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Math Help - Subspace Minimization

  1. #1
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    Question Subspace Minimization

    Let f(x) be a \mathcal{C}^1 function and let M be a subspace of \mathbb{R}^n. Suppose x^\ast \in M minimizes f(x) on M. Show \nabla f(x^\ast) \in M^\perp.

    My only idea is to take any x \in M and try to show something with the function \varphi(t) = f(x^\ast + tx), but I haven't been able to get anywhere with that. Any help is greatly appreciated!
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  2. #2
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    Quote Originally Posted by syme.gabriel View Post
    Let f(x) be a \mathcal{C}^1 function and let M be a subspace of \mathbb{R}^n. Suppose x^\ast \in M minimizes f(x) on M. Show \nabla f(x^\ast) \in M^\perp.

    My only idea is to take any x \in M and try to show something with the function \varphi(t) = f(x^\ast + tx), but I haven't been able to get anywhere with that. Any help is greatly appreciated!
    Suppose the gradient did not belong to the orthogonal complement of M.

    Then the gradient at \bold{x}^* has a component \bold{u} in M, so moving form \bold{x}^* to \bold{x}^*+\delta \bold{\hat{u}}
    changes the value of the function to:

    f(\bold{x}^*)+ \delta [\nabla f(\bold{x}^*).\bold{\hat{u}}]=f(\bold{x}^*)+ \delta |\bold{u}| ,

    so \bold{x}^* will not be a mininmum in M.

    RonL
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