Date: Wed, 25 Oct 2006 08:41:07
From: S Bond
Subject: Re: constrained optimality
Jonathan W. Ray wrote: > The section of the ch6 lecture notes on p. 12-16 doesn't make sense. I tried to explain this in class with more pictures and examples. It helps to have something to draw on. We are trying to minimize f(x), but are constrained to the surface implicitly described by g(x) = 0. If we are on this implicit surface we know that in any direction along the surface g(x) = 0. Hence the directional derivative of g along the surface must be zero. Feasible directions must be ones in which this directional derivative is zero. If g is a scalar valued multivariate function, then the directional derivative of g in the direction of a unit vector, u is the dot product of u with the gradient of g http://mathworld.wolfram.com/DirectionalDerivative.html Hence all feasible directions, which are ones which do not move off the constraint surface, must be orthogonal to the gradient of g. > On p265 in the book, > What does it mean geometrically when the Hessian is positive semidefinite in > any feasable direction? First, to check to see if we are at a critical point, we check to see if the first derivative is zero in all feasible directions. Feasible directions are as described above. To check to see if we have a minimum we need to check to see if our objective function is concave up, but we only need to check in feasible directions. Hence we just want the second derivative in feasible directions, which we can use to see if it's positive definite (concave up). > what does it mean geometrically when the negative gradient of f is in the > space spanned by the constraint normals? This means that the directional derivative of the objective function, f, is zero in all feasible directions. > What does "we cannot reduce the objective function without violating the > constraints" mean? It means that the only direction in which the objective function, f, is decreasing is a direction that takes us off the constraint surface. > Is problem 6.16 asking me to apply the method on page 288? Problem 6.16(a) is asking you to apply the method in Example 6.6 on page 266. The method will fail for this problem, so you will not have to compute H_f, H_g, or B.
|
HTML 4.01 Updated: Wed, 25 Oct 2006 08:41:07 | Powered by Perl Net::NNTP |