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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.

Canonical dual solutions for fixed cost quadratic programs

- Gao, David, Ruan, Ning, Sherali, Hanif

**Authors:**Gao, David , Ruan, Ning , Sherali, Hanif**Date:**2010**Type:**Text , Book chapter**Relation:**Optimization and Optimal Control p. 139-156**Full Text:**false**Reviewed:****Description:**This chapter presents a canonical dual approach for solving a mixed-integer quadratic minimization problem with fixed cost terms. We show that this well-known NP-hard problem in R2n can be transformed into a continuous concave maximization dual problem over a convex feasible subset of R2n with zero duality gap. The resulting canonical dual problem can be solved easily, under certain conditions, by traditional convex programming methods. Both existence and uniqueness of global optimal solutions are discussed. Application to a decoupled mixed-integer problem is illustrated and analytic solutions for both a global minimizer and a global maximizer are obtained. Examples for both decoupled and general nonconvex problems are presented. Furthermore, we discuss connections between the proposed canonical duality theory approach and the classical Lagrangian duality approach. An open problem is proposed for future study.

On the triality theory for a quartic polynomial optimization problem

**Authors:**Gao, David , Wu, Changzhi**Date:**2012**Type:**Text , Journal article**Relation:**Journal of Industrial and Management Optimization Vol. 8, no. 1 (2012), p. 229-242**Full Text:**false**Reviewed:****Description:**This paper presents a detailed proof of the triality theorem for a class of fourth-order polynomial optimization problems. The method is based on linear algebra but it solves an open problem on the double-min duality. Results show that the triality theory holds strongly in the tri-duality form for our problem if the primal problem and its canonical dual have the same dimension; otherwise, both the canonical min-max duality and the double-max duality still hold strongly, but the double-min duality holds weakly in a symmetrical form. Some numerical examples are presented to illustrate that this theory can be used to identify not only the global minimum, but also the local minimum and local maximum.

Canonical dual approach to solving the maximum cut problem

- Wang, Zhenbo, Fang, Shucherng, Gao, David, Xing, Wenxun

**Authors:**Wang, Zhenbo , Fang, Shucherng , Gao, David , Xing, Wenxun**Date:**2012**Type:**Text , Journal article**Relation:**Journal of Global Optimization Vol. , no. (2012), p. 1-11**Full Text:**false**Reviewed:****Description:**This paper presents a canonical dual approach for finding either an optimal or approximate solution to the maximum cut problem (MAX CUT). We show that, by introducing a linear perturbation term to the objective function, the maximum cut problem is perturbed to have a dual problem which is a concave maximization problem over a convex feasible domain under certain conditions. Consequently, some global optimality conditions are derived for finding an optimal or approximate solution. A gradient decent algorithm is proposed for this purpose and computational examples are provided to illustrate the proposed approach. © 2012 Springer Science+Business Media, LLC.

- 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.

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