# Talk:Karush–Kuhn–Tucker conditions

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## KKT in complex spaces

Could anyone please write on validity of the KKT for real-valued functions of complex vector variables? The current revision of the article covers the real case only. (2andrewknyazev (talk) 23:48, 22 July 2010 (UTC)).

This would be a nonstandard topic, better more of research literature than an encyclopedia. (A complex vector space with complex dimension D can be viewed as a real vector space of dimension 2D.) Kiefer.Wolfowitz (talk) 11:08, 25 July 2010 (UTC)
Is there a special reason for complex being "nonstandard"? Why, e.g., minimizing a real quadratic form over a complex space would be any less "standard" than over a real space? If it is true that the KKT holds without any changes for complex vector variables, that would be an encyclopedia-level statement. If there is a special reason, why it is not true, that would also be an encyclopedia-level statement, I guess. (2andrewknyazev (talk) 16:53, 26 July 2010 (UTC))
WP is not a forum for original research. An article on KKT conditions should respect WP guidelines, particularly wrt the weight of topics in standard, leading reliable references (e.g. textbooks and research monographs, or invited review articles).
In a constructive spirit, let me suggest some references. For the complex field, you may look at literature about Hilbert's 17th problem (sums of squares representations, e.g. J-B Lasserre), plurisubharmonic functions, etc. No reliable standard reference comes to mind for complex vector spaces: It may be worthswhile to look at Karlin's book from the 1950s, e.g. Sobyczyk (sic.). Kiefer.Wolfowitz (talk) 21:52, 29 July 2010 (UTC)

Complex optimization problems can be easliy transformed to real optimization problems mechanically. Thus I do not think it have any special properties. Before transformation you need define a order on complex space, to transform it to real one. If you mean real-valued function over complex field space, it is even simpler, just analyze a vector space with 2n dimensions (real and imaginary parts). --91.213.255.7 (talk) 03:44, 10 April 2012 (UTC)

## a minus or a plus ?

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)

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)).

## typo

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)

These are equivalent statements.

The introduction says "Allowing inequality constraints, the KKT approach to nonlinear programming generalizes the method of Lagrange multipliers, which had traditionally required linear constraints." Shouldn't "linear" (the second-to-last word) be replaced with "equality"? AFAIK, KKT and Langrange multipliers both do nonlinear programming quite naturally, and the first three words of the sentence also seem to support the idea that the shortcoming of Lagrange multipliers is that they require "equality constraints." 18.214.1.205 (talk) 02:46, 31 March 2010 (UTC)

Thanks for noticing the problem with linear versus equality constraints, which I fixed. Kiefer.Wolfowitz (talk) 07:59, 31 March 2010 (UTC)

## One Condition missing

h(x) = 0 193.143.32.39 09:48, 3 January 2007 (UTC)

Isn't that implied by the stipulation that x* be a feasible point? -- Jitse Niesen (talk) 12:44, 4 January 2007 (UTC)

## Regularity section

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)

## Suggestions for article improvement

• 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)
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)
• The article refers to active constraints without defining what it means by "active". I suppose this refers to conditions with non-zero mu and lambda? Janez Demsar (talk) 19:41, 31 August 2009 (UTC)
It also doesn't define what is meant by "Primal feasibility", "Dual feasibility", and "Complementary slackness". -- KarlHallowell (talk) 13:04, 14 May 2013 (UTC)
• I think the article should reference (and compare/contrast with) the well known Fritz John conditions as well (see | Bertsekas' Fritz John paper for some background ). Also, more elaboration on the Slater condition would be great, since this is one of the most used conditions. Lavaka (talk) 16:47, 16 September 2009 (UTC)

## Sufficient conditions

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)

The only difference between the necessary/sufficient conditions is that in the sufficient case f, g are convex and h is affine.

You are correct. It's just a particular case. I've clarified this section a bit. (2andrewknyazev (talk) 18:00, 28 July 2010 (UTC)).

## math not displayed

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)

the Hessian matrix determines whether the region is convex or not, H(x)> 0 means that it is stricly convex H(x)≥ 0 the region is convex. Strict convexity ensures that the solution found is a global minimum and thus optimum otherwise it might simply be a saddle point. —Preceding unsigned comment added by 86.163.157.227 (talk) 11:18, 18 April 2009 (UTC)

## First Order Necessary Conditions (FONC)

In the book Applied Optimization (2006) from Ross Baldick, the author almost makes no mention of the KKT conditions, and instead uses terms such as First Order Necessary Conditions (FONC), and Second Order Sufficient Conditions (SOSC). Are these terms used regularly in optimization? Do they warrant a mention in this article? ---201.127.248.133 (talk) 01:34, 10 February 2010 (UTC)

Now added (2andrewknyazev (talk) 18:02, 28 July 2010 (UTC)).

## SD

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

## typo

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)

## Maximization case

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)

## Example

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)

How about some examples? Lagrange multipliers article, have multiple examples of using them to solve some optimizations. Similar simple examples could be created for this article. Does anybody have some external resources, books or articles, with sensible examples? --91.213.255.7 (talk) 03:45, 10 April 2012 (UTC)

German and even more Portugal wikipedia looks to have interesting examples. — Preceding unsigned comment added by 91.213.255.7 (talk) 03:49, 10 April 2012 (UTC)

## Mistake in the example ?

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 "=".

## Second order necessary conditions?

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)