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        riddles >> general problem-solving / chatting / whatever >> Lagrange Multiplier Conditions
        
(Message started by: william wu on Sep 7th, 2003, 11:09pm)
Title: Lagrange Multiplier Conditions
Post by william wu on Sep 7th, 2003, 11:09pm
Exactly what conditions must a maximizing/minimzing function and its constraints satisfy in order for lagrange multipliers to be applicable? I have no reference on this subject.
Title: Re: Lagrange Multiplier Conditions
Post by Icarus on Sep 8th, 2003, 9:02pm
"Calculus of Variations" by Robert Weinstock (Dover, 1974) has this to say:


"A necessary condition for a minimum (or maximum) of F(x, y, ..., z) with respect to variables x, y, ..., z that satisfy

( 7 )   Gi(x, y, ..., z) = Ci    (i = 1, 2, ..., N),


Where the Ci are given constants, is

( 8 )   [partial]F*/[partial]x = [partial]F*/[partial]y = ... = [partial]F*/[partial]z = 0,


where F* = F + [sum] [lambda]iGi.  The constants [lambda]1, [lambda]2, ..., [lambda]N - introduced as undetermined Lagrange multipliers - are evaluated, together with the minimizing (or maximizing) values of x, y, ..., z, by means of the set of equations ( 7 ) and ( 8 )."



Unfortunately, he is reticent about the assumptions that this clearly requires. At the start of the chapter he defines "piecewise differentiable", then tosses out the phrase "We eliminate consideration of any functions whose derivative undergoes infinitely many changes of sign in a finite interval."

I have to assume that piecewise differentiability along with only a finite number of sign changes for the derivative in any finite interval (presumably for each variable, holding the others constant) are sufficient conditions for lagrange multipliers to be necessary to the existance of an extrema.
Title: Re: Lagrange Multiplier Conditions
Post by Icarus on Sep 9th, 2003, 7:40pm
"Mathematical Methods for Physicists" by George Arfken (Acedemic Press, 1970) discusses only the case of a single restraint for a 3 dimensional situation. He adds this consideration: "The method will fail if all of the coefficients of [lambda] vanish at the extremum. ... It is then impossible to solve for [lambda]."

I believe the appropriate expansion of this to more variables and restraints is that for any i, not all of the [partial]Gi/[partial]xj are zero, where Gi is the ith constraint function and xj is the jth coordinate variable.


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