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Physical quantities in QCD are independent of renormalization scheme (RS),but that exact invariance is spoiled by truncations of the perturbation series."Optimization" corresponds to making the perturbative approximant, at any givenorder, locally invariant under small RS changes. A solution of the resultingoptimization equations is presented. It allows an efficient algorithm forfinding the optimized result. Example results for R(e+e-)=3(Sum q_i^2)(1+R) tofourth order (NNNLO) are given that show nice convergence, even down toarbitrarily low energies. The Q=0 "freezing" behaviour, R=0.3+/-0.3, found atthird order is confirmed and made more precise; R=0.2+/-0.1. Low-energy resultsin the MS-bar scheme, by contrast, show the typical pathologies of anon-convergent asymptotic series.

Optimization of QCD Perturbation Theory: Results for R(e+e-) at fourth order