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I would like to ask about a sufficient condition under which a solution for an infinite linear programming is unique.

In standard finite dimensional linear programmings, like $\min_x p\cdot x$ subject to $Ax\ge b$, it is known by Mangasarian (1978) that a solution $x^*$ is unique if and only if for any $p'$ sufficiently close to $p$, $x^*$ remains to be a solution of the problem $\min_x p'\cdot x$ subject to $Ax\ge b$.

Now, do we have the same result for infinite linear programming, assuming that p is a bounded linear real-valued map defined on some Banach space, and A is bounded linear operator from a Banach to another Banach, where $\ge$ is induced by some cone?

Any easy-to-check sufficient conditions for the uniqueness of solution in infinite dimensional linear programming is appreciated.

tarou2
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  • I don't know the answer offhand, but suspect that you would need some conditions on the domain (for example, a Hilbert space)? In particular, you need to characterise 'close'. – copper.hat Mar 03 '15 at 04:53
  • Sorry for insufficient explanation. Yes, what I have in my mind is Banach space and bounded linear operators. – tarou2 Mar 03 '15 at 05:15
  • Just out of curiosity, Is there some practical reason for solving this or is it just interesting? – Baby Dragon Mar 03 '15 at 05:28
  • I suspect something like $p \in (K^+)^\circ$, where $K= {h | Ah \ge 0 }$ and $\ ^+$ denotes the positive polar? Edit: This is not right, $K$ needs to correspond to the 'active' gradients, and I don't know how to characterise these with $A$. – copper.hat Mar 03 '15 at 05:42

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