The complementary variational principles and associated algonrthm are presented in this paper for solving a class of nonconvex mechanics problems governed by the Landau-Ginzburg equation. The method used here is the canonical dual transfonnation developed recently in nonconvex analysis by the author. It is shown that, by this method, certain very difficult nonconvex potential variational problems can be converted into a coupled quadratic mixed variational problem. The extremality conditions of these complementary variational problems are controlled by the triality theory discovered recently. Therefore, a complementary energy principle (i.e. the canonical duality theorem with zero duality gap) is established, which leads to a potentially useful primal-dual algorithm. Applications in finite-dimensional problems are discussed. Results show that the discretized nonconvex potential variational problems in n-dimensional space can be converted into certain simple canonical (either convex or concave) dual problems. Therefore, a complete set of solutions is obtained for the Landau-Ginzburg equation in finite-dimensional space.
[1] Atai, A. A. and Steigmann, D.Coupled deformations of elastic curves and surfaces . International Journal of Solids and Structures, 35, 1915-1952 (1998).
2.
[2] Auchmuty, G.Duality for non-convex variational principles , Journal of Difrrential Equations, 50, 80-145 (1983).
3.
[3] Auchmuty, G.Variational principles for self-adjoint elliptic eigenproblems, in Nonconvex/Nonsmooth Mechanics: Modelling, Methods and Algorithms, ed. D. Y. Gao , R. W Ogden & G. Stavroulakis , Kluwer, Dordrecht , 2001.
4.
[4] Benson, H.Concave minimization: theory, applications and algorithms, in Handbook of Global Optimization, pp. 43-148, ed. R. Horst & P Pardalos , Kluwer, Dordrecht . 1995.
5.
[5] Clarke, F. H.The dual action, optimal control, and generalized gradients. in Mathematical Control Theory, vol. 14. pp. 109-119, Banach Center Publications, PWN, Warsaw , 1985.
6.
[6] Crouzeix, J. P.Duality framework in quasiconvex programming, in Generalized Comnexity in Optimization and Economics. pp. 207-226, ed. S. Schaible & W. T. Ziemba , Academic. New York , 1981.
7.
[7] Dacorogna, D.Direct Methods in the Calculus of Variations, Springer, Berlin , 1989.
8.
[8] Ekeland, I.Legendre duality in nonconvex optimization and calculus of variations . SIAM Journal of Control and Optimization, 15, 905-934 (1977).
9.
[9] Ekeland, I.Convexity Methods in Hamiltonian Mechanics, Springer, Berlin , 1990.
10.
[10] Ekeland, I.Nonconvex Duality, to appear in Proceedings of IUTAMS symposium on Duality: Complementarityo and Symmetty in ANonlinear Mechanics, ed. D.Y. Gao , Kluwer, Dordrecht , 2003.
11.
[11] Ekeland, I. and Temam, R.Comnex Analysis and Variational Pivblems. North-Holland, Amsterdam , 1976.
12.
[12] Floudas, C. A. and Visweswaran, VQuadratic optimization. in Handbook of Global Optimiization. pp. 217-270. ed. R. Horst & P. M. Pardalos . Kluwer, Dordrecht . 1995.
13.
[13] Gao, D. Y.Dual extremum principles in finite deformation theory with applications to post-buckling analysis of extended nonlinear beam theory , Applied Mechanics Review, 50, S64-S71 (1997).
14.
[14] Gao, D. Y.Duality, triality and complementary extremum principles in nonconvex parametric variational problems with applications . IMA Journal ofApplied Mathematics, 61, 199-235 (1998).
15.
[15] Gao, D. Y.Duality mathematics , in Wiley Encyclopedia of Electronics and Electrical Engineering. vol. 6, pp. 68-77 . 1999.
16.
[16] Gao, D. Y.General analytic solutions and complementary variational principles for large deformation nonsmooth mechanics . Meccanica34, 169-198 (1999).
17.
[17] Gao, D. Y.Duality Principles in Nonconvex Systems: Theory Methods and Applications, Kluwer, Dordrecht , 2000.
18.
[18] Gao, D. Y.Analytic solution and triality theory for nonconvex and nonsmooth variational problems with applications . Nonlinear Analysis, 42, 1161-1193 (2000).
19.
[19] Gao, D. Y.Canonical dual transformation method and generalized triality theory in nonsmooth global optimization . Journal of Globel Optimization. 17. 127-160 (2000).
20.
[20] Gao, D. Y. , (2000). Finite deformnation beam models and triality theory in dynamical -post ibuckling anlalysis . International Journal of Nonlinear Mechanics, 35, 103-131 (2000).
21.
[21] Gao, D. Y. , (2000). Canonical Dual Control for Nonconvex Distributed-Parameter Systems: Theory and Method. In Control of Nonliner Distribuied Parameter System, Goong Chen , Irena Lasiecka and Jianxin Zhou eds). Marcel Dekker 2000,85-112.
22.
