Linear Optimization and Extensions pp 309-386 | Cite as

# Ellipsoid Algorithms

## Abstract

Gaius Julius Caesar (101–44 B.C.) made military use of it when his legions conquered Gaulle, Machiavelli coined it as a political doctrine for Florentine princes to improve *his* princes’ control over their subjects, and mathematicians employ it e.g. when they use *binary search* to locate the optimum of a *p*-unimodal function over a compact convex subset of ℝ^{1}.
*Divide and conquer*, or more precisely divide and reign, is what Niccolo’s doctrine translates to. The compact convex subset of ℝ^{1} is, of course, some finite interval of the real line and the basic idea of binary search consists in successively *halving* this interval — like we did in Chapter 7.5.3. Utilizing the *p*-unimodality (see below) of the function to be optimized, we then discard one half of the original interval forever and continue the search in the other half of the original interval — which is again a compact convex subset of ℝ^{1}. Consequently, the basic idea can be reapplied and we can iterate. Since the length of the left-over interval is half of the length of the previous interval, the 1-dimensional “volume” of the remaining compact convex subset of ℝ^{1} to be searched shrinks at a geometric rate and we obtain fast convergence of the iterative scheme.

### Keywords

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