Perfect duality theory and complete solutions to a class of global optimization problemsFootnote*

*This article is dedicated to Professor Gilbert Strang on the occasion of his 70th birthday. The main results of this article has been presented at the International Conference on Nonsmooth/Nonconvex Mechanics, Aristotle University of Thessaloniki (A.U.Th.), July 5–6, 2002 (keynote lecture), and the Second International Conference on Optimization and Control with Applications, August 18–22, 2002, Yellow Mountains, Anhui, China (plenary lecture).

References

• Atai , A.A. and Steigmann , D. 1998 . Coupled deformations of elastic curves and surfaces . Int. J. Solids and Structures , 35 : 1915 – 1952 .
   Google Scholar
• Auchmuty , G. 1983 . Duality for non-convex variational principles . J. Diff. Equations , 50 : 80 – 145 .
   Web of Science ®Google Scholar
• Auchmuty G. 2001 Variational principles for self-adjoint elliptic eigenproblems In: D.Y. Gao, R.W. Ogden and G. Stavroulakis (Eds.) Nonconvex/Nonsmooth Mechanics: Modelling, Methods and Algorithms p. 478 Kluwer Academic Publishers
   Google Scholar
• Benson H. 1995 Concave minimization: theory, applications and algorithms In R. Horst and P. Pardalos (Eds.) Handbook of Global Optimization pp. 43–148 Kluwer Academic Publishers
   Google Scholar
• Clarke F.H. 1985 The dual action, optimal control, and generalized gradients Mathematical Control Theory pp. 109–119. Banach Center Publ. 14, PWN Warsaw
   Google Scholar
• Crouzeix J.P. 1981 Duality framework in quasiconvex programming In: S. Schaible and W.T. Ziemba (Eds.) Generalized Convexity in Optimization and Economics pp. 207–226 Academic Press
   Google Scholar
• Dacorogna D. 1989 Direct Methods in the Calculus of Variations Springer-Verlag
   Google Scholar
• Ekeland , I. 1977 . Legendre duality in nonconvex optimization and calculus of variations . SIAM J. Control and Optimization , 15 : 905 – 934 .
   Google Scholar
• Ekeland I. 1990 Convexity Methods in Hamiltonian Mechanics p. 247 Springer-Verlag
   Google Scholar
• Ekeland I. 2003 Nonconvex duality In: D.Y. Gao (Ed.) Proceedings of IUTAM Symposium on Duality, Complementarity and Symmetry in Nonlinear Mechanics (to appear) Kluwer Academic Publishers Dordrecht/Boston/London
   Google Scholar
• Ekeland I. Temam R. 1976 Convex Analysis and Variational Problems North-Holland
   Google Scholar
• Floudas C.A. Visweswaran V. 1995 Quadratic optimization In: R. Horst and P.M. Pardalos (Eds.) Handbook of Global Optimization pp. 217–270 Kluwer Academic Publishers Dordrecht/Boston/London
   Google Scholar
• Gao , D.Y. 1997 . Dual extremum principles in finite deformation theory with applications to post-buckling analysis of extended nonlinear beam theory . Applied Mechanics Reviews , 50 ( 11 ) : S64 – S71 .
   Google Scholar
• Gao , D.Y. 1998 . Duality, triality and complementary extremum principles in nonconvex parametric variational problems with applications . IMA J. Appl. Math. , 61 : 199 – 235 .
   Google Scholar
• Gao , D.Y. 1999a . Duality mathematics . Wiley Encyclopedia of Electronical and Electronical Engineering , 6 : 68 – 77 .
   Google Scholar
• Gao , D.Y. 1999b . General analytic solutions and complementary variational principles for large deformation nonsmooth mechanics . Meccanica , 34 : 169 – 198 .
   Web of Science ®Google Scholar
• Gao D.Y. 2000a Duality Principles in Nonconvex Systems: Theory, Methods and Applications xviii p. 454 Kluwer Academic Publishers Dordrecht/Boston/London
   Google Scholar
• Gao , D.Y. 2000b . Analytic solution and triality theory for nonconvex and nonsmooth variational problems with applications . Nonlinear Analysis , 42 ( 7 ) : 1161 – 1193 .
   Google Scholar
• Gao , D.Y. 2000 . Canonical dual transformation method and generalized triality theory in nonsmooth global optimization . J. Global Optimization , 17 ( 1/4 ) : 127 – 160 .
   Web of Science ®Google Scholar
• Gao D.Y. 2001a Bi-duality in nonconvex optimization In: C.A. Floudas and P.D. Pardalos (Eds.) Encyclopedia of Optimization 1 pp. 477–482 Kluwer Academic Publishers Dordrecht/Boston/London
   Google Scholar
• Gao D.Y. 2001b Tri-duality in global optimization In: C.A. Floudas and P.D. Pardalos (Eds.) Encyclopedia of Optimization 1 pp. 485–491 Kluwer Academic Publishers Dordrecht/Boston/London
   Google Scholar
• Gao , D.Y. 2001c . Complementarity, polarity and triality in nonsmooth, nonconvex and nonconservative Hamilton systems . Philosophical Transactions of the Royal Society: Mathematical, Physical and Engineering Sciences , 359 : 2347 – 2367 .
   Google Scholar
• Gao D.Y. 2002 Duality and triality in non-smooth, nonconvex and nonconservative systems: a survey, new phenomena and new results In: C. Baniotopoulos (Ed.) Nonsmooth/Nonconvex Mechanics with Applications in Engineering pp. 1–14 Thessaloniki Greece
