TY - JOUR AU - Anorld, V. I. PY - 1998 DA - 1998// TI - On teaching mathematics JO - Russ. Math. Surv. VL - 53 UR - https://doi.org/10.1070/RM1998v053n01ABEH000005 DO - 10.1070/RM1998v053n01ABEH000005 ID - Anorld1998 ER - TY - CHAP AU - Ali, E. AU - Gao, D. Y. ED - Gao, D. Y. ED - Latorre, V. ED - Ruan, N. PY - 2017 DA - 2017// TI - Improved canonical dual finite element method and algorithm for post-buckling analysis of nonlinear gao beam BT - Canonical Duality Theory: Unified Methodology for Multidisciplinary Study PB - Springer CY - New York UR - https://doi.org/10.1007/978-3-319-58017-3_14 DO - 10.1007/978-3-319-58017-3_14 ID - Ali2017 ER - TY - JOUR AU - Bierlaire, M. AU - Crittin, F. PY - 2006 DA - 2006// TI - Solving noisy, large-scale fixed-point problems and systems of nonlinear equations JO - Transp. Sci. VL - 40 UR - https://doi.org/10.1287/trsc.1050.0119 DO - 10.1287/trsc.1050.0119 ID - Bierlaire2006 ER - TY - BOOK AU - Border, K. C. PY - 1985 DA - 1985// TI - Fixed Point Theorems with Applications to Economics and Game Theory PB - Cambridge University Press CY - New York UR - https://doi.org/10.1017/CBO9780511625756 DO - 10.1017/CBO9780511625756 ID - Border1985 ER - TY - JOUR AU - Chen, Y. AU - Gao, D. Y. PY - 2016 DA - 2016// TI - Global solutions to nonconvex optimization of 4th-order polynomial and log-sum-exp functions JO - J. Glob. Optim. VL - 64 UR - https://doi.org/10.1007/s10898-014-0244-5 DO - 10.1007/s10898-014-0244-5 ID - Chen2016 ER - TY - BOOK AU - Ciarlet, P. G. PY - 2013 DA - 2013// TI - Linear and Nonlinear Functional Analysis with Applications PB - SIAM CY - Philadelphia ID - Ciarlet2013 ER - TY - JOUR AU - Eaves, B. C. PY - 1972 DA - 1972// TI - Homotopies for computation of fixed points JO - Math. Program. VL - 3 UR - https://doi.org/10.1007/BF01584975 DO - 10.1007/BF01584975 ID - Eaves1972 ER - TY - JOUR AU - Gao, D. Y. PY - 1996 DA - 1996// TI - Nonlinear elastic beam theory with applications in contact problem and variational approaches JO - Mech. Res. Commun. VL - 23 UR - https://doi.org/10.1016/0093-6413(95)00071-2 DO - 10.1016/0093-6413(95)00071-2 ID - Gao1996 ER - TY - JOUR AU - Gao, D. Y. PY - 1999 DA - 1999// TI - General analytic solutions and complementary variational principles for large deformation nonsmooth mechanics JO - Meccanica VL - 34 ID - Gao1999 ER - TY - JOUR AU - Gao, D. Y. PY - 1999 DA - 1999// TI - Pure complementary energy principle and triality theory in finite elasticity JO - Mech. Res. Commun. VL - 26 UR - https://doi.org/10.1016/S0093-6413(98)00096-2 DO - 10.1016/S0093-6413(98)00096-2 ID - Gao1999 ER - TY - BOOK AU - Gao, D. Y. PY - 2000 DA - 2000// TI - Duality Principles in Nonconvex Systems: Theory, Methods and Applications PB - Springer CY - New York UR - https://doi.org/10.1007/978-1-4757-3176-7 DO - 10.1007/978-1-4757-3176-7 ID - Gao2000 ER - TY - JOUR AU - Gao, D. Y. PY - 2000 DA - 2000// TI - Analytic solution and triality theory for nonconvex and nonsmooth variational problems with applications JO - Nonlinear Anal. VL - 42 UR - https://doi.org/10.1016/S0362-546X(99)00129-7 DO - 10.1016/S0362-546X(99)00129-7 ID - Gao2000 ER - TY - JOUR AU - Gao, D. Y. PY - 2000 DA - 2000// TI - Finite deformation beam models and triality theory in dynamical post-buckling analysis JO - Int. J. Non-Linear Mech. VL - 5 UR - https://doi.org/10.1016/S0020-7462(98)00091-2 DO - 10.1016/S0020-7462(98)00091-2 ID - Gao2000 ER - TY - JOUR AU - Gao, D. Y. PY - 2000 DA - 2000// TI - Canonical dual transformation method and generalized triality theory in nonsmooth