Critical point approximation through exact regularization
Authors:
Enrique Fernández Cara and Carlos Moreno
Journal:
Math. Comp. 50 (1988), 139153
MSC:
Primary 65K10; Secondary 65N30
MathSciNet review:
917822
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Abstract: We present several iterative methods for finding the critical points and/or the minima of a functional which is essentially the difference between two convex functions. The underlying idea relies upon partial and exact regularization of the functional, which allows us to preserve the local feature in a large number of applications, as well as to obtain some convergence results. These methods are further applied to some differential problems of the semilinear elliptic type arising in plasma physics and fluid mechanics.
 [1]
Herbert
Amann, Existence and stability of solutions for semilinear
parabolic systems, and applications to some diffusion reaction
equations, Proc. Roy. Soc. Edinburgh Sect. A 81
(1978), no. 12, 35–47. MR 529375
(80b:35078), http://dx.doi.org/10.1017/S0308210500010428
 [2]
Herbert
Amann, On the existence of positive solutions of nonlinear elliptic
boundary value problems, Indiana Univ. Math. J. 21
(1971/72), 125–146. MR 0296498
(45 #5558)
 [3]
Antonio
Ambrosetti and Giovanni
Mancini, On some free boundary problems, Recent contributions
to nonlinear partial differential equations, Res. Notes in Math.,
vol. 50, Pitman, Boston, Mass.London, 1981, pp. 24–36. MR 639743
(84b:35120)
 [4]
Giles
Auchmuty, Duality for nonconvex variational principles, J.
Differential Equations 50 (1983), no. 1,
80–145. MR
717869 (86a:49053), http://dx.doi.org/10.1016/00220396(83)900852
 [5]
T.
Brooke Benjamin, The alliance of practical and analytical insights
into the nonlinear problems of fluid mechanics, Applications of
methods of functional analysis to problems in mechanics (Joint Sympos.,
IUTAM/IMU, Marseille, 1975) Springer, Berlin, 1976, pp. 8–29.
Lecture Notes in Math., 503. MR 0671099
(58 #32375)
 [6]
H. Berestycki, Thesis, University of Paris VI, 1980.
 [7]
Henri
Berestycki and Haïm
Brézis, On a free boundary problem arising in plasma
physics, Nonlinear Anal. 4 (1980), no. 3,
415–436. MR
574364 (83b:76096), http://dx.doi.org/10.1016/0362546X(80)900838
 [8]
Henri
Berestycki, Enrique
Fernández Cara, and Roland
Glowinski, A numerical study of some questions in vortex rings
theory, RAIRO Anal. Numér. 18 (1984),
no. 1, 7–85 (English, with French summary). MR 727602
(85i:76018)
 [9]
A.
Bermúdez and C.
Moreno, Duality methods for solving variational inequalities,
Comput. Math. Appl. 7 (1981), no. 1, 43–58. MR 593554
(82e:49016), http://dx.doi.org/10.1016/08981221(81)900067
 [10]
Frank
H. Clarke, Generalized gradients and
applications, Trans. Amer. Math. Soc. 205 (1975), 247–262.
MR
0367131 (51 #3373), http://dx.doi.org/10.1090/S00029947197503671316
 [11]
W.
Fenchel, On conjugate convex functions, Canadian J. Math.
1 (1949), 73–77. MR 0028365
(10,435b)
 [12]
E. FernándezCara & C. Moreno, "Exact regularization and critical point approximation," in Contributions to Nonlinear Partial Differential Equations, Vol. II (I. Díaz and P. L. Lions, eds.). (To appear.)
 [13]
Michel
Fortin, Minimization of some nondifferentiable functionals by the
augmented Lagrangian method of Hestenes and Powell, Appl. Math. Optim.
