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Math Help - Multivariable Calculus Decent Direction Proof

  1. #1
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    Multivariable Calculus Decent Direction Proof

    I am reading the proof of the following lemma:
    Let f: \mathbb{R}^n \rightarrow \mathbb{R} be differentiable at \bar{\mathbf{x}}}. If \nabla f(\bar{\mathbf{x}})^{\mathrm{T}} \mathbf{d} < 0, then \mathbf{d} is a decent direction for f at \bar{\mathbf{x}}}.

    I don't understand the first line of the proof, which states: because f is differentiable at \bar{\mathbf{x}}} then

    f(\bar{\mathbf{x}} + \lambda \mathbf{d}) = f(\bar{\mathbf{x}}) + \lambda\nabla f(\bar{\mathbf{x}})^{\mathrm{T}} \mathbf{d} + \lambda ||\mathbf{d}||\alpha (\lambda \mathbf{d})

    where
    \lim_{\lambda \rightarrow 0}\alpha (\lambda \mathbf{d}) = 0

    They define a decent direction as:
    \exists \delta > 0: f(\bar{\mathbf{x}} + \lambda \mathbf{d}) < f(\bar{\mathbf{x}}) \; \forall \lambda \in (0, \delta)

    I don't understand what \alpha is exactly. Is it just the Fréchet derivative?
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  2. #2
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    Re: Multivariable Calculus Decent Direction Proof

    Hey fobos3.

    You should look at order functions in mathematics - namely big-O and little-o.

    Big O notation - Wikipedia, the free encyclopedia

    Linearization - Wikipedia, the free encyclopedia
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  3. #3
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    Re: Multivariable Calculus Decent Direction Proof

    Chiro: I don't see what that has to do with the question.

    - Hollywood
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  4. #4
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    Re: Multivariable Calculus Decent Direction Proof

    To me this looks like a multi-variate taylor series expansion in which one is approximating it through a linearization.

    Its like how we use taylor series to expand f(x+a) for some arbitrary x or a with one of them known and the other is considered to be small. In the above case, we are doing it with a multivariable function using a vector instead of a scalar.

    The Big-O notation gives us how the error behaves in regard to its parameters (namely lambda and the d vector). When these approach zero in magnitude and/or length then you get the result discussed above.

    Big-O notation is used a lot in mathematics to look at how residuals or errors behave under certain conditions.
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