Canonical duality theory and algorithm for solving bilevel knapsack problems with applications
- Authors: Gao, David
- Date: 2021
- Type: Text , Journal article
- Relation: IEEE Transactions on Systems, Man, and Cybernetics: Systems Vol. 51, no. 2 (2021), p. 893-904
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- Description: A novel canonical duality theory (CDT) is presented for solving general bilevel mixed integer nonlinear optimization governed by linear and quadratic knapsack problems. It shows that the challenging knapsack problems can be solved analytically in term of their canonical dual solutions. The existence and uniqueness of these analytical solutions are proved. NP-hardness of the knapsack problems is discussed. A powerful CDT algorithm combined with an alternative iteration and a volume reduction method is proposed for solving the NP-hard bilevel knapsack problems. Application is illustrated by benchmark problems in optimal topology design. The performance and novelty of the proposed method are compared with the popular commercial codes. © 2013 IEEE.
On topology optimization and canonical duality method
- Authors: Gao, David
- Date: 2018
- Type: Text , Journal article
- Relation: Computer Methods in Applied Mechanics and Engineering Vol. 341, no. (2018), p. 249-277
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- Description: Topology optimization for general materials is correctly formulated as a bi-level knapsack problem, which is considered to be NP-hard in global optimization and computer science. By using canonical duality theory (CDT) developed by the author, the linear knapsack problem can be solved analytically to obtain global optimal solution at each design iteration. Both uniqueness, existence, and NP-hardness are discussed. The novel CDT method for general topology optimization is refined and tested by both 2-D and 3-D benchmark problems. Numerical results show that without using filter and any other artificial technique, the CDT method can produce exactly 0-1 optimal density distribution with almost no checkerboard pattern. Its performance and novelty are compared with the popular SIMP and BESO approaches. Additionally, some mathematical and conceptual mistakes in literature are explicitly addressed. A brief review on the canonical duality theory for modeling multi-scale complex systems and for solving general nonconvex/discrete problems are given in Appendix. This paper demonstrates a simple truth: elegant designs come from correct model and theory. © 2018
Discrete state transition algorithm for unconstrained integer optimization problems
- Authors: Zhou, Xiaojun , Gao, David , Yang, Chunhua , Gui, Weihua
- Date: 2016
- Type: Text , Journal article
- Relation: Neurocomputing Vol. 173, no. (2016), p. 864-874
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- Description: A recently new intelligent optimization algorithm called discrete state transition algorithm is considered in this study, for solving unconstrained integer optimization problems. Firstly, some key elements for discrete state transition algorithm are summarized to guide its well development. Several intelligent operators are designed for local exploitation and global exploration. Then, a dynamic adjustment strategy "risk and restoration in probability" is proposed to capture global solutions with high probability. Finally, numerical experiments are carried out to test the performance of the proposed algorithm compared with other heuristics, and they show that the similar intelligent operators can be applied to ranging from traveling salesman problem, boolean integer programming, to discrete value selection problem, which indicates the adaptability and flexibility of the proposed intelligent elements. (C) 2015 Elsevier B.V. All rights reserved.
Optimal design of water distribution networks by a discrete state transition algorithm
- Authors: Zhou, Xiaojun , Gao, David , Simpson, Angus
- Date: 2016
- Type: Text , Journal article
- Relation: Engineering Optimization Vol. 48, no. 4 (2016), p. 603-628
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- Description: In this study it is demonstrated that, with respect to model formulation, the number of linear and nonlinear equations involved in water distribution networks can be reduced to the number of closed simple loops. Regarding the optimization technique, a discrete state transition algorithm (STA) is introduced to solve several cases of water distribution networks. Firstly, the focus is on a parametric study of the 'restoration probability and risk probability' in the dynamic STA. To deal effectively with head pressure constraints, the influence is then investigated of the penalty coefficient and search enforcement on the performance of the algorithm. Based on the experience gained from training the Two-Loop network problem, a discrete STA has successfully achieved the best known solutions for the Hanoi, triple Hanoi and New York network problems. © 2015 Taylor & Francis.
Canonical dual solutions to nonconvex radial basis neural network optimization problem
- Authors: Latorre, Vittorio , Gao, David
- Date: 2014
- Type: Text , Journal article
- Relation: Neurocomputing Vol. 134, no. Special issue (2014), p. 189-197
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- Description: Radial Basis Functions Neural Networks (RBFNNs) are tools widely used in regression problems. One of their principal drawbacks is that the formulation corresponding to the training with the supervision of both the centers and the weights is a highly non-convex optimization problem, which leads to some fundamental difficulties for the traditional optimization theory and methods. This paper presents a generalized canonical duality theory for solving this challenging problem. We demonstrate that by using sequential canonical dual transformations, the nonconvex optimization problem of the RBFNN can be reformulated as a canonical dual problem (without duality gap). Both global optimal solution and local extrema can be classified. Several applications to one of the most used Radial Basis Functions, the Gaussian function, are illustrated. Our results show that even for a one-dimensional case, the global minimizer of the nonconvex problem may not be the best solution to the RBFNNs, and the canonical dual theory is a promising tool for solving general neural networks training problems. © 2014 Elsevier B.V.
Stable trajectory of logistic map
- Authors: Li, Chaojie , Zhou, Xiaojun , Gao, David
- Date: 2014
- Type: Text , Journal article
- Relation: Nonlinear Dynamics Vol. 78, no. 1 (2014), p. 209-217
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- Description: In this paper, the stable trajectory of Logistic Map has been investigated by canonical duality theory from the perspective of global optimization. Numerical result of our method shows that it totally differs from traditional chaotic solution solved by Euler method. In addition, we have applied our method to three well-known standard benchmarks in global optimization. Numerical simulations are given to illustrate the effectiveness of the main results.
A direct optimization method for low group delay FIR filter design
- Authors: Wu, Changzhi , Gao, David , Lay Teo, Kok
- Date: 2013
- Type: Text , Journal article
- Relation: Signal Processing Vol. 93, no. 7 (2013), p. 1764-1772
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- Description: This paper studies the design of FIR filter with low group delay, where the desired phase response is not being approximated. It is formulated as a constrained optimization problem, which is then solved globally. Numerical experiments show that our design method can produce a filter with smaller group delay than that obtained by the existing convex optimization method used in conjunction with a minimum phase spectral factorization method under the same design criteria. Furthermore, our formulation offers us the flexibility for the trade-off between the group delay and the magnitude response directly. It also allows the feasibility of imposing constraints on the group delay. © 2013 Elsevier B.V.
- Description: 2003011019