2 (1975/76), no. 3, 236–250. MR 0417897
(54 #5945)
 [14]
Michel
Fortin and Roland
Glowinski, Augmented Lagrangian methods, Studies in
Mathematics and its Applications, vol. 15, NorthHolland Publishing
Co., Amsterdam, 1983. Applications to the numerical solution of boundary
value problems; Translated from the French by B. Hunt and D. C. Spicer. MR 724072
(85a:49004)
 [15]
L.
E. Fraenkel and M.
S. Berger, A global theory of steady vortex rings in an ideal
fluid, Acta Math. 132 (1974), 13–51. MR 0422916
(54 #10901)
 [16]
Avner
Friedman and Bruce
Turkington, Asymptotic estimates for an axisymmetric rotating
fluid, J. Funct. Anal. 37 (1980), no. 2,
136–163. MR
578929 (81h:76058), http://dx.doi.org/10.1016/00221236(80)900385
 [17]
D. Gabay, "Minimizing the difference of two convex functions: Algorithms based on exact regularization." (To appear.)
 [18]
D. Gabay, Chapter 9 in Reference 14.
 [19]
D. Gabay & B. Mercier, "A dual algorithm for the solution of nonlinear variational problems via finite element approximations," Comput. Math. Appl., v. 2, 1976, pp. 1740.
 [20]
David
Gilbarg and Neil
S. Trudinger, Elliptic partial differential equations of second
order, SpringerVerlag, BerlinNew York, 1977. Grundlehren der
Mathematischen Wissenschaften, Vol. 224. MR 0473443
(57 #13109)
 [21]
Roland
Glowinski, Numerical methods for nonlinear variational
problems, Springer Series in Computational Physics, SpringerVerlag,
New York, 1984. MR 737005
(86c:65004)
 [22]
R.
Glowinski and A.
Marrocco, Sur l’approximation, par éléments
finis d’ordre un, et la résolution, par
pénalisationdualité, d’une classe de problèmes
de Dirichlet non linéaires, Rev. Française Automat.
Informat. Recherche Opérationnelle \jname RAIRO Analyse
Numérique 9 (1975), no. R2, 41–76
(French, with Loose English summary). MR 0388811
(52 #9645)
 [23]
J.B.
HiriartUrruty, Tangent cones, generalized gradients and
mathematical programming in Banach spaces, Math. Oper. Res.
4 (1979), no. 1, 79–97. MR 543611
(80i:58011), http://dx.doi.org/10.1287/moor.4.1.79
 [24]
D.
Kinderlehrer, L.
Nirenberg, and J.
Spruck, Regularity in elliptic free boundary problems, J.
Analyse Math. 34 (1978), 86–119 (1979). MR 531272
(83d:35060), http://dx.doi.org/10.1007/BF02790009
 [25]
P.L.
Lions and B.
Mercier, Splitting algorithms for the sum of two nonlinear
operators, SIAM J. Numer. Anal. 16 (1979),
no. 6, 964–979. MR 551319
(81g:47070), http://dx.doi.org/10.1137/0716071
 [26]
Bernard
Martinet, Détermination approchée d’un point
fixe d’une application pseudocontractante. Cas de
l’application prox, C. R. Acad. Sci. Paris Sér. AB
274 (1972), A163–A165 (French). MR 0290213
(44 #7397)
 [27]
C. Mercier, The Magnetohydrodynamic Approach to the Problem of Plasma Confinement in Closed Magnetic Configurations, EURATOMCEA, Comm. of the European Communities, Luxembourg, 1974 (Rep. EUR 5127/1).
 [28]
Wei
Ming Ni, On the existence of global vortex rings, J. Analyse
Math. 37 (1980), 208–247. MR 583638
(81i:76017), http://dx.doi.org/10.1007/BF02797686
 [29]
J.
Norbury, Steady planar vortex pairs in an ideal fluid, Comm.
Pure Appl. Math. 28 (1975), no. 6, 679–700. MR 0399645
(53 #3488)
 [30]
J. M. Puel, "Sur un problème de valeur propre nonlinéaire et de frontiere libre," C. R. Acad. Sci. Paris Sér. A, v. 284, 1977, pp. 861863.
