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Claude Lemaréchal

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Claude Lemaréchal
Claude Lemaréchal in 2005
NationalityFrench
Known forBundle methods of descent for convex minimization and nonsmooth optimization
AwardsDantzig Prize of SIAM and MPS 1994
Scientific career
FieldsMathematical optimization
Operations research
Scientific computing
InstitutionsINRIA

Claude Lemaréchal is a French applied mathematician, and former senior researcher (directeur de recherche) at INRIA[1] near Grenoble, France.

In mathematical optimization, Claude Lemaréchal is known for his work in numerical methods for nonlinear optimization, especially for problems with nondifferentiable kinks. Lemaréchal and Philip Wolfe pioneered bundle methods of descent for convex minimization.[2]

Awards

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In 1994, Claude Lemaréchal and Roger J-B Wets were each awarded the George B. Dantzig Prize. Recognizing "original research that has had a major impact on the field of mathematical programming", the Dantzig Prize is awarded by the Society for Industrial and Applied Mathematics (SIAM) and the Mathematical Programming Society (MPS).[2]

Lagrangian duality and nonconvex primal problems

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See also: Lagrangian multiplier, Convex analysis, and Convex optimization

Soon after joining INRIA (then named "IRIA"), Lemaréchal had the assignment of helping a glass-manufacturer with a problem of scheduling its production, a problem whose first formulation required minimizing a non-convex function. For this non-convex minimization problem, Lemaréchal applied the theory of Lagrangian duality that was described in Lasdon's Optimization Theory for Large Systems.[3][4] Because the primal problem was non-convex, there was no guarantee that a solution to the dual problem would provide useful information about the primal. Nonetheless, the dual problem did furnish useful information.[5] Lemaréchal's success with Lagrangian dual methods on nonlinear programming problems with nonconvexities interested Ivar Ekeland and Jean–Pierre Aubin, who applied the Shapley–Folkman lemma to explain Lemaréchal's success.[6][7] The Aubin–Ekeland analysis of duality gaps considered the convex closure of a nonconvex minimization problem — that is, the problem defined by the closed convex hull of the epigraph of the original problem. Following Ekeland and Aubin, similar applications of the Shapley–Folkman lemma are described in optimization monographs[7][8] and textbooks.[9] These developments were catalyzed by Lemaréchal's demonstration that Lagrangian-dual methods were useful on some optimization problems that lacked convexity.

Bundle methods of descent

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See also: Convex optimization and Subgradient method

Lemaréchal's research also led to his work on (conjugate) subgradient methods and on bundle methods of descent for convex minimization problems.

Notes

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  1. ^ INRIA is the acronym for the National Institute for Research in Computer Science and Control, in the original French, Institut national de recherche en informatique et en automatique (INRIA).
  2. ^ a b Citation of Claude Lemaréchal for the George Dantzig Prize in 1994 in Optima, Issue 44 (1994) pages 4-5.
  3. ^
    • Lasdon, Leon S. (1970). Optimization theory for large systems. Macmillan series in operations research. New York: The Macmillan Company. pp. xi+523. MR 0337317.
    • Lasdon, Leon S. (2002). Optimization theory for large systems (reprint of the 1970 Macmillan ed.). Mineola, New York: Dover Publications, Inc. pp. xiii+523. MR 1888251.
  4. ^ Aardal, Karen (March 1995). "Optima interview Claude Lemaréchal" (PDF). Optima: Mathematical Programming Society Newsletter: 2–4.
  5. ^
    • Lemaréchal, Claude (April 1973). Utilisation de la dualité dans les problémes non convexes [Use of duality for non-convex problems] (Report). Domaine de Voluceau, Rocquencourt, 78150 Le Chesnay, France: IRIA (Laboratoire de recherche en informatique et automatique). p. 41.{{cite report}}: CS1 maint: location (link)
    • Lemaréchal's experiments were discussed in later publications:
      • Aardal, Karen (March 1995). "Optima interview Claude Lemaréchal" (PDF). Optima: Mathematical Programming Society Newsletter: 2–4.
      • Hiriart-Urruty, Jean-Baptiste; Lemaréchal, Claude (1993). "XII Abstract duality for practitioners". Convex analysis and minimization algorithms, Volume II: Advanced theory and bundle methods. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Vol. 306. Berlin: Springer-Verlag. pp. 136–193 (and Bibliographical comments on pp. 334–335). ISBN 978-3-540-56852-0. MR 1295240.
  6. ^ Aubin, J.P.; Ekeland, I. (1976). "Estimates of the duality gap in nonconvex optimization". Mathematics of Operations Research. 1 (3): 225–245. doi:10.1287/moor.1.3.225. JSTOR 3689565. MR 0449695.
  7. ^ a b
    • Page 373: Ekeland, Ivar (1976). "Appendix I: An a priori estimate in convex programming". In Ekeland, Ivar; Temam, Roger (eds.). Convex analysis and variational problems. Studies in mathematics and its applications. Vol. 1 (translated, with new appendices, from the (1973) French ed.). Amsterdam: North-Holland Publishing Co. pp. 357–373. MR 0463994.
    • Page 373: Ekeland, Ivar (1999). "Appendix I: An a priori estimate in convex programming". In Ekeland, Ivar; Temam, Roger (eds.). Convex analysis and variational problems. Classics in applied mathematics. Vol. 28 (Corrected reprinting of the (1976) North–Holland ed.). Philadelphia, PA: Society for Industrial and Applied Mathematics (SIAM). pp. 357–373. ISBN 978-0-89871-450-0. MR 1727362.
  8. ^
    • Aubin, Jean-Pierre (2007). "14.2 Duality in the case of non-convex integral criterion and constraints, pages 458-476 (especially 14.2.3 The Shapley-Folkman theorem, pages 463-465)". Mathematical methods of game and economic theory (Reprint with a new author's preface of 1982 revised English ed.). Mineola, NY: Dover Publications, Inc. pp. xxxii+616. ISBN 978-0-486-46265-3. MR 2449499.
    • Besides presenting Ekeland-style analysis of duality gaps (acknowledgment on page 381), Bertsekas (1982) applies Lagrangian dual methods to the scheduling of electrical power plants ("unit commitment problems"), where nonconvexity appears because of integer constraints: Bertsekas, Dimitri P. (1982). "5.6 Large scale separable integer programming problems and the exponential method of multipliers". Constrained optimization and Lagrange multiplier methods. Computer Science and Applied Mathematics (first [Reprinted 1996 Athena Scientific, Belmont, MA., 1-886529-04-3] ed.). New York: Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers]. pp. 364–381. Bibcode:1982colm.book.....B. ISBN 978-0-12-093480-5. MR 0690767.
  9. ^
    • See Figure 5.1.9 (page 496): Bertsekas, Dimitri P. (1999). "5.1.6 Separable problems and their geometry". Nonlinear Programming (Second ed.). Cambridge, MA.: Athena Scientific. pp. 494–498. ISBN 978-1-886529-00-7.
    • Pages 267–279: Hiriart-Urruty, Jean-Baptiste (1998). "6 Ensembles et fonctions convexes. Projection sur un convexe fermé". Optimisation et analyse convexe. Mathématiques. Paris: Presses Universitaires de France. pp. 247–306. ISBN 978-2-13-048983-2. MR 1613914.

Bibliography

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Biographical

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Scientific publications

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