14Ruan, Ning
8Zhou, Xiaojun
5Yang, Chunhua
4Fang, Shucherng
4Latorre, Vittorio
4Li, Chaojie
4Nahavandi, Saeid
4Xing, Wenxun
3Chen, Yi
3Zhu, Jinghao
2Ali, Elaf
2Bhatti, Asim
2Cai, Kun
2Creighton, Douglas
2Hanoun, Samer
2Lim, Cheepeng
2Lin, Gang-Xuan
2Liu, Chao
2Morales-Silva, Daniel

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33Global optimization
250102 Applied Mathematics
170103 Numerical and Computational Mathematics
15Canonical duality theory
9Canonical duality
709 Engineering
7Canonical duality theories
608 Information and Computing Sciences
60802 Computation Theory and Mathematics
6Optimization
401 Mathematical Sciences
4Duality theory
4Global minimizers
4Maximization problem
4NP-hardness
4Perturbation method
4Polynomial optimization
4Problem solving
30105 Mathematical Physics
3Canonical dual finite element method

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A comparative study of state transition algorithm with harmony search and artificial bee colony

- Zhou, Xiaojun, Gao, David, Yang, Chunhua

**Authors:**Zhou, Xiaojun , Gao, David , Yang, Chunhua**Date:**2013**Type:**Text , Journal article**Relation:**Advances in Intelligent Systems and Computing Vol. 212, no. (2013), p. 651-659**Full Text:**false**Reviewed:****Description:**We focus on a comparative study of three recently developed nature-inspired optimization algorithms, including state transition algorithm, harmony search and artificial bee colony. Their core mechanisms are introduced and their similarities and differences are described. Then, a suit of 27 well-known benchmark problems are used to investigate the performance of these algorithms and finally we discuss their general applicability with respect to the structure of optimization problems. Â© Springer-Verlag Berlin Heidelberg 2013.**Description:**2003011220

A direct optimization method for low group delay FIR filter design

- Wu, Changzhi, Gao, David, Lay Teo, Kok

**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**Full Text:**false**Reviewed:****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

A multiobjective state transition algorithm for single machine scheduling

- Zhou, Xiaojun, Hanoun, Samer, Gao, David, Nahavandi, Saeid

**Authors:**Zhou, Xiaojun , Hanoun, Samer , Gao, David , Nahavandi, Saeid**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. 79-88**Full Text:**false**Reviewed:****Description:**In this paper, a discrete state transition algorithm is introduced to solve a multiobjective single machine job shop scheduling problem. In the proposed approach, a non-dominated sort technique is used to select the best from a candidate state set, and a Pareto archived strategy is adopted to keep all the non-dominated solutions. Compared with the enumeration and other heuristics, experimental results have demonstrated the effectiveness of the multiobjective state transition algorithm. © Springer International Publishing Switzerland 2015.

A novel canonical dual computational approach for prion AGAAAAGA amyloid fibril molecular modeling

- Zhang, Jiapu, Gao, David, Yearwood, John

**Authors:**Zhang, Jiapu , Gao, David , Yearwood, John**Date:**2011**Type:**Text , Journal article**Relation:**Journal of Theoretical Biology Vol. 284, no. 1 (2011), p. 149-157**Full Text:**false**Reviewed:****Description:**Many experimental studies have shown that the prion AGAAAAGA palindrome hydrophobic region (113-120) has amyloid fibril forming properties and plays an important role in prion diseases. However, due to the unstable, noncrystalline and insoluble nature of the amyloid fibril, to date structural information on AGAAAAGA region (113-120) has been very limited. This region falls just within the N-terminal unstructured region PrP (1-123) of prion proteins. Traditional X-ray crystallography and nuclear magnetic resonance (NMR) spectroscopy experimental methods cannot be used to get its structural information. Under this background, this paper introduces a novel approach of the canonical dual theory to address the 3D atomic-resolution structure of prion AGAAAAGA amyloid fibrils. The novel and powerful canonical dual computational approach introduced in this paper is for the molecular modeling of prion AGAAAAGA amyloid fibrils, and that the optimal atomic-resolution structures of prion AGAAAAGA amyloid fibils presented in this paper are useful for the drive to find treatments for prion diseases in the field of medicinal chemistry. Overall, this paper presents an important method and provides useful information for treatments of prion diseases. Â© 2011.

A study on concave optimization via canonical dual function

- Zhu, Jinghao, Tao, Shiming, Gao, David

**Authors:**Zhu, Jinghao , Tao, Shiming , Gao, David**Date:**2009**Type:**Text , Journal article**Relation:**Elsevier Vol. 224, no. 2 (2009), p. 459-464**Full Text:**false**Reviewed:****Description:**In this study we find a global minimizer of a concave function over a sphere. By introducing a differential equation, we obtain the invariant characteristics for a given optimization problem by constructing a canonical dual function. We present two theorems concerning the global optimality of an extrema of the optimization problem.

