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For example, when I am trying to minimize the function $f(x) = x^2 - 6x +12$ where $x\geq$ are non negative real number. If by some way, I was able to obtaining the upper bound (only) or lower bound (only) of this problem but without knowing the corresponding $x$, for example like knowing that min $f(x)$ is less than $3.5$.

Would there be any methods that could help exploit this information to reach the optimal point quickly than usual.

In this case, the optimal solution for this problem would be that $Min \ f(x) = 3$ at $x=3$

Thank you very much

Tuong Nguyen Minh
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  • I don't think having an upper bound on the minimum tells you much. Any feasible point gives you an upper bound simply by evaluating the objective there. –  Sep 05 '19 at 05:54
  • Oh then how about a lower bound ? – Tuong Nguyen Minh Sep 05 '19 at 07:58
  • You could look into the theory of duality in convex optimization (see e.g. Ch. 5 of Boyd and Vandenberghe). Every feasible value of the dual problem is a lower bound on the optimal value of the original problem. –  Sep 08 '19 at 07:41

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