Critical point approximation through exact regularization

Authors:
Enrique Fernández Cara and Carlos Moreno

Journal:
Math. Comp. **50** (1988), 139-153

MSC:
Primary 65K10; Secondary 65N30

MathSciNet review:
917822

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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 semi-linear parabolic systems, and applications to some diffusion reaction equations*, Proc. Roy. Soc. Edinburgh Sect. A**81**(1978), no. 1-2, 35–47. MR**529375**, 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****[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****[4]**Giles Auchmuty,*Duality for nonconvex variational principles*, J. Differential Equations**50**(1983), no. 1, 80–145. MR**717869**, 10.1016/0022-0396(83)90085-2**[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****[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**, 10.1016/0362-546X(80)90083-8**[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****[9]**A. Bermúdez and C. Moreno,*Duality methods for solving variational inequalities*, Comput. Math. Appl.**7**(1981), no. 1, 43–58. MR**593554**, 10.1016/0898-1221(81)90006-7**[10]**Frank H. Clarke,*Generalized gradients and applications*, Trans. Amer. Math. Soc.**205**(1975), 247–262. MR**0367131**, 10.1090/S0002-9947-1975-0367131-6**[11]**W. Fenchel,*On conjugate convex functions*, Canadian J. Math.**1**(1949), 73–77. MR**0028365****[12]**E. Fernández-Cara & 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 non-differentiable functionals by the augmented Lagrangian method of Hestenes and Powell*, Appl. Math. Optim.**2**(1975/76), no. 3, 236–250. MR**0417897****[14]**Michel Fortin and Roland Glowinski,*Augmented Lagrangian methods*, Studies in Mathematics and its Applications, vol. 15, North-Holland 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****[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****[16]**Avner Friedman and Bruce Turkington,*Asymptotic estimates for an axisymmetric rotating fluid*, J. Funct. Anal.**37**(1980), no. 2, 136–163. MR**578929**, 10.1016/0022-1236(80)90038-5**[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. 17-40.**[20]**David Gilbarg and Neil S. Trudinger,*Elliptic partial differential equations of second order*, Springer-Verlag, Berlin-New York, 1977. Grundlehren der Mathematischen Wissenschaften, Vol. 224. MR**0473443****[21]**Roland Glowinski,*Numerical methods for nonlinear variational problems*, Springer Series in Computational Physics, Springer-Verlag, New York, 1984. MR**737005****[22]**R. Glowinski and A. Marrocco,*Sur l’approximation, par éléments finis d’ordre un, et la résolution, par pénalisation-dualité, d’une classe de problèmes de Dirichlet non linéaires*, Rev. Française Automat. Informat. Recherche Opérationnelle Sér. Rouge Anal. Numér.**9**(1975), no. R-2, 41–76 (French, with Loose English summary). MR**0388811****[23]**J.-B. Hiriart-Urruty,*Tangent cones, generalized gradients and mathematical programming in Banach spaces*, Math. Oper. Res.**4**(1979), no. 1, 79–97. MR**543611**, 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**, 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**, 10.1137/0716071**[26]**Bernard Martinet,*Détermination approchée d’un point fixe d’une application pseudo-contractante. Cas de l’application prox*, C. R. Acad. Sci. Paris Sér. A-B**274**(1972), A163–A165 (French). MR**0290213****[27]**C. Mercier,*The Magnetohydrodynamic Approach to the Problem of Plasma Confinement in Closed Magnetic Configurations*, EURATOM-CEA, 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**, 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****[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. 861-863.**[31]**R. Tyrrell Rockafellar,*Monotone operators and the proximal point algorithm*, SIAM J. Control Optimization**14**(1976), no. 5, 877–898. MR**0410483****[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****[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****[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****[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****[36]**David G. Schaeffer,*Non-uniqueness in the equilibrium shape of a confined plasma*, Comm. Partial Differential Equations**2**(1977), no. 6, 587–600. MR**0602535****[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****[39]**R. Temam,*A non-linear eigenvalue problem: the shape at equilibrium of a confined plasma*, Arch. Rational Mech. Anal.**60**(1975/76), no. 1, 51–73. MR**0412637****[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****[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**, 10.1007/BF00250669**[42]**J. F. Toland,*Duality in nonconvex optimization*, J. Math. Anal. Appl.**66**(1978), no. 2, 399–415. MR**515903**, 10.1016/0022-247X(78)90243-3

Retrieve articles in *Mathematics of Computation*
with MSC:
65K10,
65N30

Retrieve articles in all journals with MSC: 65K10, 65N30

Additional Information

DOI:
https://doi.org/10.1090/S0025-5718-1988-0917822-3

Keywords:
Nonconvex optimization,
exact regularization,
semilinear elliptic problems with discontinuous nonlinearities,
plasma confinement,
vortex rings

Article copyright:
© Copyright 1988
American Mathematical Society