(Nonconvexity of Sr in LMRP Algorithm) Theorem 12.1 allows us to solve

problem (P

′

j

) by first sorting a subset of the i’s and then computing the partial sums given

by (12.23), choosing the r that minimizes Sr and setting yi = 1 for i = 1, . . . , r. It is

tempting to think that Sr is convex with respect to r, since then we could consider each r

in turn as long as Sr is decreasing, and then stop as soon as Sr increases (or use an even

more efficient method like binary search). Unfortunately, this claim is not true. Provide a

counterexample with four variables such that ai < 0="" and="" bi=""> 0 for all i, and such that

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