global optimization JO - J. Glob. Optim. VL - 17 UR - https://doi.org/10.1023/A:1026537630859 DO - 10.1023/A:1026537630859 ID - Gao2000 ER - TY - JOUR AU - Gao, D. Y. PY - 2005 DA - 2005// TI - Sufficient conditions and perfect duality in nonconvex minimization with inequality constraints JO - J. Ind. Manag. Optim. VL - 1 ID - Gao2005 ER - TY - JOUR AU - Gao, D. Y. PY - 2006 DA - 2006// TI - Complete solutions and extremality criteria to polynomial optimization problems JO - J. Glob. Optim. VL - 35 UR - https://doi.org/10.1007/s10898-005-3068-5 DO - 10.1007/s10898-005-3068-5 ID - Gao2006 ER - TY - JOUR AU - Gao, D. Y. PY - 2007 DA - 2007// TI - Solutions and optimality to box constrained nonconvex minimization problems JO - J. Ind. Manag. Optim. VL - 3 UR - https://doi.org/10.3934/jimo.2007.3.293 DO - 10.3934/jimo.2007.3.293 ID - Gao2007 ER - TY - CHAP AU - Gao, D. Y. ED - Sergeyev, Y. D. ED - Kvasov, D. E. ED - Mukhametzhanov, M. S. PY - 2016 DA - 2016// TI - On unified modeling, theory, and method for solving multi-scale global optimization problems BT - Proceedings of the 2nd International Conference Numerical Computations: Theory and Algorithms UR - https://doi.org/10.1063/1.4965311 DO - 10.1063/1.4965311 ID - Gao2016 ER - TY - STD TI - Gao, D.Y.: On unified modeling, canonical duality-triality theory, challenges and breakthrough in optimization. https://arxiv.org/abs/1605.05534 (2016) UR - https://arxiv.org/abs/1605.05534 ID - ref19 ER - TY - CHAP AU - Gao, D. Y. ED - Gao, D. Y. ED - Latorre, V. ED - Ruan, N. PY - 2017 DA - 2017// TI - Analytic solutions to large deformation problems governed by generalized neo-Hookean model BT - Canonical Duality Theory: Unified Methodology for Multidisciplinary Study PB - Spinger CY - New York UR - https://doi.org/10.1007/978-3-319-58017-3_2 DO - 10.1007/978-3-319-58017-3_2 ID - Gao2017 ER - TY - JOUR AU - Gao, D. Y. PY - 2018 DA - 2018// TI - On topology optimization and canonical duality method JO - Comput. Methods Appl. Mech. Eng. VL - 341 UR - https://doi.org/10.1016/j.cma.2018.06.027 DO - 10.1016/j.cma.2018.06.027 ID - Gao2018 ER - TY - CHAP AU - Gao, D. Y. ED - Singh, V. K. ED - Gao, D. Y. ED - Fisher, A. PY - 2018 DA - 2018// TI - Canonical duality-triality: Unified understanding modeling, problems, and NP-hardness in multi-scale optimization BT - Emerging Trends in Applied Mathematics and Hi-Perfermance Computingy PB - Springer CY - New York ID - Gao2018 ER - TY - CHAP AU - Gao, D. Y. AU - Hajilarov, E. ED - Gao, D. Y. ED - Latorre, V. ED - Ruan, N. PY - 2017 DA - 2017// TI - Analytic solutions to 3-d finite deformation problems governed by St. Venant–Kirchhoff material BT - Canonical Duality Theory: Unified Methodology for Multidisciplinary Study PB - Spinger CY - New York UR - https://doi.org/10.1007/978-3-319-58017-3_3 DO - 10.1007/978-3-319-58017-3_3 ID - Gao2017 ER - TY - BOOK AU - Gao, D. Y. AU - Latorre, V. AU - Ruan, N. PY - 2017 DA - 2017// TI - Canonical Duality Theory: Unified Methodology for Multidisciplinary Study PB - Springer CY - New York UR - https://doi.org/10.1007/978-3-319-58017-3 DO - 10.1007/978-3-319-58017-3 ID - Gao2017 ER - TY - JOUR AU - Gao, D. Y. AU - Ogden, R. W. PY - 2008 DA - 2008// TI - Multi-solutions to non-convex variational problems with implications for phase transitions and numerical computation JO - Q. J. Mech. Appl. Math. VL - 61 UR - https://doi.org/10.1093/qjmam/hbn014 DO - 10.1093/qjmam/hbn014 ID - Gao2008 ER - TY - JOUR AU - Gao, D. Y. AU - Ruan, N. PY - 2010 DA - 2010// TI - Solutions to quadratic minimization