[22] Gao. D. . Bi-duality in nonconvex optimization, in Encyclopedia of Opotimization, vol. 1, pp.477-482. ed. C. A. Floidas & P. D. Pardalos . Kluwer, Dordrecht , 2001.
23.
[23] Gao, D. Y.Tri-duality in global optimization, in Encyclopedia of Optitnimazation Vol. 1, pp. 485-491 ed. C A Floudas & P D. Paidalos , Kluwei, Dordrecht , 2001.
24.
[24] Gao. D. Y.Complementarity, polarity and triality in nonsmooths, nonconvex and nonconservative Hamilton systems . Philosophical Transactions of the Royal Society: Mathematical, Physical and Engineering Sciences, 359, 2347-2367 (2001).
25.
[25] Gao, D. Y.Duality and triality in non-smooth, nonconvex and nonconservative systems: a survey, new phenomena and new results, in Nonsmooth/Nonconvex Mechanics with Applications in Engineering, pp. 1-14. ed. C. Baniotopoulos , Thessaloniki, Greece , 2002.
26.
[26] Gao, D. Y.Complete solutions to nonconvex quadratic optimization problems , to appear in Journal of Global Optimization, special issue on Duality, ed. D. Y. Gao & K. L. Teo, 2004.
27.
[27] Gao, D. Y.Perfect duality theory and complete solutions to a class of nonconvex optimization problems . Optimization, 52(4-5), 467-493 , (2003).
28.
[28] Gao, D. Y. , Jie-Fang Li and Viehland, D.Complete solutions and triality theory to Landau-Ginzburg equation in imperfect ferroelectrics, Proceedings of the 4th International Conference on Nonlinear Mechanics, ed. W Z. ChieisShanghai University Press, Shanghai , 2002, submitted to Physics Review.
29.
[29] Gao, D. Y. , Ogden R. W. and Stavroulakis, G.Nonsmooth and Nonconvex Mechanics: Modelling Analysis and NVnierical Methods, Kluwer, Dordrecht , 2001.
30.
[30] Gao, D. Y. and Ogden R. WOn the complementary-dual-polar variational principles in finite deformation theory: I. Hyperelasticity (2003), in preparation.
31.
[31] Gao, D. Y. and Strang, G.Geometric nonlinearity: potential energy., complementary energy, and the gap function . Quarterly Journal of Applied Mathematics, 47, 487-504 (1989a).
32.
[32] Gao, D. Y and Strang, G.Dual extremum principles in finite deformation elastoplastic analysis . Acta Applied Mathematics, 17, 257-267 (1989b).
33.
[33] Gasimov, R. N.Augmented Lagrangian duality and nondifferentiable optimization methods in nonconvex programming . Journal of Global Optimization, 24, 187-203 (2002).
34.
[34] Goh, C. J. and Yang, X. Q.Duality in Optimization and Variational Inequalities, Taylor and Francis, London , 2002.
35.
[35] Horst, R. , Pardalos, P M. . and Nguyen Van Thoai . Introduction to Global Optimization, Kluwer, Dordrecht . 2000.
36.
[36] Rockafellar, R. T. and Wets, R. J. B.Variational Analysis, Springer, Berlin , 1997.
37.
[37] Powell, M. J. D.UOBYQA: unconstrained optimization by quadratic approximation , Mathematical Programming, Series B, 92, 555-582 (2002).
38.
[38] Rubinov, A. M. and Gasimov, R. N.Scalarization and nonlinear scalar duality for vector optimization with preferences that are not necessarily a pre-order relation . Journal of Global Optimization, special issue on Duality, ed. D. Y. Gao & K. L. Teo (2003) to appear.
39.
[39] Sewell, M. J.Maximum and Minimunm Principles, Cambridge University Press. Cambridge , 1987.
40.
[40] Singer, I.Duality for optimization and best approximation over finite intersections . Nuimerical Functional Analysus and Optimization, 19, 903-915 (1998).
41.
[41] Strang, G.Introduction to Applied Mathematics, Wellesley/Cambridge University Press, Cambridge , 1986.
42.
[42] Tabarrok, B. and Rimrott, F. P J.Variational Methods and Complementary Formulations in Dynamics, Kluwer, Dordrecht , 1994.
43.
[43] Thach, P. TDiewert-Crouzeix conjugation for general quasiconvex duality and applications . Journal of Optimization Theory and Applications. 86, 719-743 (1995).
44.
[44] Walk, M.Theory of Duality in Mathematical Programming, Springer, Vienna , 1989.
45.
[45] Wright, M. H.The interior-point revolution in constrained optimization, in High-Perfornmance Algorithms and Sofware in Nonlinear Optimization, pp. 359-381. ed. R. DeLeone , A. Murli , P. M. Pardalos , & G. Toraldo , Kluwer, Dordrecht , 1998.
46.
[46] Yang, Chen-NingThe law of parity conservation and other symmetry laws of physics, Nobel Lecture, December 11, 1957. PDF file can be downloaded at http://www.nobel.se/physics/laureates/1957/yang-lecture.pdf.
