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20802 Computation Theory and Mathematics
2Computational complexity
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An efficient classification using support vector machines

- Ruan, Ning, Chen, Yi, Gao, David

**Authors:**Ruan, Ning , Chen, Yi , Gao, David**Date:**2013**Type:**Text , Conference paper**Relation:**Proceedings of 2013 Science and Information Conference, SAI 2013 p. 585-589**Full Text:**false**Reviewed:****Description:**Support vector machine (SVM) is a popular method for classification in data mining. The canonical duality theory provides a unified analytic solution to a wide range of discrete and continuous problems in global optimization. This paper presents a canonical duality approach for solving support vector machine problem. It is shown that by the canonical duality, these nonconvex and integer optimization problems are equivalent to a unified concave maximization problem over a convex set and hence can be solved efficiently by existing optimization techniques. © 2013 The Science and Information Organization.

- Yuan, Y. B., Fang, Shucherng, Gao, David

**Authors:**Yuan, Y. B. , Fang, Shucherng , Gao, David**Date:**2012**Type:**Text , Journal article**Relation:**Journal of Global Optimization Vol. 52, no. 2 (2012), p. 195-209**Full Text:**false**Reviewed:****Description:**This paper studies the canonical duality theory for solving a class of quadri- nomial minimization problems subject to one general quadratic constraint. It is shown that the nonconvex primal problem in Rn can be converted into a concave maximization dual problem over a convex set in R2 , such that the problem can be solved more efficiently. The existence and uniqueness theorems of global minimizers are provided using the triality theory. Examples are given to illustrate the results obtained. © 2011 Springer Science+Business Media, LLC.

Canonical duality theory and algorithm for solving challenging problems in network optimisation

**Authors:**Ruan, Ning , Gao, David**Date:**2012**Type:**Text , Conference paper**Relation:**19th International Conference on Neural Information Processing, ICONIP 2012 Vol. 7665 LNCS, p. 702-709**Full Text:****Reviewed:****Description:**This paper presents a canonical dual approach for solving a general nonconvex problem in network optimization. Three challenging problems, sensor network location, traveling salesman problem, and scheduling problem are listed to illustrate the applications of the proposed method. It is shown that by the canonical duality, these nonconvex and integer optimization problems are equivalent to unified concave maximization problem over a convex set and hence can be solved efficiently by existing optimization techniques. © 2012 Springer-Verlag.**Description:**2003010653

**Authors:**Ruan, Ning , Gao, David**Date:**2012**Type:**Text , Conference paper**Relation:**19th International Conference on Neural Information Processing, ICONIP 2012 Vol. 7665 LNCS, p. 702-709**Full Text:****Reviewed:****Description:**This paper presents a canonical dual approach for solving a general nonconvex problem in network optimization. Three challenging problems, sensor network location, traveling salesman problem, and scheduling problem are listed to illustrate the applications of the proposed method. It is shown that by the canonical duality, these nonconvex and integer optimization problems are equivalent to unified concave maximization problem over a convex set and hence can be solved efficiently by existing optimization techniques. © 2012 Springer-Verlag.**Description:**2003010653

- Gao, David, Watson, Layne, Easterling, David, Thacker, William, Billups, Stephen

**Authors:**Gao, David , Watson, Layne , Easterling, David , Thacker, William , Billups, Stephen**Date:**2013**Type:**Text , Journal article**Relation:**Optimization Methods and Software Vol. 28, no. 2 (2013), p. 313-326**Full Text:**false**Reviewed:****Description:**This paper presents a massively parallel global deterministic direct search method (VTDIRECT) for solving nonconvex quadratic minimization problems with either box or1 integer constraints. Using the canonical dual transformation, these well-known NP-hard problems can be reformulated as perfect dual stationary problems (with zero duality gap). Under certain conditions, these dual problems are equivalent to smooth concave maximization over a convex feasible space. Based on a perturbation method proposed by Gao, the integer programming problem is shown to be equivalent to a continuous unconstrained Lipschitzian global optimization problem. The parallel algorithm VTDIRECT is then applied to solve these dual problems to obtain global minimizers. Parallel performance results for several nonconvex quadratic integer programming problems are reported. © 2013 Copyright Taylor and Francis Group, LLC.**Description:**2003010580