 [31]
R.
Tyrrell Rockafellar, Monotone operators and the proximal point
algorithm, SIAM J. Control Optimization 14 (1976),
no. 5, 877–898. MR 0410483
(53 #14232)
 [32]
R.
T. Rockafellar, Augmented Lagrangians and applications of the
proximal point algorithm in convex programming, Math. Oper. Res.
1 (1976), no. 2, 97–116. MR 0418919
(54 #6954)
 [33]
R.
T. Rockafellar, The multiplier method of Hestenes and Powell
applied to convex programming, J. Optimization Theory Appl.
12 (1973), 555–562. MR 0334953
(48 #13271)
 [34]
R.
T. Rockafellar, Convex functions and duality in optimization
problems and dynamics, Mathematical systems theory and economics, I,
II (Proc. Internat. Summer School, Varenna, 1967) Springer, Berlin, 1969,
pp. 117–141. Lecture Notes in Operations Research and
Mathematical Economics, Vols. 11, 12. MR 0334940
(48 #13258)
 [35]
D.
H. Sattinger, Monotone methods in nonlinear elliptic and parabolic
boundary value problems, Indiana Univ. Math. J. 21
(1971/72), 979–1000. MR 0299921
(45 #8969)
 [36]
David
G. Schaeffer, Nonuniqueness in the equilibrium shape of a confined
plasma, Comm. Partial Differential Equations 2
(1977), no. 6, 587–600. MR 0602535
(58 #29210)
 [37]
M. Sermange, Thesis, University of Paris XI, 1982.
 [38]
Ivar
Stakgold, Estimates for some free boundary problems, Ordinary
and partial differential equations (Proc. Sixth Conf., Univ. Dundee,
Dundee, 1980) Lecture Notes in Math., vol. 846, Springer, Berlin,
1981, pp. 333–346. MR 610658
(82f:35189)
 [39]
R.
Temam, A nonlinear eigenvalue problem: the shape at equilibrium of
a confined plasma, Arch. Rational Mech. Anal. 60
(1975/76), no. 1, 51–73. MR 0412637
(54 #759)
 [40]
R.
Temam, Remarks on a free boundary value problem arising in plasma
physics, Comm. Partial Differential Equations 2
(1977), no. 6, 563–585. MR 0602544
(58 #29213)
 [41]
J.
F. Toland, A duality principle for nonconvex optimisation and the
calculus of variations, Arch. Rational Mech. Anal. 71
(1979), no. 1, 41–61. MR 522706
(82b:49031), http://dx.doi.org/10.1007/BF00250669
 [42]
J.
F. Toland, Duality in nonconvex optimization, J. Math. Anal.
Appl. 66 (1978), no. 2, 399–415. MR 515903
(80a:49025), http://dx.doi.org/10.1016/0022247X(78)902433
 [1]
 H. Amann, "Existence and stability of solutions for semilinear parabolic systems and applications to some diffusionreaction equations," Proc. Roy. Soc. Edinburgh Sect. A, v. 81, 1978, pp. 3747. MR 529375 (80b:35078)
 [2]
 H. Amann, "On the existence of positive solutions of nonlinear elliptic boundary value problems," Indiana Univ. Math. J., v. 21, 1971, pp. 125146. MR 0296498 (45:5558)
 [3]
 A. Ambrosetti & G. Mancini, "Remarks on some free boundary problems," in Recent Contributions to Nonlinear Partial Differential Equations (H. Berestycki and H. Brézis, eds.), Pitman, London, 1981. MR 639743 (84b:35120)
 [4]
 G. Auchmuty, "Duality for nonconvex variational principles," J. Differential Equations, v. 50, 1983, pp. 80145. MR 717869 (86a:49053)
 [5]
 T. B. Benjamin, The Alliance of Practical and Analytical Insights into the Nonlinear Problems of Fluid Mechanics, Lecture Notes in Math., vol. 503, SpringerVerlag, Berlin and New York, 1976, pp. 829. MR 0671099 (58:32375)
 [6]
