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Solutions and optimality criteria to box constrained nonconvex minimization problems
 J. Industrial and Management Optimization
"... (Communicated by K.L. Teo) Abstract. This paper presents a canonical duality theory for solving nonconvex polynomial programming problems subjected to box constraints. It is proved that under certain conditions, the constrained nonconvex problems can be converted to the socalled canonical (perfect) ..."
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(Communicated by K.L. Teo) Abstract. This paper presents a canonical duality theory for solving nonconvex polynomial programming problems subjected to box constraints. It is proved that under certain conditions, the constrained nonconvex problems can be converted to the socalled canonical (perfect) dual problems, which can be solved by deterministic methods. Both global and local extrema of the primal problems can be identified by a triality theory proposed by the author. Applications to nonconvex integer programming and Boolean least squares problems are discussed. Examples are illustrated. A conjecture on NPhard problems is proposed. 1. Primal problem and its dual form. The box constrained nonconvex minimization problem is proposed as a primal problem (P) given below: (P) : min {P (x) = Q(x) + W (x)} (1) x∈Xa where Xa = {x ∈ R n  ℓ l ≤ x ≤ ℓ u} is a feasible space, Q(x) = 1 2 xT Ax − c T x is a quadratic function, A = A T ∈ R n×n is a given symmetric matrix, ℓ l, ℓ u, and c are three given vectors in R n, W (x) is a nonconvex function. In this paper, we simply assume that W (x) is a socalled doublewell fourth order polynomial function defined by W (x) = 1 1 2 2 Bx2 �2 − α, (2) where B ∈ Rm×n is a given matrix and α> 0 is a given parameter. The notation x  used in this paper denotes the Euclidean norm of x. Problems of the form (1) appear frequently in many applications, such as semilinear nonconvex partial differential equations [15], structural limit analysis, discretized optimal control problems with distributed parameters, information theory, and network communication. Particularly, if W (x) = 0, the problem (P) is directly related to certain successive quadratic programming methods ([9, 10, 18]).
On the triality theory for a quartic polynomial optimization problem
 J. IND. MANAG. OPTIM
, 2012
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Solutions and optimality criteria for nonconvex constrained global optimization problems with connections between canonical and Lagrangian duality
 J GLOB OPTIM
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Global optimal solutions to a general sensor network localization problem
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ADVANCES IN CANONICAL DUALITY THEORY WITH APPLICATIONS TO GLOBAL OPTIMIZATION
 FOCAPO 2008
, 2008
"... Canonical duality theory is a potentially powerful methodology, which can be used to solve a wide class of discrete and continuous global optimization problems. This paper presents a brief review and recent developments of this theory with applications to some wellknow problems including polynomial ..."
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Cited by 3 (1 self)
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Canonical duality theory is a potentially powerful methodology, which can be used to solve a wide class of discrete and continuous global optimization problems. This paper presents a brief review and recent developments of this theory with applications to some wellknow problems including polynomial minimization, mixed integer and fractional programming, nonconvex minimization with nonconvex quadratic constraints, etc. Results shown that under certain conditions, these difficult problems can be solved by deterministic methods within polynomial times, and the NPhard problems can be transformed to a minimal stationary problem in dual space. Concluding remarks and open problems are presented in the end.
Double Well Potential Function and Its Optimization in The ndimensional Real Space – Part I, submitted working paper
, 2012
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Analytical solutions to general antiplane shear problems in finite elasticity
, 2014
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Introduction to Canonical Duality Theory
, 2009
"... Canonical Duality Theory is a versatile and potentially powerful methodology which is composed mainly of a canonical dual transformation, a complementarydual principle, and an associated triality theory. The canonical dual transformation can be used for modeling complex systems and to formulate ..."
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Canonical Duality Theory is a versatile and potentially powerful methodology which is composed mainly of a canonical dual transformation, a complementarydual principle, and an associated triality theory. The canonical dual transformation can be used for modeling complex systems and to formulate perfect dual problems without duality gap; the complementarydual principle presents a unified analytic solution form for general problems in continuous and discrete systems; the triality theory is comprised by a saddle minmax duality and two pairs of doublemin, doublemax dualities. This theory reveals an intrinsic duality pattern in complex phenomena and can be used to solve a very large class of challenging problems in complex systems. This lecture presents, within a unified framework, a selfcontained comprehensive introduction and some new developments on canonical duality theory for complex systems with emphasis on methods and applications in nonlinear analysis and optimization. Intrinsic relations among the popular semipositive programming, semiinfinite programming, complementarity theory, variational inequality, penalty methods, and the Lagrangian duality theory are revealed within the unified framework of the canonical duality theory. Applications are illustrated by a class of challenging (NPhard) problems in global optimization and nonconvex analysis. It is shown that by the use of the canonical dual transformation, nonconvex constrained primal problems can be converted into certain simple canonical dual problems, which can be solved to obtain all extremal points, and NPhard discrete optimization problems can be transformed to certain minimal stationary problems in continuous space. Optimality conditions (both local and global) for these extrema can be identified by the triality theory. This lecture brings some fundamentally new insights into nonconvex analysis, global optimization, and computational science.
DOI: 10.1177/1081286513482483
, 2013
"... Postbuckling solutions of hyperelastic beam by canonical dual finite element method ..."
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Postbuckling solutions of hyperelastic beam by canonical dual finite element method