Quadratic FunctionsConsider the strictly convex quadratic function f ( x ) = 1 2 x t P x − q t x + r ,, where P ∈ R n × n is a positive definite matrix, q ∈ R n and r ∈ R . Let H := { H : H is a k − dimensional subspace in R n } . Clearly, the restriction of f to any H ∈ H is again a strictly convex function. For any H ∈ H , we will use x H to denote the unique optimal point of the following problem min. x ∈ H f ( x ) .Now consider the map, ψ ( H ) = x H .Prove / Disprove: The map ψ is bijective.Remark: It is assumed that P is invertible and q ≠ 0 .

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Quadratic FunctionsConsider the strictly convex quadratic function   f  (  x  )  =      1    2        x    t    P  x  −      q    t    x  +  r  ,, where   P  ∈            R              n      ×      n       is a positive definite matrix,   q  ∈            R        n   and   r  ∈      R    . Let       H    :=  {  H  :  H   is a   k  −  dimensional subspace in             R        n    }  . Clearly, the restriction of f to any   H  ∈      H   is again a strictly convex function. For any   H  ∈      H    , we will use       x    H   to denote the unique optimal point of the following problem      min.          x      ∈      H          f  (  x  )  .Now consider the map,   ψ  (  H  )  =      x    H    .Prove / Disprove: The map   ψ is bijective.Remark: It is assumed that P is invertible and   q  ≠      0    .

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Step 1Whether   ψ is surjective depends on the definition of its codomain. For example, if   n  =  2 and   k  =  1, the image of   ψ is a curve on             R        2  . So, if the codomain of   ψ is             R        2  ,   ψ is certainly not surjective.Step 2At any rate,   ψ may not be injective even if the origin is not a global minimum. Consider   a  =  (  1  ,  0  ,  0      )    T   and   f  (  x  )  =  ‖  x  −  a      ‖    2  . Let   {      e    1    ,      e    2    ,      e    3    } be the canonical basis of             R        3  . Then       x    H    =  a when   H  =  span  ⁡  {      e    1    ,      e    2    } or   span  ⁡  {      e    1    ,      e    3    }.

Quadratic Functions. Consider the strictly convex quadratic function f(x)=1/2 x:t Px-q^t x+r, where P in R^{n times n} is a positive definite matrix, q in R^n and r in R.

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