   Google Scholar
• Gao D.Y. 2003a Perfect duality theory complete solutions for constrained optimization problems In: D.Y. Gao and K.L. Teo (Eds.) J. Global Optimisation special issue on duality (to be published)
   Google Scholar
• Gao D.Y. 2003b Canonical dual principle, algorithm, and complete solutions to Landau-Ginzburg equation with applications In: D. Steigmann (Ed.) Journal of Mathematics and Mechanics of Solids special issue dedicated to Professor Ray Ogden for the occasion of his 60th birthday
   Google Scholar
• Gao D.Y. Li Jie-Fang Viehland D. 2002 Complete solutions and triality theory to Landau-Ginzburg equation in imperfect ferroelectrics In: W.Z. Chien (Ed.) Proceedings of the 4th International Conference on Nonlinear Mechanics Submitted to Physics Review Shanghaui University Press
   Google Scholar
• Gao D.Y. Ogden R.W. Stavroulakis G. 2001 Nonsmooth and Nonconvex Mechanics:Modelling, Analysis and Numerical Methods xliv p. 471 Kluwer Academic Publishers Boston/Dordrecht/London
   Google Scholar
• Gao , D.Y. and Strang , G. 1989a . Geometric nonlinearity: potential energy, complementary energy, and the gap function . Quart. Appl. Math , 47 ( 3 ) : 487 – 504 .
   Google Scholar
• Gao , D.Y. and Strang , G. 1989b . Dual extremum principles in finite deformation elastoplastic analysis . Acta Appl. Math. , 17 : 257 – 267 .
   Google Scholar
• Gasimov , R.N. 2002 . Augmented Lagrangian duality and nondifferentiable optimization methods in nonconvex programming . J. Global Optimization , 24 : 187 – 203 .
   Web of Science ®Google Scholar
• Goh C.J. Yang X.Q. 2002 Duality in Optimization and Variational Inequalities p. 329 Taylor and Francis
   Google Scholar
• Horst R. Pardalos M. Panos Thoai Nguyen Van 2000 Introduction to Global Optimization Kluwer Academic Publishers
   Google Scholar
• Rockafellar R.T. Wets R.J.B. 1997 Variational Analysis Springer Berlin, New York
   Google Scholar
• Powell , M.J.D. 2002 . UOBYQA: unconstrained optimization by quadratic approximation . Mathematical Programming, Series B , 92 ( 3 ) : 555 – 582 .
   Web of Science ®Google Scholar
• Rubinov A.M. Gasimov R.N. 2003 Scalarization and nonlinear scalar duality for vector optimization with preferences that are not necessarily a pre-order relation J. Global Optimization (special issue on Duality edited by D.Y. Gao and K.L. Teo), to appear
   Google Scholar
• Rubinov A.M. Yang X.Q. 2003 Lagrange-type Functions in Constrained Non-convex Optimization p. 285 Kluwer Academic Publishers Boston/Dordrecht/London
   Google Scholar
• Rubinov , A.M. , Yang , X.Q. and Glover , B.M. 2001 . Extended Lagrange and penalty functions in optimization . J. Optim. Theory Appl. , 111 ( 2 ) : 381 – 405 .
   Google Scholar
• Sewell M.J. 1987 Maximum and Minimum Principles p. 468 Cambridge Univ. Press
   Google Scholar
• Singer , I. 1998 . Duality for optimization and best approximation over finite intersections . Numer. Funct. Anal. Optim. , 19 ( 7–8 ) : 903 – 915 .
   Web of Science ®Google Scholar
• Strang G. 1986 Introduction to Applied Mathematics p. 758 Wellesley-Cambridge Press
   Google Scholar
• Tabarrok B. Rimrott F.P.J. 1994 Variational Methods and Complementary Formulations in Dynamics Kluwer Academic Publishers Dordrecht
   Google Scholar
• Thach , P.T. 1993 . Global optimality criterion and a duality with a zero gap in nonconvex optimization . SIAM J. Math. Anal. , 24 ( 6 ) : 1537 – 1556 .
   Web of Science ®Google Scholar
• Thach , P.T. 1995 . Diewert-Crouzeix conjugation for general quasiconvex duality and applications . J. Optim. Theory Appl. , 86 ( 3 ) : 719 – 743 .
   Web of Science ®Google Scholar
• Thach , P.T. , Konno , H. and Yokota , D. 1996 . Dual approach to minimization on the set of Pareto-optimal solutions . J. Optim. Theory Appl. , 88 ( 3 ) : 689 – 707 .
   Google Scholar
• Toland , J.F. 1978 . Duality in nonconvex optimization . J. Mathematical Analysis and Applications , 66 : 399 – 415 .
   Web of Science ®Google Scholar
• Tuy H. 1995 D.C. optimization: theory, methods and algorithms In: R. Horst and P. Pardalos (Eds.) Handbook of Global Optimization pp. 149–216 Kluwer Academic Publishers
   Google Scholar
• Walk M. 1989 Theory of Duality in Mathematical Programming Springer-Verlag Wien/New York
   Google Scholar
• Wright M.H. 1998 The interior-point revolution in constrained optimization In: R. DeLeone, A. Murli, P.M. Pardalos and G. Toraldo (Eds.) High-performance Algorithms and Software in Nonlinear Optimization pp. 359–381 Kluwer Academic Publishers Dordrecht
   Google Scholar
• Rockaffellar R.T. 1974 Conjugate Duality and Optimization SIAM Philadelphia
   Google Scholar