Parametrized linear optimization problem

Submitted by Anonymous on Mon, 08/27/2007 - 15:29.

Algebra questions

Let $p_0, p_1, \ldots p_{L-1}$ be L different probability distributions over M elements:

$p_i = (p_i^0, p_i^1, \ldots p_i^{M-1})$ where $p_i^j \in (0,1), \forall i=0\ldots L-1, \forall j=0\ldots M-1$ , and $\sum_{j=0}^{M-1} p_i^j = 1, \forall i = 0\ldots L-1$ .

Let $c_0, c_1, \ldots c_{L-1}$ be L real, positive constants. Let $a$ be a real positive constant, and let $\delta \in (0,1)$ .

We have the following linear optimization problem:

Maximize $f =\delta p_0 x$
Such that:
- $x^T = (x_0,\ldots, x_{M-1})$ are M positive real variables;

- $x_i - \delta p_0 x \leq (1-\delta) a, \forall i= 0,\ldots, M-1;$

- $\delta (p_j - p_0) x \leq (1-\delta)(c_j - c_0), \forall j=1,\ldots,L-1;$

We know that for some value of $\delta = \delta^*$ , the optimization problem accepts a solution with the maximum objective $f = f^*$ . I would like to know the minimum value of $\delta$ such that the problem still accepts a solution.

Thank you for your help.

Navigation

User login

Parametrized linear optimization problem