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 εk-global minimization of the Augmented Lagrangian with simple constraints, where εk → ε. Global convergence to an ε-global minimizer of the original problem is proved. The subproblems are solved using the αBB method. Numerical experiments are presented.

This paper presents a brief review and some new developments on the canonical duality theory with applications to a class of variational problems in nonconvex mechanics and global optimization. These nonconvex problems are directly related to a large class of semi-linear partial differential equations in mathematical physics including phase transitions, post-buckling of large deformed beam model, chaotic dynamics, nonlinear field theory, and superconductivity. Numerical discretizations of these equations lead to a class of very difficult global minimization problems in finite dimensional space. It is shown that by the use of the canonical dual transformation, these nonconvex constrained primal problems can be converted into certain very simple canonical dual problems. The criticality condition leads to dual algebraic equations which can be solved completely. Therefore, a complete set of solutions to these very difficult primal problems can be obtained. The extremality of these solutions are controlled by the so-called triality theory. Several examples are illustrated including the nonconvex constrained quadratic programming. Results show that these very difficult primal problems can be converted into certain simple canonical (either convex or concave) dual problems, which can be solved completely. Also some very interesting new phenomena, i.e. trio-chaos and meta-chaos, are discovered in post-buckling of nonconvex systems. The author believes that these important phenomena exist in many nonconvex dynamical systems and deserve to have a detailed study.

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 primal-dual methods for solving difficult nonconvex minimization problems. Results shown that the difficult quadratic minimization problem with quadratic constraint can be converted into a one-dimensional 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)

The dual conservation laws of elasticity are systematically re-examined by using both Noether’s variational approach and Coleman–Noll–Gurtin’s thermodynamics approach. These dual conservation laws can be interpreted as the dual configurational force, and therefore they provide the dual energy–momentum tensor. Some previously unknown and yet interesting results in elasticity theory have been discovered. As an example, we note the following duality condition between the configuration force (energy–momentum tensor) P and the dual configuration force (dual energy–momentum tensor) L, P LðP: FÞ1 rðP xÞ: This and other results derived in this paper may lead to a better understanding of configurational mechanics and therefore of mechanics of defects.

(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 so-called 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 NP-hard 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 so-called double-well fourth order polynomial function defined by W (x) = 1 1 2 2 |Bx|2 �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]).

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 well-know 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 NP-hard problems can be transformed to a minimal stationary problem in dual space. Concluding remarks and open problems are presented in the end.

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Solutions and optimality criteria for nonconvex constrained global optimization problems with connections between canonical and Lagrangian duality

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 well-know 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 NP-hard discrete optimization problems can be transformed to certain minimal stationary problems in continuous space. Concluding remarks and open problems are presented in the end.

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. Key words: deterministic global optimization, augmented Lagrangians, nonlinear programming, algorithms, numerical experiments. 1