Talk:Karush–Kuhn–Tucker conditions

message: The KKT theorem seems to refer to a different formulation of the problem that is actually there: The KKT theorem makes no reference to \lambda or h, and the Lagrangian in the theorem is L(x,\mu) whereas in the problem statement it is L(x,\mu,\lambda). Also, the term "optimal vector" is not defined, but if it means "globally optimal" then the theorem is wrong since the saddle point condition at most gives local optimality (and no convexity or the like is assumed here). In short, this central element in the page seems to be a mixture of different formulations. UffeHThygesen (talk) 09:47, 9 December 2020 (UTC)
thread_title:  Error in the main KKT theorem 

message: In other sources, there is a minus (instead of a plus) sign in front the multiplier (or it is to the right of the equality sign). Is the English Wikipedia article wrong? See for instance the Swedish article. /Anton  —Preceding unsigned comment added by 83.249.20.170 (talk) 20:00, 2 September 2009 (UTC)
replies: text: I have edited http://sv.wikipedia.org/wiki/Karush-Kuhn-Tucker-villkor to make the sign correct.  (2andrewknyazev (talk) 14:12, 29 July 2010 (UTC)
thread_title:  a minus or a plus ? 

message: The condition seems to be g_i(x)=<0 and g and f are concave  I think at least g should be convex see   e.g.  http://www2.imm.dtu.dk/courses/02711/lecture3.pdf
There seems to be a typo here: should the complementarity condition be
$\sum_i \mu_i g_i(x^\ast) = 0$, not $\mu_i g_i(x^\ast) = 0$ for each $i$?
Tveldhui 21:05, 17 March 2006 (UTC)
replies: text: Thanks for noticing the problem with linear versus equality constraints, which I fixed. Kiefer.Wolfowitz (talk) 07:59, 31 March 2010 (UTC)
thread_title:  typo 

message: h(x) = 0
193.143.32.39 09:48, 3 January 2007 (UTC)
replies: text: Isn't that implied by the stipulation that x* be a feasible point? -- Jitse Niesen (talk) 12:44, 4 January 2007 (UTC)
thread_title:  One Condition missing 

message: Does the $\lambda$ under "regularity" mean the $\mu$ and $\nu$ in the section before that? In this case this should be corrected, because the $\lambda$ for the dual multipliers doesn't appear anywhere else in the article.  —Preceding unsigned comment added by 84.75.149.134 (talk) 22:17, 30 January 2008 (UTC)
thread_title:  Regularity section 

message: * The article needs historical background.
* It needs to show how the Lagrange Method can be an special case of KKT, when one uses two inequality constraints as an equivalent of one equality constraint.
* From an economics student's point of view, the article should show more results related to the problem of cost minimization subject to constraints. Also, it would be a good idea to explain the different steps involved in applying the method to particular problems. For example, minimization problems imply a different set up of conditions than maximization problems. A reliable reference on this is Chiang and Wainwright's textbook of mathematical economics.
* The article needs a graphical ilustration of an example of functions defined over a set of inequality constraints.--Forich (talk) 03:13, 6 May 2008 (UTC)
replies: text: How about finding a max or min value of f(x) on an interval [a, b] or perhaps a two dimensional simplex? The inequalities would be constraints on the domain of x. You could have similar constraints on the range (say f has to be less than or equal to another function g). That should yield an interesting problem graphically. -- KarlHallowell (talk) 13:04, 14 May 2013 (UTC)

text: It also doesn't define what is meant by "Primal feasibility", "Dual feasibility", and "Complementary slackness". -- KarlHallowell (talk) 13:04, 14 May 2013 (UTC)
thread_title:  Suggestions for article improvement 

message: Someone correct me if I'm wrong, but shouldn't the sufficient conditions involve the Hessian? I think it's something like: the Hessian is positive definate with respect to feasible search directions? It appears the sufficient conditions are just copy/pasted necessary conditions... Alec.N (talk) 08:11, 27 October 2008 (UTC)
replies: text: You are correct. It's just a particular case. I've clarified this section a bit. (2andrewknyazev (talk) 18:00, 28 July 2010 (UTC)
thread_title:  Sufficient conditions 