 H. Berestycki, Thesis, University of Paris VI, 1980.
 [7]
 H. Berestycki & H. Brézis, "On a free boundary problem arising in plasma physics," Nonlinear Anal. T.M.A., v. 4, 1980, pp. 415436. MR 574364 (83b:76096)
 [8]
 H. Berestycki, E. FernándezCara & R. Glowinski, "A numerical study of some questions in vortex ring theory," RAIRO Anal. Numér., v. 18, 1984, pp. 785. MR 727602 (85i:76018)
 [9]
 A. Bermúdez & C. Moreno, "Duality methods for solving variational inequalities," Comput. Math. Appl., v. 7, 1981, pp. 4358. MR 593554 (82e:49016)
 [10]
 F. H. Clarke, "Generalized gradients and applications," Trans. Amer. Math. Soc., v. 205, 1975, pp. 247262. MR 0367131 (51:3373)
 [11]
 W. Fenchel, "On conjugate convex functions," Canad. J. Math., v. 1, 1949, pp. 7377. MR 0028365 (10:435b)
 [12]
 E. FernándezCara & C. Moreno, "Exact regularization and critical point approximation," in Contributions to Nonlinear Partial Differential Equations, Vol. II (I. Díaz and P. L. Lions, eds.). (To appear.)
 [13]
 M. Fortin, "Minimization of some nondifferentiable functionals by the augmented Lagrangian method of Hestenes and Powell," Appl. Math. Optim., v. 2, 1976, pp. 236250. MR 0417897 (54:5945)
 [14]
 M. Fortin & R. Glowinski, Augmented Lagrangian Methods: Applications to the Numerical Solution of BoundaryValue Problems, NorthHolland, Amsterdam, 1983. MR 724072 (85a:49004)
 [15]
 L. E. Fraenkel & M. S. Berger, "On the global theory of vortex rings in an ideal fluid," Acta Math., v. 132, 1974, pp. 1351. MR 0422916 (54:10901)
 [16]
 A. Friedman & B. Turcktngton, "Asymptotic estimates for an axisymmetric rotating fluid," J. Funct. Anal., v. 37, 1980, pp. 136163. MR 578929 (81h:76058)
 [17]
 D. Gabay, "Minimizing the difference of two convex functions: Algorithms based on exact regularization." (To appear.)
 [18]
 D. Gabay, Chapter 9 in Reference 14.
 [19]
 D. Gabay & B. Mercier, "A dual algorithm for the solution of nonlinear variational problems via finite element approximations," Comput. Math. Appl., v. 2, 1976, pp. 1740.
 [20]
 D. Gilbarg & N. S. Trudinger, Elliptic Partial Differential Equations of the Second Order, SpringerVerlag, Berlin and New York, 1977. MR 0473443 (57:13109)
 [21]
 R. Glowinski, Numerical Methods for Nonlinear Variational Problems, 2nd ed., SpringerVerlag, Berlin and New York, 1984. MR 737005 (86c:65004)
 [22]
 R. Glowinski & A. Marrocco, "Sur l'approximation par éléments finis d'ordre un et la résolution par pénalisationdualité d'une classe de problèmes de Dirichlet non linéaires," RAIRO Anal. Numér. Ser. Rouge, v. 2, 1975, pp. 4176. MR 0388811 (52:9645)
 [23]
 J. B. HiriartUrruty, "Tangent cones, generalized gradients and mathematical programming in Banach spaces," Math. Oper. Res., v. 4, 1979, pp. 7997. MR 543611 (80i:58011)
 [24]
 D. Kinderlehrer, L. Nirenberg & J. Spruck, "Regularity in elliptic free boundary problems. II," J. Analyse Math., v. 34, 1978, pp. 86119. MR 531272 (83d:35060)
 [25]
 P. L. Lions & B. Mercier, "Splitting algorithms for the sum of two nonlinear operators," SIAM J. Numer. Anal., v. 16, 1979, pp. 964979. MR 551319 (81g:47070)
 [26]
 B. Martinet, "Détermination approchée d'un point fixe d'une application pseudocontractante. Cas de l'application proximale," C. R. Acad. Sci. Paris Sér. A, v. 274, 1972, pp. 163165. MR 0290213 (44:7397)
 [27]
 C. Mercier, The Magnetohydrodynamic Approach to the Problem of Plasma Confinement in Closed Magnetic Configurations, EURATOMCEA, Comm. of the European Communities, Luxembourg, 1974 (Rep. EUR 5127/1).
