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History-2

History-2. Boris Polyak Institute for Control Science Moscow, Russia Luminy, April 2007. History of mathematical programming in the USSR: analyzing the phenomenon. 17th Intern. Symp. on Math. Progr., Atlanta, August 2000 Math. Progr., ser. B, 2002, v. 91, No. 3, 401-416. History-1.

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History-2

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  1. History-2 Boris Polyak Institute for Control Science Moscow, Russia Luminy, April 2007

  2. History of mathematical programming in the USSR: analyzing the phenomenon • 17th Intern. Symp. on Math. Progr., Atlanta, August 2000 • Math. Progr., ser. B, 2002, v. 91, No. 3, 401-416.

  3. History-1 Period – from Euler through Chebyshev, Markov, Lyapunov, Kantorovich to 1950s and 1960s. Later period - verybriefly. Now it is time to survey modern history. Not surprisingly it will be strongly related to the name Arik Nemirovski Hence it will be a selected history.

  4. Starting point – middle of 1970s • Situation in MP: general picture clarified, extremum conditions elaborated, numerous numerical methods are developed, their local rate of convergence is studied. • Future directions look as minor improvements of the known ones, no breakthroughs are expected.

  5. The event • Time – 1975 • Place – economic department of MSU, seminar of Prof. Yudin • Speaker – A.Nemirovski • Listeners – students in economics and B.Polyak • Event – new look at optimization

  6. Main contributions • Notion of oracle • Notion of complexity for classes of optimization problems • Notion of efficient method • Checking efficiency of particular methods (global rate of convergence!) • New methods (ellipsoid method, mirror descent etc.)

  7. Publications • Papers in Ekonomika i matem. metody, Doklady AN SSSR, 1976 – 1979 • The book Nemirovski, Yudin 1979 (English translation – 1983)

  8. Response Some results have been recognized very fast (e.g. ellipsoid method, which led to Khachiyan’s theorem on polynomial complexity of LP and to Fulkerson prize, 1982). However most of the ideas were too revolutionary and their understanding and development was slow. Nesterov’s book Efficient methods of optimization, Moscow, 1989 (in Russian).

  9. Further results (1980s) Nemirovsky—>Nemirovskii—>Nemirovskij —>Nemirovski New directions: Nonparametric statistics Noise-corrupted optimization problems Control, stability

  10. Example Nemirovski, Polyak, Necessary conditions for stability of polynomials and their use, Autom. Remote Control, 1994, V. 55, No. 11, 1644-1649 Corollary: ai=rand, i=0,…,n – coefficients of the polynomial p(s). Tnen probability of p(s) to be stable <1/[(n+1)/2]! References: 1.Gauss 2.Newton 3.Lobachevski 4.Polya, Szegö

  11. Collaboration with Yu.Nesterov • Self-concordant functions • Damped Newton method • Global convergence and complexity • Self-concordant barriers • Interior-point methods

  12. Summarizing publication Nesterov, Nemirovskii Interior-point polynomial algorithms in convex programming,SIAM, 1994

  13. Semi-Definite Programming • LMIs and their role in control (S.Boyd) • Interior-point methods for SDP • Software (LMI toolbox) First presentation – Luminy, 1990, 1st

  14. Recent results • Robust optimization (with R.Ben-Tal) • New approach to control • Engineering applications (truss topology etc.) • Combinatorial optimization • And much more…

  15. Some personal comments…

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