Advances in canonical duality theory with applications to global optimization

**Authors:**Gao, David**Date:**2008**Type:**Text , Conference proceedings**Relation:**FOCAPO 2008, Boston, June 29th-July 02, Published in Proceedings of the Fifth International Conference Foundations of Computer-Aided Process Operations pg. 73-82 p. 73-81**Full Text:**false**Reviewed:**

Advances in Global Optimization

- Gao, David, Ruan, Ning, Xing, Wenxun

**Authors:**Gao, David , Ruan, Ning , Xing, Wenxun**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**Full Text:**false**Reviewed:****Description:**This proceedings volume addresses advances in global optimization - a multidisciplinary research field that deals with the analysis, characterization, and computation of global minima and/or maxima of nonlinear, non-convex, and nonsmooth functions in continuous or discrete forms. The volume contains selected papers from the third biannual World Congress on Global Optimization in Engineering & Science (WCGO), held in the Yellow Mountains, Anhui, China on July 8-12, 2013. The papers fall into eight topical sections: mathematical programming; combinatorial optimization; duality theory; topology optimization; variational inequalities and complementarity problems; numerical optimization; stochastic models and simulation; and complex simulation and supply chain analysis.

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.

Analytical solutions to general anti-plane shear problems in finite elasticity

**Authors:**Gao, David**Date:**2016**Type:**Text , Journal article**Relation:**Continuum Mechanics and Thermodynamics Vol. 28, no. 1-2 (2016), p. 175-194**Full Text:****Reviewed:****Description:**This paper presents a pure complementary energy variational method for solving a general anti-plane shear problem in finite elasticity. Based on the canonical dualityâ€“triality theory developed by the author, the nonlinear/nonconvex partial differential equations for the large deformation problem are converted into an algebraic equation in dual space, which can, in principle, be solved to obtain a complete set of stress solutions. Therefore, a general analytical solution form of the deformation is obtained subjected to a compatibility condition. Applications are illustrated by examples with both convex and nonconvex stored strain energies governed by quadratic-exponential and power-law material models, respectively. Results show that the nonconvex variational problem could have multiple solutions at each material point, the complementary gap function and the triality theory can be used to identify both global and local extremal solutions, while the popular convexity conditions (including rank-one condition) provide mainly local minimal criteria and the Legendre-Hadamard condition (i.e., the so-called strong ellipticity condition) does not guarantee uniqueness of solutions. This paper demonstrates again that the pure complementary energy principle and the triality theory play important roles in finite deformation theory and nonconvex analysis. © 2015, Springer-Verlag Berlin Heidelberg.

**Authors:**Gao, David**Date:**2016**Type:**Text , Journal article**Relation:**Continuum Mechanics and Thermodynamics Vol. 28, no. 1-2 (2016), p. 175-194**Full Text:****Reviewed:****Description:**This paper presents a pure complementary energy variational method for solving a general anti-plane shear problem in finite elasticity. Based on the canonical dualityâ€“triality theory developed by the author, the nonlinear/nonconvex partial differential equations for the large deformation problem are converted into an algebraic equation in dual space, which can, in principle, be solved to obtain a complete set of stress solutions. Therefore, a general analytical solution form of the deformation is obtained subjected to a compatibility condition. Applications are illustrated by examples with both convex and nonconvex stored strain energies governed by quadratic-exponential and power-law material models, respectively. Results show that the nonconvex variational problem could have multiple solutions at each material point, the complementary gap function and the triality theory can be used to identify both global and local extremal solutions, while the popular convexity conditions (including rank-one condition) provide mainly local minimal criteria and the Legendre-Hadamard condition (i.e., the so-called strong ellipticity condition) does not guarantee uniqueness of solutions. This paper demonstrates again that the pure complementary energy principle and the triality theory play important roles in finite deformation theory and nonconvex analysis. © 2015, Springer-Verlag Berlin Heidelberg.

Anticipating synchronization through optimal feedback control

- Huang, Tingwen, Gao, David, Li, Chuandong, Xiao, MingQing

**Authors:**Huang, Tingwen , Gao, David , Li, Chuandong , Xiao, MingQing**Date:**2012**Type:**Text , Journal article**Relation:**Journal of Global Optimization Vol. 52, no. 2 (2012), p. 281-290**Full Text:**false**Reviewed:****Description:**In this paper, we investigate the anticipating synchronization of a class of coupled chaotic systems through discontinuous feedback control. The stability criteria for the involved error dynamical system are obtained by means of model transformation incorporated with Lyapunov functional and linear matrix inequality. Also, we discuss the optimal designed controller based on the obtained criteria. The numerical simulation is presented to demonstrate the theoretical results. © 2011 Springer Science+Business Media, LLC.

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.

