Talk:Lagrange multipliers

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[edit] Notation change

Hi,

I just changed looking for an extremum of g to looking for an extremum of h although I'm not absolutely sure. But I think it is the right term.

Thanks for catching that; that occurrence of g seems to have been missed when the notation was changed in December.--Steuard 20:51, Jan 28, 2005 (UTC)
Strictly looking for an extremum of h(\vec x, \vec \lambda) also implies the original \vec g(\vec x) = \vec 0 via \frac{\partial \vec h}{\partial \vec \lambda} = \vec 0 . 84.160.236.56 19:29, 6 Feb 2005 (UTC)

[edit] Additional Sections for Lagrange Multipliers

Economic Interpretation

An important addition to the discussion on Lagrange multipliers, might be the economic interpretation of such variables, wherein they act as a quantity of marginal utility of a specific constraint to the objective function.

Primal-Dual Relationship

They also are fundamental in understanding the relationship between a primal and dual optimization problem. This is related to the economic interpretation since in the dual problem the Lagrangian multipliers become the decision variables and the primal decision variables have the role of the Lagrange mulipliers.

Complimentary Slackness When constraints are inequalities they are variables that indicate whether a constraint is binding. If binding then the multiplier is strictly greater than zero.

Trade-off between multiple objectives

In multi-objective optimization, one can formulate using the epsilon-constraint method, then the Lagrange multiplier can be interpreted as the tradeoff between mutliple objectives, which becomes an important value in decision analysis to understand the topology of the Pareto optimal frontier.

How to handle inequality

A discussion of how to introduce Lagrange multipliers in case of inequality constraints would be nice, since the only change required is to impose a sign requirement of the Lagrange multiplier. A figure explaining why this is so would be even nicer.

Please add to this list, or post information in the article and delete from this list, otherwise I will try to get my facts together and some references and post, using the established notation.--Kgcrowther 01:00, 16 August 2005 (UTC)

I had roughly the same things to say... this article seriously needs expansion. --02:28, 21 November 2007 (UTC)

[edit] Treatment of Lagrange Multipliers as a method ONLY

The treatment of Lagrange multipliers as a method only is narrow in scope. Although commonly used in some modern text as a method, they are also a variable with an important role, otherwise referred to as the shadow price or dual variable. --Kgcrowther 15:36, 16 August 2005 (UTC)

[edit] Latest change

There was an inconsistency with how I defined the Lagrangian (and how h was defined previously) in the article and the way it is defined in "without permanent scarring", which meant that I had left out a minus sign. I think the equations as written now are right, but I'm not sure which form is most standard. M0nkey 03:02, 10 February 2006 (UTC) (formerly 68.238.90.222)

[edit] This sentence does not make much sense...

"Only when the contour we are crossing actually touches tangentially the contour g = c we are following will this not be possible"

Perhaps it should be changed to:

"Only when the contour we are crossing touches tangentially the value of f does not change."

??

I've changed it. How do you like it now? -lethe talk + 18:53, 9 April 2006 (UTC)

[edit] Proofs?

I believe this section requires more information in regarding to proving that the coordinates obtained from the method are indeed maxima or minimas, ie more examples & proofs for examples?

Proofs are not really encyclopedic, and they may stay in the way of reading the article. More examples never hurt, and if proofs are really necessary at some point and they are helpful and short, one could also put in a proof maybe. Oleg Alexandrov (talk) 19:00, 3 May 2006 (UTC)
Theorems are only right when they are proven so, one can understand how to use the theorem, but it is only when he understands the proof does he truely understand the meaning of the theorem. I do appreciate how long proofs can do more harm than good to the article, but for instance adding the proof to a specific example, no matter how short, would greatly aid the reader in comprehending the text. RZ heretic 10:51, 5 May 2006 (UTC)
Existence of lagrange multipliers is a necessity condition, not a sufficiency condition. In any case I think some sort of proof would be helpful, specifically I think that geometric intuition suffices for the case of one constraint but is less clear when considering multiple constraints (with several multipliers). M0nkey 00:34, 15 June 2006 (UTC)

[edit] Simplicity

This article barely introduces a non mathematician to Lagrange multipliers. The first few sentences could do with a little English.

The article doesnt also mention the 'real world' applications of Lagrange multipliers. Kendirangu 07:32, 23 January 2007 (UTC)

I have to disagree; the lead has only words and a picture with no algebra. The remainder of the article only uses school-level algebra, and so should be comprehensible to many non-mathematicians. Applications in mechanics and economics are mentioned (in addition to the worked examples). --catslash 23:00, 18 February 2007 (UTC)
I agree with the simplicity comment, and added a gentler statement of the idea behind Lagrange multipliers. Having "only words" does not make the intro simple... the additional vocab such as "stationary points", "constrained functions", "Fermat's theorem", etc. make it a difficult read for the non-expert. Also, articles should provide intuition as well as mechanics. Triathematician (talk) 16:25, 1 January 2008 (UTC)
OK, yes a more intuitive/graphic description is desirable, and that was a valiant attempt, (and here comes the but) but the use of set of points is pretty technical, and I found it hard to relate the second sentence to the concept in question. Surely the point is that we can't just go anywhere we want on the mountain, but have to find the highest point we can reach whilst staying on the footpaths? Sorry not to be offering constructive suggestions! --catslash (talk) 21:16, 1 January 2008 (UTC)
My assumption was that "peak" implies the very highest point on the mountain. It's an inexact analogy, but I wanted to get the point of tangent surfaces across somehow without bringing in objective, constraint functions, etc. Triathematician (talk) 13:37, 4 January 2008 (UTC)

