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In mathematical programming, one popular problem is the quadratic programming problem \begin{align} &{\rm maximize}~~~1/2x^TQx+c^Tx \\ & {\rm subject to}~~~Ax\leq b, \end{align} where Q and A are real finite-dimensional matrices and x,c, and b are vectors in $\mathbb{R}^n$.

My question is about this problem where Q and A are infinite-dimensional linear operators and x,c, and b are vectors in an arbitrary real Hilbert space H. By a simple net surfing I have found some results on necessary and sufficient conditions of this problem but I do not know how to solve it. In other words I do not know is there any algorithm in infinite-dimensional quadratic programming!

I would be grateful if you could help me.

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  • $\begingroup$ Hmm, if the problem is truly infinite-dimensional, won't it be difficult to even represent the inputs and outputs of any algorithm? $\endgroup$ Commented Aug 15, 2016 at 7:23
  • $\begingroup$ Your opinion seems logical. Do you mean there is no way? $\endgroup$ Commented Aug 15, 2016 at 7:32

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There are several problems behind this question:

  • How do you define inequalities $x\ge0$ in an infinite-dimensional space?

  • It might happen that constraint qualifications are generically not satisfied. For instance, it is not guaranteed that $\{x\ge0 \}$ has non-empty interior.

If you have a specific setting in mind, then you can try to apply traditional finite-dimensional methods: interior point, penalty methods, semi-smooth Newton methods, etc.

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