In this paper, we combine the new global optimization method proposed by Jiao [H. Jiao, A branch and bound algorithm for globally solving a class of nonconvex programming problems, Nonlinear Anal. 70 (2) (2008) 1113-1...
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In this paper, we combine the new global optimization method proposed by Jiao [H. Jiao, A branch and bound algorithm for globally solving a class of nonconvex programming problems, Nonlinear Anal. 70 (2) (2008) 1113-1123] with a suitable deleting technique to propose a new accelerating global optimization algorithm for solving a class of nonconvex programming problems (NP). This technique offers a possibility to cut away a large part of the currently investigated region in which the global optimal solution of NP does not exist, and can be seen as an accelerating device for the global optimization algorithm of the nonconvex programming problems. Compared with the method in the above cited reference, numerical results show that the computational efficiency is obviously improved by using this new technique in the number of iterations, the required list length and the overall execution time of the algorithm. (C) 2009 Elsevier Ltd. All rights reserved.
A branch and bound algorithm is proposed for globally solving a class of nonconvex programming problems (NP). For minimizing the problem, linear lower bounding functions (LLBFs) of objective function and constraint fu...
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A branch and bound algorithm is proposed for globally solving a class of nonconvex programming problems (NP). For minimizing the problem, linear lower bounding functions (LLBFs) of objective function and constraint functions are constructed, then a relaxation linear programming is obtained which is solved by the simplex method and which provides the lower bound of the optimal value. The proposed algorithm is convergent to the global minimum through the successive refinement of linear relaxation of the feasible region and the solutions of a series of linear programming problems. And finally the numerical experiment is reported to show the feasibility and effectiveness of the proposed algorithm. (C) 2008 Elsevier Ltd. All rights reserved.
Efficient techniques for discrete event simulation and sensitivity analysis of production systems are used to accelerate the solution of nonconvex optimization problems which are quite often encountered in practice. T...
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Efficient techniques for discrete event simulation and sensitivity analysis of production systems are used to accelerate the solution of nonconvex optimization problems which are quite often encountered in practice. The design algorithm is of the branch and bound type and solves a sequence of convex problems resulting from successive partitions of the feasible set. Examples are given, and computational considerations are discussed.
A combined homotopy between the Karush-Kuhn-Tucker (K-K-T) system on a nonconvex programming problem and a trivial system is constructed. Under the so-called normal cone condition of the constraints, the pathway from ...
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A combined homotopy between the Karush-Kuhn-Tucker (K-K-T) system on a nonconvex programming problem and a trivial system is constructed. Under the so-called normal cone condition of the constraints, the pathway from almost any interior point of the feasible set of K-K-T point of the problem is proved to exist. This smooth pathway can be traced by some numerical continuation algorithm. A theoretical foundation of interior point methods for nonconvex programming is presented.
In this paper, we consider nonconvex differentiable programming under linear and nonlinear differentiable constraints. A reduced gradient and GRG (generalized reduced gradient) descent methods involving stochastic per...
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In this paper, we consider nonconvex differentiable programming under linear and nonlinear differentiable constraints. A reduced gradient and GRG (generalized reduced gradient) descent methods involving stochastic perturbation are proposed and we give a mathematical result establishing the convergence to a global minimizer. Numerical examples are given in order to show that the method is effective to calculate. Namely, we consider classical tests such as the statistical problem, the octagon problem, the mixture problem and an application to the linear optimal control servomotor problem. (C) 2013 Elsevier Inc. All rights reserved.
This paper presents a robust canonical duality-triality theory for solving nonconvex programming problems under data uncertainty. This theory includes a robust canonical saddle-point theorem and robust canonical optim...
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This paper presents a robust canonical duality-triality theory for solving nonconvex programming problems under data uncertainty. This theory includes a robust canonical saddle-point theorem and robust canonical optimality conditions, which can be used to identify both robust global and local extrema of the primal problem. Two numerical examples are presented to illustrate that the robust Triality theory is particularly powerful for solving nonconvex optimization problems with data uncertainty.
