex problem, KKT conditions may be neither necessary nor sufficient conditions for primal-dual optimal solutions. ${\bf counter-example4}$ For a convex problem, even strong duality holds, there could be no solution for the KKT condition, thus no solution for Lagrangian multipliers.

2018年12月30日 · Stack Exchange Network. Stack Exchange network consists of 183 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

2017年9月25日 · Stack Exchange Network. Stack Exchange network consists of 183 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

2021年12月27日 · Since stationarity of $(X', y_i')$ alone is sufficient for its equality-constrained problem, whereas inequality-constrained problems require all KKT conditions to be fulfilled, it is not surprising that fulfilling some of the KKT conditions for $(X, y_i)$ does not imply fulfilling the condition for $(X', y_i')$.

That is, do I need to discern the set of active constraints ahead of time to setup the KKT conditions? If so, how would I without knowing the optimal solution apriori? Obviously, because of complementarity I know that the lagrange multipliers of inactive constraints will inevitably become zero, but is there a way to know which will be inactive ...