Canonical Dual Transformation Method and Generalized Triality Theory in Nonsmooth Global Optimization

This paper presents, within a unified framework, a potentially powerful canonical dual transformation method and associated generalized duality theory in nonsmooth global optimization. It is shown that by the use of this method, many nonsmooth/nonconvex constrained primal problems in \realn can be reformulated into certain smooth/convex unconstrained dual problems in \realm with m≤slant n and without duality gap, and some NP-hard concave minimization problems can be transformed into unconstrained convex minimization dual problems. The extended Lagrange duality principles proposed recently in finite deformation theory are generalized suitable for solving a large class of nonconvex and nonsmooth problems. The very interesting generalized triality theory can be used to establish nice theoretical results and to develop efficient alternative algorithms for robust computations.

[1]  Hoang Tuy,et al.  D.C. Optimization: Theory, Methods and Algorithms , 1995 .

[2]  Harold P. Benson,et al.  Concave Minimization: Theory, Applications and Algorithms , 1995 .

[3]  Ivan Singer.  A general theory of dual optimization problems , 1986 .

[4]  P. Pardalos,et al.  Handbook of global optimization , 1995 .

[5]  Margaret H. Wright,et al.  The interior-point revolution in constrained optimization , 1998 .

[6]  P. Panagiotopoulos,et al.  Quasidifferentiability and Nonsmooth Modelling in Mechanics, Engineering and Economics , 1996 .

[7]  Gilbert Strang,et al.  Introduction to applied mathematics , 1988 .

[8]  J. Toland.  A duality principle for non-convex optimisation and the calculus of variations , 1979 .

[9]  G. Strang,et al.  Geometric nonlinearity: potential energy, complementary energy, and the gap function , 1989 .

[10]  Michael L. Overton,et al.  A Primal-dual Interior Method for Nonconvex Nonlinear Programming , 1998 .

[11]  Giulio Maier.  Complementary plastic work theorems in piecewise-linear elastoplasticity , 1969 .

[12]  P. T. Thach,et al.  Dual approach to minimization on the set of pareto-optimal solutions , 1996 .

[13]  Phan Thien Thach,et al.  Diewert-Crouzeix conjugation for general quasiconvex duality and applications , 1995 .

[14]  David Yang Gao,et al.  Duality, triality and complementary extremum principles in non-convex parametric variational problems with applications , 1998 .

[15]  Gao Yang.  Panpenalty finite element programming for plastic limit analysis , 1988 .

[16]  David Yang Gao,et al.  Minimax and Triality Theory in Nonsmooth Variational Problems , 1998 .

[17]  Enrico Gobbetti,et al.  Encyclopedia of Electrical and Electronics Engineering , 1999 .

[18]  Stephen J. Wright.  Primal-Dual Interior-Point Methods , 1997, Other Titles in Applied Mathematics.

[19]  I. Singer.  Some further duality theorems for optimization problems with reverse convex constraint sets , 1992 .

[20]  J. Sobieszczanski-Sobieski,et al.  Nonlinear Rescaling in Discrete Minimax , 2001 .

[21]  Jean-Paul Penot,et al.  On Quasi-Convex Duality , 1990, Math. Oper. Res..

[22]  P. Panagiotopoulos.  Inequality problems in mechanics and applications , 1985 .

[23]  Frank H. Clarke.  The dual action, optimal control, and generalized gradients , 1985 .

[24]  V. F. Demʹi︠a︡nov.  Quasidifferentiability and nonsmooth modelling in mechanics, engineering, and economics , 1996 .

[25]  David Yang Gao,et al.  Bi-Complementarity and Duality: A Framework in Nonlinear Equilibria with Applications to the Contact Problem of Elastoplastic Beam Theory☆☆☆ , 1998 .

[26]  R. Rockafellar.  Conjugate Duality and Optimization , 1987 .

[27]  Juan Enrique Martínez-Legaz,et al.  On φ-convexity of convex functions , 1998 .

[28]  I. Ekeland,et al.  Convex analysis and variational problems , 1976 .

[29]  Gao Yang,et al.  On the complementary bounding theorems for limit analysis , 1988 .

[30]  G. Stavroulakis,et al.  Nonconvex Optimization in Mechanics: Algorithms, Heuristics and Engineering Applications , 1997 .

[31]  Ivan Singer.  On dualities between function spaces , 1996, Math. Methods Oper. Res..

[32]  Ivar Ekeland,et al.  Legendre Duality in Nonconvex Optimization and Calculus of Variations , 1977 .

[33]  Giles Auchmuty,et al.  Duality algorithms for nonconvex variational principles , 1989 .

[34]  J. Toland.  Duality in nonconvex optimization , 1978 .

[35]  Giles Auchmuty.  Min-max problems for non-potential operator equations , 1996 .

[36]  Hoang Tuy,et al.  Polyhedral annexaton, dualization and dimension reduction technique in global optimization , 1991, J. Glob. Optim..

[37]  Manfred Walk.  Theory of Duality in Mathematical Programming , 1989 .

[38]  Phan Thien Thach.  Global optimality criterion and a duality with a zero gap in nonconvex optimization , 1993 .

[39]  D. Gao.  Duality Principles in Nonconvex Systems: Theory, Methods and Applications , 2000 .

[40]  J. Hiriart-Urruty.  Generalized Differentiability / Duality and Optimization for Problems Dealing with Differences of Convex Functions , 1985 .

[41]  M. J. Sewell,et al.  Maximum and minimum principles , 1989, The Mathematical Gazette.

[42]  J. Moreau,et al.  Topics in Nonsmooth Mechanics , 1988 .

[43]  Ivan Singer.  Duality for optimization and best approximation over finite intersections , 1998 .

[44]  I. Ekeland.  Convexity Methods In Hamiltonian Mechanics , 1990 .

[45]  H. Weinert.  Ekeland, I. / Temam, R., Convex Analysis and Variational Problems. Amsterdam‐Oxford. North‐Holland Publ. Company. 1976. IX, 402 S., Dfl. 85.00. US $ 29.50 (SMAA 1) , 1979 .

[46]  J. Moreau,et al.  La notion de sur-potentiel et les liaisons unilatérales en élastostatique , 1968 .

[47]  Juan Enrique Martínez-Legaz,et al.  Dualities Associated To Binary Operations On R , 1995 .

[48]  J. Ericksen,et al.  Equilibrium of bars , 1975 .

[49]  D. Gao.  Dual Extremum Principles in Finite Deformation Theory With Applications to Post-Buckling Analysis of Extended Nonlinear Beam Model , 1997 .

[50]  Giles Auchmuty.  Duality for non-convex variational principles , 1983 .

[51]  Raffaele Casciaro,et al.  A mixed formulation and mixed finite elements for limit analysis , 1982 .

[52]  Dumitru Motreanu,et al.  Minimax Theorems and Qualitative Properties of the Solutions of Hemivariational Inequalities , 1998 .