Re: variable metric method automatic gradient yields Indeterminate
- To: mathgroup at smc.vnet.net
- Subject: [mg70426] Re: [mg70397] variable metric method automatic gradient yields Indeterminate
- From: Andrzej Kozlowski <akoz at mimuw.edu.pl>
- Date: Mon, 16 Oct 2006 02:34:28 -0400 (EDT)
- References: <200610150419.AAA12698@smc.vnet.net>
On 15 Oct 2006, at 13:19, Chris Chiasson wrote: > A user-defined augmented lagrange multiplier method for NMinimize > drives a (user defined) variable metric method for FindMinimum. The > NMinimize routine ends up creating penalty functions that have > "discontinuous" first order derivatives due to the presence of > functions like Max. At the points of discontinuity in the derivative, > the Indeterminate result usually ends up multiplied by zero. The limit > of the derivative exists. > > for example, the function passed to FindMinimum is > func=Max[0,-X[1]]^2+Max[0,-1+X[1]+X[2]]^2+(-1+X[1])^2+(-1+X[2])^2 > > its derivative with respect to X[1] is > 2*(-1+Max[0,-1+X[1]+X[2]]*Piecewise[{{1,X[1]+X[2]>1}},0]+ > Max[0,-X[1]]*Piecewise[{{-1,X[1]<0},{0,X[1]>0}},Indeterminate]+X[1]) > > Notice that when X[1] is zero, the Piecewise returns Indeterminate, > which is multiplied by zero from the nearby Max function. However, > 0*Indeterminate is still Indeterminate in Mathematica. When the > derivative is evaluated at X[1]=zero, the answer returned is > Indeterminate. This totally messes up the numerical routine. > > the limit of D[func,X[1]] as X[1]->0 is > Piecewise[{{-2,X[2]<1}},2*(-2 +X[2])] > > I am tempted to check the gradient vector for Indeterminate results, > look up the "corresponding" variable (heh, how do I really know which > one is responsible?), and take the limit as that variable approaches > the value I wanted to evaluate at. I don't know how well that will > work in practice. > > So, does anyone have any suggestions? > -- > http://chris.chiasson.name/ > It interesting to note that the following two methods give different looking answers: p = Block[{Indeterminate}, D[func, X[1]] /. X[1] -> 0] 2*Max[0, X[2] - 1]*Piecewise[{{1, X[2] > 1}}] - 2 q = Limit[D[func, X[1]], X[1] -> 0] Piecewise[{{-2, X[2] < 1}}, 2*(X[2] - 2)] but are, of course, equivalent: Simplify[p == q] True (This is also meant as a suggestion of an alternative method, though I would not recommend its unthinking use in other situations where Indeterminate occurs). Andrzej Kozlowski
- References:
- variable metric method automatic gradient yields Indeterminate
- From: "Chris Chiasson" <chris@chiasson.name>
- variable metric method automatic gradient yields Indeterminate