Applying the canonical dual theory in optimal control problems

- Zhu, Jinghao, Wu, Dan, Gao, David

**Authors:**Zhu, Jinghao , Wu, Dan , Gao, David**Date:**2012**Type:**Text , Journal article**Relation:**Journal of global optimization Vol. 54, no. 2 (2012), p. 221-233**Full Text:**false**Reviewed:****Description:**This paper presents some applications of the canonical dual theory in optimal control problems. The analytic solutions of several nonlinear and nonconvex problems are investigated by global optimizations. It turns out that the backward differential flow defined by the KKT equation may reach the globally optimal solution. The analytic solution to an optimal control problem is obtained via the expression of the co-state. Some examples are illustrated.

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.

Canonical dual approach to binary factor analysis

- Ke, Sun, Shikui, Tui, Gao, David, Xu, Lei

**Authors:**Ke, Sun , Shikui, Tui , Gao, David , Xu, Lei**Date:**2009**Type:**Text , Conference paper**Relation:**8th International conference on Independent Component Analysis and Signal Separation. p. 346-353**Full Text:**false**Reviewed:****Description:**Binary Factor Analysis (BFA) is a typical problem of Independent Component Analysis (ICA) where the signal sources are binary. Parameter learning and model selection in BFA are computationally intractable because of the combinatorial complexity. This paper aims at an efficient approach to BFA. For parameter learning, an unconstrained binary quadratic programming (BQP) is reduced to a canonical dual problem with low computational complexity; for model selection, we adopt the Bayesian Ying-Yang (BYY) framework to make model selection automatically during learning. In the experiments, the proposed approach cdual shows superior performance. Another BQP approximation round is also good in model selection and is more efficient. Two other methods, greedy and enum, are more accurate in BQP but fail to compete with cdual and round in BFA. We conclude that a good optimization is essential in a learning process, but the key task of learning is not simply optimization and an over-accurate optimization may not be preferred.

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.

**Authors:**Santos, Hugo , Gao, David**Date:**2012**Type:**Text , Journal article**Relation:**International Journal of Non-Linear Mechanics Vol. 47, no. 2 (2012), p. 240-247**Full Text:**false**Reviewed:****Description:**This paper presents a canonical dual mixed finite element method for the post-buckling analysis of planar beams with large elastic deformations. The mathematical beam model employed in the present work was introduced by Gao in 1996, and is governed by a fourth-order non-linear differential equation. The total potential energy associated with this model is a non-convex functional and can be used to study both the pre- and the post-buckling responses of the beams. Using the so-called canonical duality theory, this non-convex primal variational problem is transformed into a dual problem. In a proper feasible space, the dual variational problem corresponds to a globally concave maximization problem. A mixed finite element method involving both the transverse displacement field and the stress field as approximate element functions is derived from the dual variational problem and used to compute global optimal solutions. Numerical applications are illustrated by several problems with different boundary conditions. © 2011 Elsevier Ltd. All rights reserved.

Canonical dual least square method for solving general nonlinear systems of quadratic equations

**Authors:**Ruan, Ning , Gao, David**Date:**2010**Type:**Text , Journal article**Relation:**Computational Optimization and Applications Vol. 47, no. (2010), p. 335-347**Full Text:**false**Reviewed:****Description:**This paper presents a canonical dual approach for solving general non- linear algebraic systems. By using least square method, the nonlinear system of m -quadratic equations in n -dimensional space is first formulated as a nonconvex opti- mization problem. We then proved that, by the canonical duality theory developed by the second author, this nonconvex problem is equivalent to a concave maximization problem in R, which can be solved easily by well-developed convex optimization techniques. Both existence and uniqueness of global optimal solutions are discussed, and several illustrative examples are presented.**Description:**C1

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.

Canonical dual solutions to nonconvex radial basis neural network optimization problem

- Latorre, Vittorio, Gao, David

**Authors:**Latorre, Vittorio , Gao, David**Date:**2014**Type:**Text , Journal article**Relation:**Neurocomputing Vol. 134, no. Special issue (2014), p. 189-197**Full Text:**false**Reviewed:****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.

- Gao, David, Ruan, Ning, Pardalos, Panos

**Authors:**Gao, David , Ruan, Ning , Pardalos, Panos**Date:**2012**Type:**Text , Book chapter**Relation:**Sensors: Theory, Algorithms, and applications optimization and its applications p. 37-56**Full Text:**false**Reviewed:****Description:**This chapter presents a canonical dual approach for solving a general sum of fourth-order polynomial minimization problem. This problem arises extensively in engineering and science, including database analysis, computational biology, sensor network communications, nonconvex mechanics, and ecology. We first show that this global optimization problem is actually equivalent to a discretized minimal potential variational problem in large deformation mechanics. Therefore, a general analytical solution is proposed by using the canonical duality theory developed by the first author. Both global and local extremality properties of this analytical solution are identified by a triality theory. Application to sensor network localization problem is illustrated. Our results show when the problem is not uniquely localizable, the “optimal solution” obtained by the SDP method is actually a local maximizer of the total potential energy. However, by using a perturbed canonical dual approach, a class of Euclidean distance problems can be converted to a unified concave maximization dual problem with zero duality gap, which can be solved by well-developed convex minimization methods. This chapter should bridge an existing gap between nonconvex mechanics and global optimization.

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