Consider a linear programming problem of the form min cx, subject to constraints Ax≥b and x_i≥0 for all i. A is an m×n matrix, representing m constraints on n variables.

Suppose there is a single optimal value x^*, which minimizes cx.

You are a consulting god. You know x^*. You sit there, and twiddle your thumbs charging your Very Important Client loads of money, by the hour, and then just as they’re getting really anxious and maybe a little upset even and leaving messages on your phonemail asking “OK, now: x^*? Inspectors! Government agents! Lurking. IRS here soon. So: x^*?!”…

You get a call from a low-grade engineer over at Very Important Client HQ. Actually, they work in the basement of a bank in rural Illinois, but still, you pick up the phone accidentally, thinking they were in fact the VIC Inside Champion, with whom you have been in cahoots, and start to tell the low-grade engineer the value of x^*, when she interrupts to blurt out something.

What?

She blurts it out again.

Silence. The only thing moving in your office is the little Executive Toy magnetic top thing on your desk, which ominously and symbolically winds down and tips over.

Zoom in your eyes. They are blank. Swerve around your head, close to your phone as it presses against your hairy old ear, and then swoop into the headset, down a vaguely fibrous channel, and out into the fluourescent-lit filing-cabinet warren the low-grade engineer inhabits, down there in the basement of the old bank in Illinois.

“I said, ‘b is a random variable.’ Hello? Are you listening to me? The numbers you have for b, well, those are Gaussian random numbers, not real values. When Humbertfulson sent them to you, he must have missed sending the standard deviation column. But that’s OK, because the numbers you have are the means, and the standard deviations are simply 1/10 the value of the mean. So if it says ‘100′ in a row of b, that means that entry is a Gaussian with mean 100, and standard deviation 10. What? No, I thought you knew! No! Why — Well, fine. You bastard!”

She slams the phone down.

Cut back to you, standing in your corner-office aerie overlooking the sunlit Urban River, your forehead pressed against the window, your palms spread out against the glass above your head, in an unconscious figure of crucifixion, of surrender. Seagulls fly by, hundreds of feet below. A hawk swoops down into their midst, carries one off to the top of a nearby piling, and starts to shred it.

Two simple questions. Not the questions you answered when you found x^*. No, these are questions that your job-killing, reputation-destroying, money-blocking fool “Champion” must have intentionally kept from you. He never told you what they really wanted. He just kept feeding you rope, yards and yards of rope, and then —

But wait!

Two simple questions! What the hell? Nothing worse can happen to you at this point. Give it a go. May not be easy… but neither is it impossible.

First, they want you to delineate the 90% confidence surface around x^*. Assume that the problem remains feasible and bounded within this range of values of b. What are the possible shapes of the region? What will be the longest axis? What will be the direction of the largest uncertainty?

Second, a proportion p of the rows of the original A matrix you were given were redundant constraints, as you originally understood them. That is, with no variation, those constraints were never active. But now, but now… any one of those p rows might intersect the feasible region,might become active. How might you determine whether any (or all) of them would change the basis of the optimal solution?

Far below, the hawk pivots its head as you turn away from the window, gazing up. Appraisingly.