problems with box and integer constraints JO - J. Glob. Optim. VL - 47 UR - https://doi.org/10.1007/s10898-009-9469-0 DO - 10.1007/s10898-009-9469-0 ID - Gao2010 ER - TY - CHAP AU - Gao, D. Y. AU - Ruan, N. AU - Latorre, V. ED - Gao, D. Y. ED - Latorre, V. ED - Ruan, N. PY - 2017 DA - 2017// TI - Canonical duality-triality theory: bridge between nonconvex analysis/mechanics and global optimization in complex systems BT - Canonical Duality Theory: Unified Methodology for Multidisciplinary Study PB - Springer CY - New York UR - https://doi.org/10.1007/978-3-319-58017-3 DO - 10.1007/978-3-319-58017-3 ID - Gao2017 ER - TY - JOUR AU - Gao, D. Y. AU - Ruan, N. AU - Sherali, H. PY - 2009 DA - 2009// TI - Solutions and optimality criteria for nonconvex constrained global optimization problems with connections between canonical and Lagrangian duality JO - J. Glob. Optim. VL - 45 UR - https://doi.org/10.1007/s10898-009-9399-x DO - 10.1007/s10898-009-9399-x ID - Gao2009 ER - TY - CHAP AU - Gao, D. Y. AU - Ruan, N. AU - Sherali, H. D. ED - Chinchuluun, A. ED - Pardalos, P. ED - Enkhbat, R. ED - Tseveendorj, I. PY - 2010 DA - 2010// TI - Canonical duality solutions for fixed cost quadratic program BT - Optimization and Optimal Control. Springer Optimization and Its Applications PB - Springer CY - New York UR - https://doi.org/10.1007/978-0-387-89496-6_7 DO - 10.1007/978-0-387-89496-6_7 ID - Gao2010 ER - TY - JOUR AU - Gao, D. Y. AU - Strang, G. PY - 1989 DA - 1989// TI - Geometric nonlinearity: potential energy, complementary energy, and the gap function JO - Q. J. Mech. Appl. Math. VL - XLVII UR - https://doi.org/10.1090/qam/1012271 DO - 10.1090/qam/1012271 ID - Gao1989 ER - TY - CHAP AU - Gao, D. Y. AU - Wu, C. ED - Gao, D. Y. ED - Latorre, V. ED - Ruan, N. PY - 2017 DA - 2017// TI - Triality theory for general unconstrained global optimization problems BT - Canonical Duality Theory: Unified Methodology for Multidisciplinary Study PB - Springer CY - New York UR - https://doi.org/10.1007/978-3-319-58017-3_6 DO - 10.1007/978-3-319-58017-3_6 ID - Gao2017 ER - TY - JOUR AU - Gao, D. Y. AU - Yu, H. F. PY - 2008 DA - 2008// TI - Multi-scale modelling and canonical dual finite element method in phase transitions of solids JO - Int. J. Solids Struct. VL - 45 UR - https://doi.org/10.1016/j.ijsolstr.2007.08.027 DO - 10.1016/j.ijsolstr.2007.08.027 ID - Gao2008 ER - TY - JOUR AU - Hirsch, M. D. AU - Papadimitriou, C. AU - Vavasis, S. PY - 1989 DA - 1989// TI - Exponential lower bounds for finding Brouwer fixed points JO - J. Complex. VL - 5 UR - https://doi.org/10.1016/0885-064X(89)90017-4 DO - 10.1016/0885-064X(89)90017-4 ID - Hirsch1989 ER - TY - JOUR AU - Huang, Z. AU - Khachiyan, L. AU - Sikorski, K. PY - 1999 DA - 1999// TI - Approximating fixed points of weakly contracting mappings JO - J. Complex. VL - 15 UR - https://doi.org/10.1006/jcom.1999.0504 DO - 10.1006/jcom.1999.0504 ID - Huang1999 ER - TY - JOUR AU - Jin, Z. AU - Gao, D. Y. PY - 2017 DA - 2017// TI - On modeling and global solutions for d.c. optimization problems by canonical duality theory JO - Appl. Math. Comput. VL - 296 UR - https://doi.org/10.1016/j.amc.2016.10.010 DO - 10.1016/j.amc.2016.10.010 ID - Jin2017 ER - TY - JOUR AU - Latorre, V. AU - Gao, D. Y. PY - 2016 DA - 2016// TI - Canonical duality for solving general nonconvex constrained problems JO - Optim. Lett. VL - 10 UR - https://doi.org/10.1007/s11590-015-0860-0 DO - 10.1007/s11590-015-0860-0 ID - Latorre2016 ER - TY - JOUR AU - Latorre, V. AU - Gao, D. Y. PY - 2016 DA - 2016// TI - Global optimal