I'm not sure I understand your question. You have a seperate linear programming problem for each choice of . Since the constraint is changing with you may conceivably get a different optimal value.
For eachb inRm, let ξ(b) denote the optimal objective function value for
the following linear program:
maximize c' x
subject toAx ≤ b
x ≥ 0.
Suppose thatξ(b) < ∞ for all b. Show that the function ξ(b) is concave (a
function f on Rm is called concave if f(tx+(1−t)y) ≥ tf (x)+(1−t)f(y)
for all x and y in Rm and all 0 < t < 1). Hint: Consider the dual problem.
Isn't there only one objective value? Not one for each b value?
That looks correct. I'm assuming that this question came from Vanderbei's book on Linear Programming. It's given at the end of section 10 which is on convexity analysis. My guess is that the concavity of the optimal value function for the primal prioblem is equivalent to the convexity of the optimal value function for the dual problem. So, how are the optimal values of these two problems related?
That's right. Therefore is the optimal value for the dual problem
Now, what can you say about ? This is the minimum of for 's lying in some fixed region.
By the way, forget my earlier remark concerning convex functions; it's not correct.