 [28]
 WeiMing Ni, "On the existence of global vortex rings," J. Analyse Math., v. 37, 1980, pp. 208247. MR 583638 (81i:76017)
 [29]
 J. Norbury, "Steady planar vortex pairs in an ideal fluid," Comm. Pure Appl. Math., v. 28, 1975, pp. 679700. MR 0399645 (53:3488)
 [30]
 J. M. Puel, "Sur un problème de valeur propre nonlinéaire et de frontiere libre," C. R. Acad. Sci. Paris Sér. A, v. 284, 1977, pp. 861863.
 [31]
 R. T. Rockafellar, "Monotone operators and the proximal point algorithm," SIAM J. Control Optim., v. 14, 1976, pp. 877898. MR 0410483 (53:14232)
 [32]
 R. T. Rockafellar, "Augmented Lagrangians and applications of the proximal point algorithm in convex programming," Math. Oper. Res., v. 1, 1976, pp. 97116. MR 0418919 (54:6954)
 [33]
 R. T. Rockafellar, "The multipliers method of Hestenes and Powell applied to convex programming," J. Optim. Theory Appl., v. 12, 1973, pp. 555562. MR 0334953 (48:13271)
 [34]
 R. T. Rockafellar, Convex Functions and Duality in Optimization Problems and Dynamics, Lecture Notes Op. Res. Math. Ec., Vol. II, SpringerVerlag, Berlin, 1969. MR 0334940 (48:13258)
 [35]
 D. Sattinger, "Monotone methods in nonlinear elliptic and parabolic equations," Indiana Univ. Math. J., v. 21, 1972, pp. 9791000. MR 0299921 (45:8969)
 [36]
 D. F. Schaeffer, "Nonuniqueness in the equilibrium shape of a confined plasma," Comm. Partial Differential Equations, v. 2, 1977, pp. 587600. MR 0602535 (58:29210)
 [37]
 M. Sermange, Thesis, University of Paris XI, 1982.
 [38]
 I. Stakgold, "Estimates for some free boundary problems," in Ordinary and Partial Differential Equations Proc., Dundee 1980 (W. N. Everitt and B. D. Sleeman, eds.), Lecture Notes in Math., vol. 846, SpringerVerlag, Berlin, 1981. MR 610658 (82f:35189)
 [39]
 R. Témam, "A nonlinear eigenvalue problem: The shape at equilibrium of a confined plasma," Arch. Rational Mech. Anal., v. 60, 1975, pp. 5173. MR 0412637 (54:759)
 [40]
 R. Témam, "Remarks on a free boundary problem arising in plasma physics," Comm. Partial Differential Equations, v. 2, 1977, pp. 563585. MR 0602544 (58:29213)
 [41]
 J. F. Toland, "A duality principle for nonconvex optimization and the calculus of variations," Arch. Rational Mech. Anal., v. 71, 1979, pp. 4161. MR 522706 (82b:49031)
 [42]
 J. F. Toland, "Duality in nonconvex optimization," J. Math. Anal. Appl., v. 66, 1978, pp. 399415. MR 515903 (80a:49025)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00255718198809178223
PII:
S 00255718(1988)09178223
Keywords:
Nonconvex optimization,
exact regularization,
semilinear elliptic problems with discontinuous nonlinearities,
plasma confinement,
vortex rings
Article copyright:
© Copyright 1988
American Mathematical Society
