convex optimization Alice argues for building a new system to randomly display different ads to visitors, with each ad j appearing for a fraction j of the time. Bob disagrees and believes that the old system, which shows the same ad every time, is sufficient. Under a randomized policy, the revenue per view xmix follows the discrete distribution p = P, where Rn , 1T = 1, and P(xmix = ci) = pi. The start-up will lose money on the campaign if xmix < 0, and will consider the campaign successful if xmix > L. Alice and Bob's goal is to maximize the probability of a successful campaign while ensuring the probability of a loss is no more than