BASIC ALGORITHMS 299 of f at x. On integrating from x to x 0 , this gives D(x 0 - x) < f (x 0 ) - f (x) < E(x 0 - x). This can be rewritten as the required inequality (x 0 - as E = (1 + )C and D = (1 - )C. 2 Conclusion. The modified Newton method is one of the two pillars of the necessary conditions for optimization problems (the other one is "separation of convex sets"). Moreover, this algorithm is of practical value, giving a cheaper variant of the Newton method. Conclusion to section 6.4. If you want to do a line search for a unimodular function, then it depends on the situation which one is the best. If you want to do only function evaluations, then it is f (x 0 ) ) - x < |x 0 - x|, C