[edit] The second total derivative of f(x, y)

What's that (very simple example)? - could somebody clarify this bit for me. Thanks --catslash 23:00, 18 February 2007 (UTC)

I'm not quite sure what the author meant, but it probably has something to do with total derivative. Anyway, I rewrote that part of the example so that it does not refer anymore to total derivatives. If anybody want to give the general procedure for classifying constrained critical points, please go ahead, but for a simple example I think we should use the simple procedure. -- Jitse Niesen (talk) 13:17, 10 March 2007 (UTC)

[edit] Constrained Systems of Equations

Does anyone want to contribute a section and an example with polynomials that a high school algebra student could comprehend? Larry R. Holmgren 04:17, 24 March 2007 (UTC)

The section "Simple example" contains an example with polynomials. It's hard to come up with a simpler example, though the explanation in that section can probably be improved. Perhaps you can tell where you lose the plot when reading the section, and we can work on improving it?
PS: Please do not use ~~~~ (four tildes) when you fill in the edit summary. It does no harm, but it looks a bit silly (look up your edit in the history to see what I mean). -- Jitse Niesen (talk) 12:46, 25 March 2007 (UTC)


How does one interpret Lagrange Multiplier which ar produced on the Sensitive Report of Solver ® as an add-in to Excel®? Specifically how does not interpret 0% compared to 0.00?

[edit] g(...) is the gradient?

g(x) is often used as notation for the gradient of the function f(x). Initially, I wasn't sure if that was what was meant in this article. Could it maybe be defined a bit clearer somewhere? Maybe given explicit

"Suppose we have a function, f(x,y), to maximize subject to (add:) the constraint (/add)

g\left( x,y \right) = c,

where c is a constant. "

Im very new to contributing to Wikipedia, so I would rather not make the change myself - especially because im not sure its correct. :)

--Pellarin 10:11, 18 September 2007 (UTC)

Hi, I don't think the notation g(f) for \operatorname{grad}(f) or \nabla f is very common; it's not mentioned in the gradient article, and I've not come across it myself (perhaps it used in some particular field of study?). Conversely, I do think that the use of g() for an arbitrary function (having already used f()), is pretty common and widely understood. Yes, adding the constraint would make it read better; I will do it, but be bold next time! --catslash 10:40, 18 September 2007 (UTC)

[edit] Unique solutions?

"a number of unique equations totaling the length of x plus the length of λ. Thus, it is possible to obtain unique values for every x and λk, without inverting the gk" - unless I've misunderstood something, this is inaccurate. This claim cites Mathworld's article on LMs, but that article makes no mention of uniqueness. Having n equations and n unknowns does not guarantee uniqueness.

For example: f(x1,x2) = x1^2 + x2^2, constraint g(x1,x2) = x1^2 + 2x2^2 -1 = 0. It's trivial to see that (1,0) and (-1,0) both maximise f on g, and (0,1/sqrt(2)) and (0,-1/sqrt(2)) both minimise it, so solutions most certainly aren't unique. --144.53.251.2 06:57, 15 October 2007 (UTC)

[edit] grad gi

The phrase below seems to apply to the case with multiple constraints when the current formulation is for a single constraint:

"grad f is a linear combination of grad gi."

so I am replacing it with

"grad f = λgrad g for some real λ."

Also using \nabla for consistency with the remaining of the article. —Preceding unsigned comment added by 128.32.39.137 (talk) 06:14, 16 October 2007 (UTC)

Rodrigo de Salvo Braz 06:21, 16 October 2007 (UTC)

[edit] (n+k) Variables?

The opening paragraph, shouldn't it be (n-k) variables? —Preceding unsigned comment added by 134.10.122.120 (talk) 05:22, 4 December 2007 (UTC)

No. Each constraint adds an unknown multiplier. In the first example, f(x,y) with one constraint results in solving for Λ(x,y,λ). Two variables and one constraint gives you three variables unconstrained. Robben (talk) 08:49, 11 January 2008 (UTC)

[edit] Describing the Gradient Vectors as Parallel Vectors

In the "Introduction" the gradient vectors are said to be parallel at a extrema. I think that this is not entirely correct. Please correct me if I am wrong, but I think that in the example given the gradient vectors of the surfaces in question are only parallel when projected onto the x,y plane. Perhaps this is implied in the description, but regardless, this is a point I am confused on. —Preceding unsigned comment added by Jfischoff (talk • contribs) 20:50, 22 February 2008 (UTC)