We saw in the last lecture that the Ellipsoid Algorithm can solve the optimization problem max s.t. c ⊤ x Ax ≤ b in time poly(〈A〉, 〈b〉, 〈c〉), where by 〈z 〉 we mean the representation size of z. In proving this, we mostly disregarded numerical issues, such as how irrational numbers should be dealt with. In addition, we mostly stated the algorithm in terms of a general convex set K rather than just a polytope, but then we neglected the whole host of issues surrounding general convex sets. In this lecture, we will fill in the remaining details. Largely these are either numerical or conceptual issues that require great length to be treated formally. As a result, this lecture will be relatively informal; a more precise and complete treatment is provided by Grötschel, Lovász, and Shrijver in [GS88]. 9.1 LP runtime One loose end that we would like to mention quickly first is that it seems we should b
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