I have an optimization problem of the form max γ ⟨ f ∘ γ , w f ⟩ ⟨ g ∘ γ , w g ⟩where f ∘ γ is the composition of f and γ and the inner product is defined as ⟨ f ∘ γ , w f ⟩ = ∫ ( f ∘ γ ) ( t ) w f ( t ) d tIn other words I am trying to find a γ that maximizes the product of the two inner products above. I know one could pursue a gradient approach, which would be slow. Is there a more direct optimization method?

Question

I have an optimization problem of the form      max    γ    ⟨  f  ∘  γ  ,      w    f    ⟩  ⟨  g  ∘  γ  ,      w    g    ⟩where   f  ∘  γ is the composition of   f and   γ and the inner product is defined as  ⟨  f  ∘  γ  ,      w    f    ⟩  =  ∫  (  f  ∘  γ  )  (  t  )      w    f    (  t  )    d  tIn other words I am trying to find a   γ that maximizes the product of the two inner products above. I know one could pursue a gradient approach, which would be slow. Is there a more direct optimization method?

Paxton James · Accepted Answer

MathJax(?): Can&#039;t find handler for document MathJax(?): Can&#039;t find handler for document If   f and   g are differentiable, you could do something like the following.Define   J  (  &amp;#x03B3;  ) as your objective function. Then, the variation of   J in direction   h is given by

dourtuntellorvl · Accepted Answer

Hmm, the functions can be assume to be differentiable and smooth

I have an optimization problem of the form <munder> <mo movablelimits="true" form="prefi

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