Minimizing within convex bodies using a convex hull method

SIAM Journal on Optimization

Volumn 16, Issue 2, 2006, Pages 368-379

  Lachand Robert, Thomas ^a   Oudet, Édouard ^a  

^a UNIVERSITÉ DE SAVOIE   (France)

Author keywords

Alexandrov; Cheeger; Convex bodies; Convex functions; Newton's problem of the body of minimal resistance; Numerical schemes; Optimization

Indexed keywords

ALGORITHMS; FUNCTIONS; GEOMETRY; MATHEMATICAL MODELS; NUMERICAL METHODS; PROBLEM SOLVING;

ALEXANDROV; CHEEGER; CONVEX BODIES; CONVEX FUNCTIONS; NEWTON'S PROBLEM OF THE BODY OF MINIMAL RESISTANCE; NUMERICAL SCHEMES;

OPTIMIZATION;

EID: 33646728935     PISSN: 10526234     EISSN: None     Source Type: Journal    
DOI: 10.1137/040608039     Document Type: Article

References (22)

1. A. D. ALEXANDROV, Theory of mixed volumes for convex bodies, Mathem. Sb. USSR, 2 (1937), pp. 947-972.
2. N, J. Differential Equations, 184 (2002), pp. 475-525.
3. F. BROCK, V. FERONE, AND B. KAWOHL, A symmetry problem in the calculus of variations, Calc. Var. Partial Differential Equations, 4 (1996), pp. 593-599.
4. G. BUTTAZZO AND P. GUASONI, Shape optimization problems over classes of convex domains, J. Convex Anal., 4 (1997), pp. 343-351.
5. G. BUTTAZZO, V. FERONE, AND B. KAWOHL, Minimum problems oversets of concave functions and related questions, Math. Nachr., 173 (1993), pp. 71-89.
6. G. CARLIER, On a theorem of Alexandrov, J. Nonlinear Convex. Anal., 5 (2004), pp. 49-58.
7. G. CARLIER AND T. LACHAND-ROBERT, Convex bodies of optimal shape, J. Convex Anal., 10 (2003), pp. 265-273.
8. G. CARLIER AND T. LACHAND-ROBERT, Regularity of solutions for some variational problems subject to convexity constraint, Comm. Pure Appl. Math., 54 (2001), pp. 583-594.
9. 1-projection into sets of convex functions: A saddle point formulation, in CEMRACS 1999 (Orsay), ESAIM Proc. 10, Soc. Math. Appl. Indust., Paris, 1999, pp. 277-289.
10. G. CARLIER, T. LACHAND-ROBERT, AND B. MAURY, A numerical approach to variational problems subject to convexity constraint, Numer. Math., 88 (2001), pp. 299-318.
11. J. CHEEGER, A lower bound for the smallest eigenvalue of the Laplacian, in Problems in Analysis, A Symposium in Honor of Salomon Bochner, R. C. Gunning, ed., Princeton University Press, Princeton, NJ, 1970, pp. 195-199.
12. P. CHONÉ AND J.-C. ROCHET, Ironing, sweeping and multidimensional screening, Econometrica, 66 (1998), pp. 783-826.
13. P. CHONÉ AND H. LE MEUR, Non-convergence result for conformal approximation of variational problems subject to a convexity constraint, Numer. Funct. Anal. Optim., 22 (2001), pp. 529-547.
14. V. FRIDMAN AND B. KAWOHL, Isoperimetric estimates for the first eigenvalue of the p-Laplace operator and the Cheeger constant, Comment. Math. Univ. Carolin., 44 (2003), pp. 659-667.
15. J. HOLLAND, Adaptation in Natural and Artificial Systems, University of Michigan Press, Ann Arbor, MI, 1975.
16. I. IONESCU AND T. LACHAND-ROBERT, Generalized Cheeger sets related to landslides, Calc. Var. Partial Differential Equations, to appear.
17. B. KAWOHL, On a family of torsional creep problems, J. Reine Angew. Math., 410 (1990), pp. 1-22.
18. B. KAWOHL AND T. LACHAND-ROBERT, Characterization of Cheeger sets for convex subsets of the plane, Pacific J. Math., to appear.
19. T. LACHAND-ROBERT AND M. A. PELETIER, Newton's problem of the body of minimal resistance in the class of convex developable functions, Math. Nachr., 226 (2001), pp. 153-176.
20. P.-L. LIONS, Identification du cône dual des fonctions convexes et applications, C. R. Acad. Sci. Paris Sér. I Math., 326 (1998), pp. 1385-1390.
21. H. MINKOWSKI, Allgemeine Lehrsätze über die Konvexen Polyeder, Nach. Ges. Wiss., Göttingen, 1897, pp. 198-219.
22. R. T. ROCKAFELLAR, Convex Analysis, Princeton University Press, Princeton, NJ, 1970.