1. Vladimir Protasov (Moscow State University) Fifty years of the ‘Russian method’

2. The method of central cross-sections (the cutting plane algorithm) A.Yu.Levin 1961-1965, D.Newman, 1965

3. ``Small Russian town’’ Voronezh Voronezh school of mathematics (Krasnosel’sky, Kreyn, Rutitsky, Mirolyubov, Soldatov, Levin)

4. In the case of A.Yu.Levin the problem was How to solve ?

5. The first approach Karush-Kuhn-Tucker theorem ( stationarity ) ( nonnegativity ) (complementary sluckness) cases Lagrangean

6. The second approach The gradient relaxation method

7. Central cross-sections: the key idea For any convexity (well known) We can remove the part and concentrate oninstead of Next iteration: take a point compute draw a cross-section obtain a convex bodyand so on...

8. How to choose points Afrer sufficiently many iterations one can localize the minimum ? Answer: is the center of gravity of The ``Oracle’’ concept As a result we obtain a bogy of a small volume that contains a point of minimum of

9. may be large, although the volume is small ? How large should N be so that We take

10. Problem: How to find the center of gravity ? N grows exponentially with the dimension d and with the number of vertices ``curse of dimensiality’’ Computing gr(G) is NP-hard

11. A.Levin was delaying with publishing his result for 4 years (1961-1965), attempting to make the method implementable. Circumscribed ellipsoid method A.Nemirovsky (1977), N.Shor (1977) Circumscribed simplices method L.Levin, D.Yamitsky (1980) Inscribed ellipsoid method L.Khachiyan, S.Tarasov, A.Erlikh (1988)

12. Ellipsoid method D.B.Yudin, A.S.Nemirovsky (1976) N.Z.Shor (1977)

13. The crucial idea of the Nemirovsky-Shor ellipsoid method: Lemma. A half of ellipsoid can be put in an ellipsoid of a smaller volume. The algorithm: Ellipsoid method has made the cutting plane algorithm applicable In 1979 L.Khachiyan involved the ellipsoid method to prove the polynomial solvability of linear programming problems.

14. Nonsmooth problems. Derivative-free minimization. How to produce the planes of cross-sections ifis not differentiable ? The oraclegives the values of instead of the gradient Kuzovkin-Tikhomirov (1967) Nemirovsky-Yudin (1979) V.P (1996)

15. The k th iteration of the ellipsoid method We cannot draw a separating hyperplane. We will construct a sufficiently wide cone. 1) By changing the coordinates transfer this ellipsoid to a Euclidean ball. 2) Construct a Euclidean cone centered at 3) Draw a hyperplane and separate

16. How to construct a sufficiently wide cone ? The first idea (Kuzovkin-Tikhomirov, 1967) Let us start with the case d=2 Compute the values f(x) at vertices of some polygon. Find a vertex, whose value is not less than the values at both its neighbors: For higher dimensions d the complexity becomes enormous.

17. Kuzovkin-Tikhomirov (1967) Nemirovsky-Yudin (1979) V.P (1996) The idea of the last algorithm

19. The method of Levin-Yudin-Nemirovsky-Shor-Khachiyan-Kuzovkin-Tikhomirov-Protasov ``Russian method’’ Thank you!