# Minimization

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 January 30, 2004, 05:25 Minimization #1 doche Guest   Posts: n/a Hello, I'm searching for a method which can evaluate the global minimum of a multivariable function f(x,y,z). But due to feedback consideration, I have only acces to values of grad(f)(no way to calculate f). How can I do?

 February 4, 2004, 02:57 Clarification Re: Minimization #3 Ananda Himansu Guest   Posts: n/a Perhaps I failed to mention the simple mathematical concept behind the bookkeeping that I suggested in the previous post. Let the cost function that you are trying to minimize be denoted by g(r). Pick an arbitrary point r0=(x0,y0,z0) in your search domain. Then we can alternatively represent g as g(r) = g(r0) + Definite Path Integral of grad(g) from r0 to r Your cost evaluation function does not return g but rather grad(g). So g(r0) is unknown. Define a gauge cost function f by f(r) = Definite Path Integral of grad(g) from r0 to r From this definition, note that if g is single-valued, then f is independent of the path of integration (in theory). Also that f(r0) = 0. And that grad(f) = grad(g), and hessian(f) = hessian(g). The definition of f is theoretically equivalent to saying f(r) = g(r) - g(r0), but algorithmically the path integral definition of f must be used, since the value of g(r) is unknown. A key fact that follows from the definition of f is that if the minimum (gmin) of g occurs at rmin=(xmin,ymin,zmin), gmin = g(rmin), then the minimum (fmin) of f also occurs at rmin, i.e., fmin = f(rmin). Also, fmin = gmin - g(r0). Apply a classical gradient-based minimization algorithm to f, taking into account that because of the integral definition, the value of f must be extrapolated from nearby point/s using the value of grad(g). The end-result of your minimization effort will be obtaining the values of rmin and fmin ( = f(rmin)-f(r0) = g(rmin)-g(r0)). Ananda Himansu

 February 4, 2004, 06:50 Re: Clarification #4 doche Guest   Posts: n/a Thank you very much for your response The gauge function should be very usefull

 February 4, 2004, 12:23 Re: Clarification #5 Ananda Himansu Guest   Posts: n/a You are quite welcome. To clarify further, the path integral definition of g is obtained as the antiderivative of the differential relation dg = grad(g) * dr where the * represents a vector dot product, which may not have been clear from the way I wrote it previously. With a classical gradient-based optimization algorithm, the construction of the gauge function f is only required in order to know how low your minimum is relative to your starting point, i.e., to know gmin-g(r0). If you did not care to know this, with pure gradient-based algorithms, you could eliminate all references to f itself, and merely find the point rmin as the endpoint of your descent path, without constructing f. If your search space is small enough to do a systematic exhaustive search on a mesh with a spacing small enough to satisfy your accuracy requirement, then you will need to construct the gauge cost f, in order to efficiently compare the values and find the minimum achieved on the selected mesh. In this case, grad(g) is used only to construct f, instead of being used to select a descent direction.