By Vandenbussche D., Nemhauser G. L.

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**Extra resources for A branch-and-cut algorithm for nonconvex quadratic programs with box constraints**

**Sample text**

Since you must have someone hired at all times, you will always hire the Þrst candidate that you interview. Goal: Determine what the price of this strategy will be. 5-2 Lecture Notes for Chapter 5: Probabilistic Analysis and Randomized Algorithms Pseudocode to model this scenario: Assumes that the candidates are numbered 1 to n and that after interviewing each candidate, we can determine if it’s better than the current ofÞce assistant. Uses a dummy candidate 0 that is worse than all others, so that the Þrst candidate is always hired.

Termination: The for loop of lines 1–4 terminates when i = n + 1, so that i − 1 = n. By the statement of the loop invariant, A[1 . i − 1] is the entire array A[1 . n], and it consists of the original array A[1 . n], in sorted order. Note: We have received requests to change the upper bound of the outer for loop of lines 1–4 to length[A] − 1. That change would also result in a correct algorithm. The loop would terminate when i = n, so that according to the loop invariant, A[1 . n − 1] would consist of the n − 1 smallest values originally in A[1 .

By the way, we interpret # of anonymous functions as = # of times the asymptotic notation appears: n O(i) OK: 1 anonymous function i=1 O(1) + O(2) + · · · + O(n) not OK: n hidden constants ⇒ no clean interpretation When on left-hand side: No matter how the anonymous functions are chosen on the left-hand side, there is a way to choose the anonymous functions on the righthand side to make the equation valid. Interpret 2n2 + (n) = (n 2 ) as meaning for all functions f (n) ∈ (n), there exists a function g(n) ∈ (n2 ) such that 2n2 + f (n) = g(n).

### A branch-and-cut algorithm for nonconvex quadratic programs with box constraints by Vandenbussche D., Nemhauser G. L.

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