trajectory in chaos and NP-hardness JO - Int. J. Bifurc. Chaos VL - 26 UR - https://doi.org/10.1142/S021812741650142X DO - 10.1142/S021812741650142X ID - Latorre2016 ER - TY - CHAP AU - Liu, G. S. AU - Gao, D. Y. AU - Wang, S. Y. ED - Gao, D. Y. ED - Latorre, V. ED - Ruan, N. PY - 2017 DA - 2017// TI - Canonical duality theory for solving non-monotone variational inequality problems BT - Canonical Duality Theory: Unified Methodology for Multidisciplinary Study PB - Springer CY - New York UR - https://doi.org/10.1007/978-3-319-58017-3_7 DO - 10.1007/978-3-319-58017-3_7 ID - Liu2017 ER - TY - CHAP AU - Lu, X. J. AU - Gao, D. Y. ED - Gao, D. Y. ED - Latorre, V. ED - Ruan, N. PY - 2017 DA - 2017// TI - Canonical duality method for solving Kantorovich mass transfer problem BT - Canonical Duality Theory: Unified Methodology for Multidisciplinary Study PB - Spinger CY - New York UR - https://doi.org/10.1007/978-3-319-58017-3_5 DO - 10.1007/978-3-319-58017-3_5 ID - Lu2017 ER - TY - JOUR AU - Machalova, J. AU - Netuka, H. PY - 2017 DA - 2017// TI - Control variational method approach to bending and contact problems for Gao beam JO - Appl. Math. VL - 62 UR - https://doi.org/10.21136/AM.2017.0168-17 DO - 10.21136/AM.2017.0168-17 ID - Machalova2017 ER - TY - BOOK AU - Marsden, J. E. AU - Hughes, T. J. R. PY - 1983 DA - 1983// TI - Mathematical Foundations of Elasticity PB - Prentice-Hall CY - New York ID - Marsden1983 ER - TY - JOUR AU - Ruan, N. AU - Gao, D. Y. PY - 2014 DA - 2014// TI - Canonical duality approach for nonlinear dynamical systems JO - IMA J. Appl. Math. VL - 79 UR - https://doi.org/10.1093/imamat/hxs067 DO - 10.1093/imamat/hxs067 ID - Ruan2014 ER - TY - JOUR AU - Scarf, H. PY - 1967 DA - 1967// TI - The approximation of fixed point of a continuous mapping JO - SIAM J. Appl. Math. VL - 35 UR - https://doi.org/10.1137/0115116 DO - 10.1137/0115116 ID - Scarf1967 ER - TY - BOOK AU - Scarf, H. E. AU - Hansen, T. PY - 1973 DA - 1973// TI - Computation of Economic Equilibria PB - Yale University Press CY - New Haven ID - Scarf1973 ER - TY - JOUR AU - Shellman, S. AU - Sikorski, K. PY - 2002 DA - 2002// TI - A two-dimensional bisection envelope algorithm for fixed points JO - J. Complex. VL - 2 UR - https://doi.org/10.1006/jcom.2001.0625 DO - 10.1006/jcom.2001.0625 ID - Shellman2002 ER - TY - JOUR AU - Shellman, S. AU - Sikorski, K. PY - 2003 DA - 2003// TI - A recursive algorithm for the infinity-norm fixed point problem JO - J. Complex. VL - 19 UR - https://doi.org/10.1016/j.jco.2003.06.001 DO - 10.1016/j.jco.2003.06.001 ID - Shellman2003 ER - TY - BOOK AU - Smart, D. R. PY - 1980 DA - 1980// TI - Fixed Point Theorems PB - Cambridge University Press CY - Cambridge ID - Smart1980 ER - TY - JOUR AU - Voisei, M. D. AU - Zalinescu, C. PY - 2011 DA - 2011// TI - Some remarks concerning Gao–Strang’s complementary gap function JO - Appl. Anal. VL - 90 UR - https://doi.org/10.1080/00036811.2010.483427 DO - 10.1080/00036811.2010.483427 ID - Voisei2011 ER - TY - JOUR AU - Wang, Z. B. AU - Fang, S. -. C. AU - Gao, D. Y. AU - Xing, W. X. PY - 2008 DA - 2008// TI - Global extremal conditions for multi-integer quadratic programming JO - J. Ind. Manag. Optim. VL - 4 UR - https://doi.org/10.3934/jimo.2008.4.213 DO - 10.3934/jimo.2008.4.213 ID - Wang2008 ER - TY - BOOK AU - Yang, Z. PY - 1999 DA - 1999// TI - Computing Equilibria and Fixed Points: The Solution of Nonlinear Inequalities PB - Kluwer Academic Publishers CY - Dordrecht UR - https://doi.org/10.1007/978-1-4757-4839-0 DO - 10.1007/978-1-4757-4839-0 ID - Yang1999 ER -