

Subject:
Min/Max with functions of two variables, subject to constraints
Category: Science > Math Asked by: trumanga List Price: $3.00 
Posted:
18 Aug 2006 12:41 PDT
Expires: 17 Sep 2006 12:41 PDT Question ID: 757434 
given two functions f(x,y) and g(x,y) [assume both are "wellbehaved" : continuous and differentiable] If we are solving a min/max problem on f subject to the constraint that g(x,y)=0 is it possible (knowing nothing more about f or g) that there could be 0 solutions? 3 solutions? infinitely many solutions? (I think the answer will be found by thinking about LaGrange Multipliers  but I'm not sure) Thanks for your help 

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Subject:
Re: Min/Max with functions of two variables, subject to constraints
From: mathtalkga on 18 Aug 2006 20:30 PDT 
Since you ask about minima and maxima (I think), I'll assume these are realvalued functions. You have an equation g(x,y) = 0 that presumably restricts the "feasible" points (x,y) to a curve of some kind [you did say to assume g is "wellbehaved"]. That is, the solutions to g(x,y) = 0 form a 1dimensional smooth manifold, with the possible exception of finitely many points of selfintersection and/or cusps (kinks) in the curve. As far as the question of whether maxima/minima exist, or whether there can be multiple occurrences, the issues of selfintersection or cusps are not really critical (no pun intended). Sure, you can artificially introduce an obstacle to the existence of a maximum or minimum by declaring g(x,y) to be mysteriously undefined at the points where the extrema would otherwise occur, but again we will assume "nice" behavior. In that case the big issue is whether the curve has bounded or unbounded extent. A closed and bounded subset of the Euclidean plane is compact, and a continuous function f(x,y) on a compact set will have both (at least one) maxima and minima. Think, for example, of g(x,y) = 0 defining a circle in the plane. The range of values as you go around must have a high and a low point (though it may have more than one of each). On the other hand the curve could be open ended (even if it crosses itself). A simple case would be a straight line, and since then y is explicitly a function of x, this is really a onedimensional optimization problem over (oo,+oo). In consequence there may not be a maximum or a minimum point, or there may be one kind of extremum but not the other, or there could be both a maximum AND a minimum. So the easiest case to discuss in theory is the compact (closed and bounded) curve, since there we are guaranteed the continuous function f will have (at least) one maximum and one minimum (and maybe more). If it's a question about computing the maximum and minimum values, this is part of the subject of numerical analysis. Googling for onedimensional numerical optimization will give you some ideas of the diverse methods that are available. regards, mathtalkga 
Subject:
Re: Min/Max with functions of two variables, subject to constraints
From: mathtalkga on 19 Aug 2006 05:15 PDT 
Let me add that the phrase "solving a min/max problem" means something to a mathematician that seems inconsistent with the data of the original Question here. Given a function of two variables, say f(x,y), one poses a "min/max" problem by asking (for example) how to choose a value of variable x, such that the range of values y gives the least possible (minimum) value of the maximum value of f(x,y): Min Max f(x,y) x y Hence the name "min/max problem". The additional constraint g(x,y) = 0, which tends to reduce the problem from two variables to one, and the mention of Lagrange multipliers (a method for imposing a constraint using techniques for otherwise unconstrained optimization) lead me to think, however, this is _not_ the kind of problem you have in mind. To briefly return to the specific questions you raised (could there be no solutions? finitely many solutions? infinitely many?), these are issues of existence and uniqueness, two more search keywords that might help you find the information you are looking for. regards, mathtalkga 
Subject:
Re: Min/Max with functions of two variables, subject to constraints
From: activealexaokiga on 26 Aug 2006 00:01 PDT 
Isn't your question just related to this page?  http://www.dpgraph.com/MCExamples/MCExamples.html "[g is tangent at one or more points to a level curve of f,] the gradients of f and g are scalar multiples of each other and the coordinates of tangency points can be determined by solving a system of 3 (usually nonlinear) equations. " I'm getting the notion that your question is conditional. Search : "calculus subject to constraint" 
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