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12
Global minimization using an Augmented Lagrangian method with variable lowerlevel constraints
, 2007
"... A novel global optimization method based on an Augmented Lagrangian framework is introduced for continuous constrained nonlinear optimization problems. At each outer iteration k the method requires the εkglobal minimization of the Augmented Lagrangian with simple constraints, where εk → ε. Global c ..."
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Cited by 39 (1 self)
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A novel global optimization method based on an Augmented Lagrangian framework is introduced for continuous constrained nonlinear optimization problems. At each outer iteration k the method requires the εkglobal minimization of the Augmented Lagrangian with simple constraints, where εk → ε. Global convergence to an εglobal minimizer of the original problem is proved. The subproblems are solved using the αBB method. Numerical experiments are presented.
Canonical dual approach for solving 01 quadratic programming problems
 J. Industrial and Management Optimization
, 2007
"... Abstract. By using the canonical dual transformation developed recently, we derive a pair of canonical dual problems for 01 quadratic programming problems in both minimization and maximization form. Regardless convexity, when the canonical duals are solvable, no duality gap exists between the prima ..."
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Cited by 22 (12 self)
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Abstract. By using the canonical dual transformation developed recently, we derive a pair of canonical dual problems for 01 quadratic programming problems in both minimization and maximization form. Regardless convexity, when the canonical duals are solvable, no duality gap exists between the primal and corresponding dual problems. Both global and local optimality conditions are given. An algorithm is presented for finding global minimizers, even when the primal objective function is not convex. Examples are included to illustrate this new approach.
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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Cited by 12 (5 self)
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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]).
Sufficient conditions and perfect duality in nonconvex minimization with inequality constraints
 J. Industrial and Management Optimization
"... Abstract. This paper presents a duality theory for solving concave minimization problem and nonconvex quadratic programming problem subjected to nonlinear inequality constraints. By use of the canonical dual transformation developed recently, two canonical dual problems are formulated, respectively. ..."
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Cited by 9 (6 self)
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Abstract. This paper presents a duality theory for solving concave minimization problem and nonconvex quadratic programming problem subjected to nonlinear inequality constraints. By use of the canonical dual transformation developed recently, two canonical dual problems are formulated, respectively. These two dual problems are perfectly dual to the primal problems with zero duality gap. It is proved that the sufficient conditions for global minimizers and local extrema (both minima and maxima) are controlled by the triality theory discovered recently [5]. This triality theory can be used to develop certain useful primaldual methods for solving difficult nonconvex minimization problems. Results shown that the difficult quadratic minimization problem with quadratic constraint can be converted into a onedimensional dual problem, which can be solved completely to obtain all KKT points and global minimizer. 1. Concave Minimization Problem and Parametrization. The concave minimization problem to be discussed in this paper is denoted as the primal problem ((P) in short)
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.
Augmented Lagrangians with possible infeasibility and finite termination for global nonlinear programming
, 2012
"... In a recent paper, Birgin, Floudas and Martínez introduced an augmented Lagrangian method for global optimization. In their approach, augmented Lagrangian subproblems are solved using the αBB method and convergence to global minimizers was obtained assuming feasibility of the original problem. In th ..."
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Cited by 2 (0 self)
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In a recent paper, Birgin, Floudas and Martínez introduced an augmented Lagrangian method for global optimization. In their approach, augmented Lagrangian subproblems are solved using the αBB method and convergence to global minimizers was obtained assuming feasibility of the original problem. In the present research, the algorithm mentioned above will be improved in several crucial aspects. On the one hand, feasibility of the problem will not be required. Possible infeasibility will be detected in finite time by the new algorithms and optimal infeasibility results will be proved. On the other hand, finite termination results that guarantee optimality and/or feasibility up to any required precision will be provided. An adaptive modification in which subproblem tolerances depend on current feasibility and complementarity will also be given. The adaptive algorithm allows the augmented Lagrangian subproblems to be solved without requiring unnecessary potentially high precisions in the intermediate steps of the method, which improves the overall efficiency. Experiments showing how the new algorithms and results are related to practical computations will be given.
Global Nonlinear Programming with possible infeasibility and finite termination
, 2012
"... In a recent paper, Birgin, Floudas and Martínez introduced an augmented Lagrangian method for global optimization. In their approach, augmented Lagrangian subproblems are solved using the αBB method and convergence to global minimizers was obtained assuming feasibility of the original problem. In th ..."
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Cited by 1 (0 self)
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In a recent paper, Birgin, Floudas and Martínez introduced an augmented Lagrangian method for global optimization. In their approach, augmented Lagrangian subproblems are solved using the αBB method and convergence to global minimizers was obtained assuming feasibility of the original problem. In the present research, the algorithm mentioned above will be improved in several crucial aspects. On the one hand, feasibility of the problem will not be required. Possible infeasibility will be detected in finite time by the new algorithms and optimal infeasibility results will be proved. On the other hand, finite termination results thatguaranteeoptimalityand/orfeasibilityuptoanyrequiredprecisionwillbeprovided. An adaptive modification in which subproblem tolerances depend on current feasibility and complementarity will also be given. The adaptive algorithm allows the augmented Lagrangian subproblems to be solved without requiring unnecessary potentially high precisions in the intermediate steps of the method, which improves the overall efficiency. Experiments showing how the new algorithms and results are related to practical computations will be given.
Global solutions to nonconvex optimization of 4thorder polynomial and logsumexp functions
 J. Global Optimization
, 2014
"... Abstract This paper presents a canonical dual approach for solving a nonconvex global optimization problem governed by a sum of fourthorder polynomial and a logsumexp function. Such a problem arises extensively in engineering and sciences. Based on the canonical dualitytriality theory, this non ..."
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Abstract This paper presents a canonical dual approach for solving a nonconvex global optimization problem governed by a sum of fourthorder polynomial and a logsumexp function. Such a problem arises extensively in engineering and sciences. Based on the canonical dualitytriality theory, this nonconvex problem is transformed to an equivalent dual problem, which can be solved easily under certain conditions. We proved that both global minimizer and the biggest local extrema of the primal problem can be obtained analytically from the canonical dual solutions. As two special cases, a quartic polynomial minimization and a minimax problem are discussed. Existence conditions are derived, which can be used to classify easy and relative hard instances. Applications are illustrated by several nonconvex and nonsmooth examples. Keywords Global optimization · canonical duality theory · doublewell function · logsumexp function · polynomial minimisation · minimax problems Mathematics Subject Classification 90C26, 90C30, 90C46 1
KKT solution and conic relaxation for solving . . .
, 2011
"... To find a global optimal solution to the quadratically constrained quadratic programming problem, we explore the relationship between its Lagrangian multipliers and related linear conic programming problems. This study leads to a global optimality condition that is more general than the known posit ..."
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To find a global optimal solution to the quadratically constrained quadratic programming problem, we explore the relationship between its Lagrangian multipliers and related linear conic programming problems. This study leads to a global optimality condition that is more general than the known positive semidefiniteness condition in the literature. Moreover, we propose a computational scheme that provides clues of designing effective algorithms for more solvable quadratically constrained quadratic programming problems.
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.