In the papers [G.C. Feng, B. Yu, Combined homotopy interior point method for nonlinear programming problems, in: H. Fujita, M. Yamaguti (Eds.), Advances in Numerical Mathematics;Proceedings of the Second Japan-China S...
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In the papers [G.C. Feng, B. Yu, Combined homotopy interior point method for nonlinear programming problems, in: H. Fujita, M. Yamaguti (Eds.), Advances in Numerical Mathematics;Proceedings of the Second Japan-China Seminar on Numerical Mathematics, in: Lecture Notes in Numerical and Applied Analysis, vol. 14, Kinokuniya, Tokyo, 1995, pp. 9-16;G.C. Feng, Z.H. Lin, B. Yu, Existence of an interior pathway to a Karush-Kuhn-Tucker point of a nonconvex programming problem, Nonlinear Analysis 32 (1998) 761-768;Z.H. Lin, B. Yu, G.C. Feng, A combined homotopy interior point method for convex programming problem, Applied Mathematics and Computation 84(1997) 193-211], a combined homotopy interior method was presented and global convergence results obtained for nonconvex nonlinear programming when the feasible set is bounded and satisfies the so called normal cone condition. However, for when the feasible set is not bounded, no result has so far been obtained. In this paper, a combined homotopy interior method for nonconvex programming problems oil the unbounded feasible set is considered. Under suitable additional assumptions, boundedness of the homotopy path, and hence global convergence, is proven. (C) 2008 Elsevier Ltd. All rights reserved.
This paper presents a linear decomposition approach for a class of nonconvex programming problems by dividing the input space into polynomially many grids. It shows that under certain assumptions the original problem ...
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This paper presents a linear decomposition approach for a class of nonconvex programming problems by dividing the input space into polynomially many grids. It shows that under certain assumptions the original problem can be transformed and decomposed into a polynomial number of equivalent linear programming subproblems. Based on solving a series of liner programming subproblems corresponding to those grid points we can obtain the near-optimal solution of the original problem. Compared to existing results in the literature, the proposed algorithm does not require the assumptions of quasi-concavity and differentiability of the objective function, and it differs significantly giving an interesting approach to solving the problem with a reduced running time.
In this paper, a combined interior point homotopy method is used to solve constrained nonconvex programming problems. Some results for the homotopy pathway are obtained. It is known that a K-K-T point can be obtained ...
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In this paper, a combined interior point homotopy method is used to solve constrained nonconvex programming problems. Some results for the homotopy pathway are obtained. It is known that a K-K-T point can be obtained from this homotopy method, and it proves that, when the homotopy map is a regular map, the K-K-T point must be a local minimum point on choosing a proper homotopy equation. (C) 2009 Elsevier Ltd. All rights reserved.
In this paper, we consider a general family of nonconvex programming problems. All of the objective functions of the problems in this family are identical, but their feasibility regions depend upon a parameter ϑ. This...
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In this paper, we consider a general family of nonconvex programming problems. All of the objective functions of the problems in this family are identical, but their feasibility regions depend upon a parameter ϑ. This family of problems is called a parametric nonconvex program (PNP). Solving (PNP) means finding an optimal solution for every program in the family. A prototype branch-and-bound algorithm is presented for solving (PNP). By modifying a prototype algorithm for solving a single nonconvex program, this algorithm solves (PNP) in one branch-and-bound search. To implement the algorithm, certain compact partitions and underestimating functions must be formed in an appropriate manner. We present an algorithm for solving a particular (PNP) which implements the prototype algorithm by forming compact partitions and underestimating functions based upon rules given by Falk and Soland. The programs in this (PNP) have the same concave objective function, but their feasibility regions are described by linear constraints with differing right-hand sides. Computational experience with this algorithm is reported for various problems.
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