Canonical duality for radial basis neural networks

- Latorre, Vittorio, Gao, David

**Authors:**Latorre, Vittorio , Gao, David**Date:**2013**Type:**Text , Journal article**Relation:**Advances in Intelligent Systems and Computing Vol. 212, no. (2013), p. 1189-1197**Full Text:**false**Reviewed:****Description:**Radial Basis Function Neural Networks (RBF NN) are a tool largely used for regression problems. The principal drawback of this kind of predictive tool is that the optimization problem solved to train the network can be non-convex. On the other hand Canonical Duality Theory offers a powerful procedure to reformulate general non-convex problems in dual forms so that it is possible to find optimal solutions and to get deep insights into the nature of the challenging problems. By combining the canonical duality theory with the RBF NN, this paper presents a potentially useful method for solving challenging problems in real-world applications. © Springer-Verlag Berlin Heidelberg 2013. Proceedings of the Eighth International Conference on Bio-Inspired Computing: Theories and Applications (BIC-TA), 2013.**Description:**2003011221

Solving facility location problem based on duality approach

**Authors:**Ruan, Ning**Date:**2015**Type:**Text , Conference paper**Relation:**3rd World Congress on Global Optimization in Engineering and Science, WCGO 2013; Anhui, China; 8th-12th July 2013 Vol. 95, p. 165-172**Full Text:**false**Reviewed:****Description:**The facility location problem is one of the most widely studied discrete location problems, whose applications arise in a variety of settings, such as routers or servers in a communication network, warehouses or distribution centres in a supply chain, hospitals or airports in a public service system. The problem involves locating a number of facilities to minimize the sum of the fixed setup costs and the variable costs of serving the market demand from these facilities. First a dual problem is developed for the facility location problem. Then general optimality conditions are also obtained, which generate sequences globally converging to a primal and dual solutions, respectively. © Springer International Publishing Switzerland 2015.

Application of canonical duality theory to fixed point problem

**Authors:**Ruan, Ning , Gao, David**Date:**2015**Type:**Text , Conference paper**Relation:**3rd World Congress on Global Optimization in Engineering and Science, WCGO 2013; Anhui, China; 8th-12th July 2013 Vol. 95, p. 157-163**Full Text:**false**Reviewed:****Description:**In this paper, we study general fixed point problem. We first rewrite the original problem in the canonical framework. Then, we proposed a canonical transformation of this problem, which leads to a convex differentiable dual problem and new iteration method. An illustrative example is presented. © Springer International Publishing Switzerland 2015.

Canonical dual approach for minimizing a nonconvex quadratic function over a sphere

**Authors:**Chen, Yi , Gao, David**Date:**2013**Type:**Text , Conference paper**Relation:**3rd World Congress on Global Optimization in Engineering and Science, WCGO 2013; Anhui, China; 8th-12th July 2013 Vol. 95, p. 149-156**Full Text:**false**Reviewed:****Description:**In this paper, we study global optimal solutions of minimizing a nonconvex quadratic function subject to a sphere constraint. The main challenge is to solve the problem when it has multiple global solutions on the boundary of the sphere, which is called hard case. By canonical duality theory, a concave maximization problem is formulated, which is one-dimensional and without duality gaps to the primal problem. Then sufficient and necessary conditions are provided to identify whether the problem is in the hard case or not. A perturbation method and associated algorithms are proposed to solve hard-case problems. Theoretical results and methods are verified by numerical examples. © Springer International Publishing Switzerland 2015.

An interesting cryptography study based on knapsack problem

**Authors:**Ruan, Ning**Date:**2013**Type:**Text , Conference paper**Relation:**Proceedings - UKSim 15th International Conference on Computer Modelling and Simulation, UKSim 2013 p. 330-334**Full Text:****Reviewed:****Description:**Cryptography is an art that has been practised through the centuries. Interest in the applications of the knapsack problem to cryptography has arisen with the advent of public key cryptography. The knapsack problem is well documented problem and all research into its properties have lead to the conjecture that it is difficult to solve. In this paper the canonical duality theory is presented for solving general knapsack problem. By using the canonical dual transformation, the integer programming problem can be converted into a continuous canonical dual problem with zero duality gap. The optimality criterion are also discussed. Numerical examples show the efficiency of the method. Â© 2013 IEEE.

**Authors:**Ruan, Ning**Date:**2013**Type:**Text , Conference paper**Relation:**Proceedings - UKSim 15th International Conference on Computer Modelling and Simulation, UKSim 2013 p. 330-334**Full Text:****Reviewed:****Description:**Cryptography is an art that has been practised through the centuries. Interest in the applications of the knapsack problem to cryptography has arisen with the advent of public key cryptography. The knapsack problem is well documented problem and all research into its properties have lead to the conjecture that it is difficult to solve. In this paper the canonical duality theory is presented for solving general knapsack problem. By using the canonical dual transformation, the integer programming problem can be converted into a continuous canonical dual problem with zero duality gap. The optimality criterion are also discussed. Numerical examples show the efficiency of the method. Â© 2013 IEEE.

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