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$f\left(\mathbf{x}\right):\mathbb{R}_+^n\rightarrow\mathbb{R}_+$ is a concave monotonically increasing function to be minimised over the feasible region $\sum_{i=1}^n x_i=1$ and $x_i\geq 0\quad\forall1\leq i\leq n$.

Given that the feasible region is a convex polytope, is it possible to say anything about the optimal $\mathbf{x}^*$? I have a hunch that at $\mathbf{x}^*$, at least one of the inequality constraints will be satisfied with equality. In other words $\mathbf{x}^*$ will be right on the edge of the feasible region. But can't prove it. Am I right there?

If it helps to consider a special case, the function is

$f\left(\mathbf{x}\right)=\log|\mathbf{I}_k+\sum_{i=1}^{n}x_i\mathbf{A}_i|$

where $\mathbf{A}_i\in\mathbb{C}^{k\times k}$ are all positive semidefinite Hermitian matrices.

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    $\begingroup$ What do you mean by "monotonically increasing"? $\endgroup$ – littleO Aug 15 '13 at 5:24
  • $\begingroup$ $f\left(\mathbf{x}\right)$ is monotonically increasing with respect to each of $x_1, x_2, \dots, x_n$ while the others are held constant. $\endgroup$ – Della Aug 15 '13 at 5:58
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um,this is obviously a economics problem.You may resort to Lagransian.If you want to know more ,confer to Microeconomic Theory by Mas-colell,Whiston and Green.

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  • $\begingroup$ Note that Barman is minimizing a concave function, rather than convex. Also, optimization problems like this arise in many areas, not just in economics. $\endgroup$ – littleO Aug 15 '13 at 5:55
  • $\begingroup$ Nope, it's not an economics problem, may be the mathematical formulation has implications in economics (I will like to know the specific area). Can you give me more specific reference (chapter number) where I can get some idea of the solution? $\endgroup$ – Della Aug 15 '13 at 6:59
  • $\begingroup$ Yes,such problems arises in many areas,but it is a typical problem of ecomomics .Many books you can resort to,forexample M.J,M.of KMATHEMATICAL APPENDIX of Microeconomic Theory by Mas-Colell,Whiston and Green.Moreover,you also can confer to more professional books ,such as M.Avriel, Nonlinear Programming analysis and methods, Prentice-Hall, 1976 $\endgroup$ – Jacky Zhang Aug 17 '13 at 4:52

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