Optimization Problem
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From: Nitesh on 23 Jun 2010 07:09 There is a linear function of two variables that I am trying to minimize under an equality constraint. But, the constraint is non linear in the variables. Is there any technique to solve this? or can I use approximations and linearize the constraint?
From: us on 23 Jun 2010 07:16 "Nitesh " <nitesh.prabhu(a)yahoo.co.in> wrote in message <hvsq0g$4pj$1(a)fred.mathworks.com>... > There is a linear function of two variables that I am trying to minimize under an equality constraint. But, the constraint is non linear in the variables. Is there any technique to solve this? or can I use approximations and linearize the constraint? don't double post(!)... rather, show CSSM what you came up with so far... us
From: Nitesh on 23 Jun 2010 07:27 "us " <us(a)neurol.unizh.ch> wrote in message <hvsqdk$1c0$1(a)fred.mathworks.com>... > "Nitesh " <nitesh.prabhu(a)yahoo.co.in> wrote in message <hvsq0g$4pj$1(a)fred.mathworks.com>... > > There is a linear function of two variables that I am trying to minimize under an equality constraint. But, the constraint is non linear in the variables. Is there any technique to solve this? or can I use approximations and linearize the constraint? > > don't double post(!)... > rather, show CSSM what you came up with so far... > > us I'm sorry..i didnt mean to double post. I havent made any progress in this.I still have the linear objective function in two variables and a nonlinear equality constraint.Is it possible to use lagrange multipliers to solve this?
From: us on 23 Jun 2010 07:35 "Nitesh " <nitesh.prabhu(a)yahoo.co.in> wrote in message <hvsr28$cfl$1(a)fred.mathworks.com>... > "us " <us(a)neurol.unizh.ch> wrote in message <hvsqdk$1c0$1(a)fred.mathworks.com>... > > "Nitesh " <nitesh.prabhu(a)yahoo.co.in> wrote in message <hvsq0g$4pj$1(a)fred.mathworks.com>... > > > There is a linear function of two variables that I am trying to minimize under an equality constraint. But, the constraint is non linear in the variables. Is there any technique to solve this? or can I use approximations and linearize the constraint? > > > > don't double post(!)... > > rather, show CSSM what you came up with so far... > > > > us > I'm sorry..i didnt mean to double post. > > I havent made any progress in this.I still have the linear objective function in two variables and a nonlinear equality constraint.Is it possible to use lagrange multipliers to solve this? well... at least you could show the ML coded function... us
From: John D'Errico on 23 Jun 2010 08:48
"Nitesh " <nitesh.prabhu(a)yahoo.co.in> wrote in message <hvsr28$cfl$1(a)fred.mathworks.com>... > "us " <us(a)neurol.unizh.ch> wrote in message <hvsqdk$1c0$1(a)fred.mathworks.com>... > > "Nitesh " <nitesh.prabhu(a)yahoo.co.in> wrote in message <hvsq0g$4pj$1(a)fred.mathworks.com>... > > > There is a linear function of two variables that I am trying to minimize under an equality constraint. But, the constraint is non linear in the variables. Is there any technique to solve this? or can I use approximations and linearize the constraint? > > > > don't double post(!)... > > rather, show CSSM what you came up with so far... > > > > us > I'm sorry..i didnt mean to double post. > > I havent made any progress in this.I still have the linear objective function in two variables and a nonlinear equality constraint.Is it possible to use lagrange multipliers to solve this? Surely it depends on the specific problem how I might attack this. Depending on the specific equality constraint, I might choose to work in the 1-manifold induced by the equality constraint. Then it could be written as an fminbnd problem, or perhaps just solved by hand. So if the constraint is x^2 + y^2 = r^2 where I wish to minimize the objective u = y - x, then I can reduce the problem to minimizing the objective u = r*sin(theta) - r*cos(theta) = r*(sin(theta) - cos(theta)) Since r is fixed, it is trivial to minimize this expression as a function of theta, then recover x and y from x = r*cos(theta), y = r*sin(theta) Just as easily, one can use Lagrange multipliers on the problem. Minimize the augmented objective function (y - x) + (x^2 + y^2 - r^2)*lambda Differentiate wrt x and then y, set them equal to zero. -1 + 2*lambda*x = 0 1 + 2*lambda*y = 0 Add and combine. lambda*(x + y) = 0 So either lambda is zero, or we know that x + y = 0. If x + y = 0, then y = - x. We can now return to the constraint. x^2 + (-x)^2 = r^2 therefore x = r/sqrt(2) or x = -r/sqrt(2) so now we know the solution must lie at one of these points. (x,y) = (+/- r/sqrt(2), +/- r/sqrt(2)) Thus y - x must be minimized at (r/sqrt(2), - r/sqrt(2)) WTP? John |