message: I've tried different encodings, but all I see at the end of the article (regularity conditions) are little squares rather than math (evidently, with different encoding you get different gibberish). E.g. "It can be shown that LICQ⇒MFCQ" etc. I went to the source, hoping that it was some simple omission, but this wasn't the case.
Bottom line, the math needs to be fixed it seems.  —Preceding unsigned comment added by 83.83.234.195 (talk) 09:53, 9 February 2009 (UTC)
replies: text: Now added (2andrewknyazev (talk) 18:02, 28 July 2010 (UTC)
thread_title:  math not displayed 

message: DSOWOLS'SE  —Preceding unsigned comment added by 82.69.107.29 (talk) 09:02, 14 March 2010 (UTC)
thread_title:  SD 

message: In the discussion of the value function, shouldn't the \lambda really be \mu?
18.78.0.83 (talk) 21:10, 28 June 2010 (UTC)
thread_title:  typo 

message: In view of the confusion regarding the signs in the Lagrangian, it would be helpful with a short discussion of the maximization case also. Isheden (talk) 08:15, 14 December 2011 (UTC)
thread_title:  Maximization case 

message: The current example that is given seems to be confusing. Can someone please update it with more details?  — Preceding unsigned comment added by 128.97.206.107 (talk) 20:33, 22 January 2013 (UTC)
thread_title:  Example 

message: I am nore sure, but the following KKT conditions in the economic example doesn't sound correct to me : 
 (\text{d} R / \text{d} Q) (1+\mu ) - \mu (\text{d} C / \text{d} Q) \le 0,
I think it should be a "=".
:The source, which I have now added, uses the inequality version along with a condition that either the inequality holds with equality or the choice variable equals zero. Loraof (talk) 17:51, 10 April 2017 (UTC)
thread_title:  Mistake in the example ? 

message: I do not know what they are called but the second order conditions should be discussed more. Second order necessary conditions:
Hessian matrix of the objective function is positive semidefinite
Second order sufficient conditions:
Hessian matrix of the objective function is positive definite
Source (a large number, but here is one: http://www.stanford.edu/class/msande211/KKTgeometry.ppt)  — Preceding unsigned comment added by 150.135.222.234 (talk) 21:30, 8 March 2013 (UTC)
thread_title:  Second order necessary conditions? 

message: What means the comma in this operation? 
:\left[ \frac{\partial g(x^*)}{\partial x},  \frac{\partial h(x^*)}{\partial x}  \right]^T 
Can someone who understand it, better explain it in the article?
:Done. Loraof (talk) 03:01, 10 April 2017 (UTC)
thread_title:  matrix notation 

message: I have the feeling, the sufficient conditions should involve a test about positivity of the hessian in more than just one direction.
That is, till now, the text states a test
:s^T\nabla ^2_{xx}L(x^*,\lambda^*,\mu^*)s \ge 0
with s being orthogonal to the gradient of the constraint functions.
But shouldn't s stem from a whole cone of admissible directions (in case of inequality constraints)?
It's hard to find references, but here is something in French (see section 5.3.4): https://fr.wikipedia.org/wiki/Conditions_d%27optimalité#Problèmes_d'optimisation_avec_contraintes_d'égalité_et_d'inégalité
2003:D5:9F20:9500:5C01:2FA:636C:3372 (talk) 12:32, 19 November 2021 (UTC)
thread_title:  Problem with second-order sufficient conditions? 

message: The current text includes:
"The solution x^*, \lambda^*, \mu^* found in the above section is a constrained local minimum if for the Lagrangian,
: L(x,\lambda,\mu) = f(x) + \sum_{i=1}^m \mu_i g_i(x) + \sum_{j=1}^\ell \lambda_j h_j(x) 
then,
:s^T\nabla ^2_{xx}L(x^*,\lambda^*,\mu^*)s \ge 0"
My strong impression is that, grammatically, the word "then" is superfluous, and it confuses (me). Also, the text reads as if the first formula is merely an apposition to or an explanation of "the Lagrangian".
Now, I understand that two formulas separated by a mere comma may make for an ugly layout. Therefore, I propose:
"The solution x^*, \lambda^*, \mu^* found in the above section is a constrained local minimum for the Lagrangian,
: L(x,\lambda,\mu) = f(x) + \sum_{i=1}^m \mu_i g_i(x) + \sum_{j=1}^\ell \lambda_j h_j(x) 
if
:s^T\nabla ^2_{xx}L(x^*,\lambda^*,\mu^*)s \ge 0"
Any motivated objection(s)?Redav (talk) 22:32, 28 January 2022 (UTC)
thread_title:  Grammar and layout