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14Ruan, Ning
8Zhou, Xiaojun
5Yang, Chunhua
4Fang, Shucherng
4Latorre, Vittorio
4Li, Chaojie
4Nahavandi, Saeid
4Xing, Wenxun
3Chen, Yi
3Zhu, Jinghao
2Ali, Elaf
2Bhatti, Asim
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2Lim, Cheepeng
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33Global optimization
250102 Applied Mathematics
170103 Numerical and Computational Mathematics
15Canonical duality theory
9Canonical duality
7Canonical duality theories
60802 Computation Theory and Mathematics
609 Engineering
6Optimization
508 Information and Computing Sciences
401 Mathematical Sciences
4Duality theory
4Global minimizers
4Maximization problem
4Perturbation method
4Polynomial optimization
4Problem solving
30105 Mathematical Physics
3Canonical dual finite element method
3Convex set

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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:**

On the convexity of nonlinear elastic energies in the right Cauchy-Green tensor

- Gao, David, Neff, Patrizio, Roventa, Ionel, Thiel, Christian

**Authors:**Gao, David , Neff, Patrizio , Roventa, Ionel , Thiel, Christian**Date:**2017**Type:**Text , Journal article**Relation:**Journal of Elasticity Vol. 127, no. 2 (2017), p. 303-308**Full Text:****Reviewed:****Description:**We present a sufficient condition under which a weak solution of the Euler-Lagrange equations in nonlinear elasticity is already a global minimizer of the corresponding elastic energy functional. This criterion is applicable to energies which are convex with respect to the right Cauchy-Green tensor , where denotes the gradient of deformation. Examples of such energies exhibiting a blow up for are given.

**Authors:**Gao, David , Neff, Patrizio , Roventa, Ionel , Thiel, Christian**Date:**2017**Type:**Text , Journal article**Relation:**Journal of Elasticity Vol. 127, no. 2 (2017), p. 303-308**Full Text:****Reviewed:****Description:**We present a sufficient condition under which a weak solution of the Euler-Lagrange equations in nonlinear elasticity is already a global minimizer of the corresponding elastic energy functional. This criterion is applicable to energies which are convex with respect to the right Cauchy-Green tensor , where denotes the gradient of deformation. Examples of such energies exhibiting a blow up for are given.

Solutions to quadratic minimization problems with box and integer constraints

**Authors:**Gao, David , Ruan, Ning**Date:**2010**Type:**Text , Journal article**Relation:**Journal of Global Optimization Vol. 47, no. 3 (2010), p. 463-484**Full Text:**false**Reviewed:**

Global optimal solutions to nonconvex euclidean distance geometry problems

**Authors:**Ruan, Ning , Gao, David**Date:**2012**Type:**Text , Conference paper**Relation:**20th International Symposium on Mathematical Theory of Networks and Systems**Full Text:**false**Reviewed:****Description:**This paper presents a canonical dual approach for solving nonconvex minimization problems in Euclidean distance geometry. The variant of this problem arises extensively in engineering and science, including computational biology, sensor network communications, database analysis, information technology, and global optimization. Due to the nonconvexity, most of these problems are NP-hard and traditional convex optimization methods can not be used directly for finding global optimal solutions. We first show that this type of nonconvex problems can be transferred to a concave maximization problem over a convex set. Then a general analytical solution is proposed by using the canonical duality theory. Applications are illustrated by network localization and minimization of Rosenbrock function. Furthermore, 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.

**Authors:**Ruan, Ning , Gao, David**Date:**2014**Type:**Text , Conference paper**Relation:**Proceedings of the 11th World Congress on Computational Mechanics (WCCM XI) p. 1-2**Full Text:**false**Reviewed:**

Impulsive control for synchronizing delayed discrete complex networks with switching topology

- Li, Chaojie, Gao, David, Liu, Chao, Chen, Guo

**Authors:**Li, Chaojie , Gao, David , Liu, Chao , Chen, Guo**Date:**2014**Type:**Text , Journal article**Relation:**Neural Computing and Applications Vol. 24, no. 1 (2014), p. 59-68**Full Text:**false**Reviewed:****Description:**In this paper, global exponential synchronization of a class of discrete delayed complex networks with switching topology has been investigated by using Lyapunov-Ruzimiki method. The impulsive scheme is designed to work at the time instant of switching occurrence. A time-varying delay-dependent criterion for impulsive synchronization is given to ensure the delayed discrete complex networks switching topology tending to a synchronous state. Furthermore, a numerical simulation is given to illustrate the effectiveness of main results Â© 2013 The Author(s).

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 primal-dual algorithm for solving fourth-order polynomial minimization problems

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

**Authors:**Zhou, Xiaojun , Gao, David , Yang, Chunhua**Date:**2014**Type:**Text , Journal article**Relation:**Applied Mathematics and Computation Vol. 227, no. (2014), p. 246-255**Full Text:**false**Reviewed:****Description:**This paper focuses on implementation of a general canonical primal-dual algorithm for solving a class of fourth-order polynomial minimization problems. A critical issue in the canonical duality theory has been addressed, i.e., in the case that the canonical dual problem has no interior critical point in its feasible space Sa+, a quadratic perturbation method is introduced to recover the global solution through a primal-dual iterative approach, and a gradient-based method is further used to refine the solution. A series of test problems, including the benchmark polynomials and several instances of the sensor network localization problems, have been used to testify the effectiveness of the proposed algorithm. © 2013 Published by Elsevier Inc. All rights reserved.

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 duality theory and triality for solving general global optimization problems in complex systems

- Morales-Silva, Daniel, Gao, David

**Authors:**Morales-Silva, Daniel , Gao, David**Date:**2015**Type:**Text , Journal article**Relation:**Mathematics and Mechanics of Complex Systems Vol. 3, no. 2 (2015), p. 139-161**Full Text:****Reviewed:****Description:**General nonconvex optimization problems are studied by using the canonical duality-triality theory. The triality theory is proved for sums of exponentials and quartic polynomials, which solved an open problem left in 2003. This theory can be used to find the global minimum and local extrema, which bridges a gap between global optimization and nonconvex mechanics. Detailed applications are illustrated by several examples. © 2015 Mathematical Sciences Publishers.

**Authors:**Morales-Silva, Daniel , Gao, David**Date:**2015**Type:**Text , Journal article**Relation:**Mathematics and Mechanics of Complex Systems Vol. 3, no. 2 (2015), p. 139-161**Full Text:****Reviewed:****Description:**General nonconvex optimization problems are studied by using the canonical duality-triality theory. The triality theory is proved for sums of exponentials and quartic polynomials, which solved an open problem left in 2003. This theory can be used to find the global minimum and local extrema, which bridges a gap between global optimization and nonconvex mechanics. Detailed applications are illustrated by several examples. © 2015 Mathematical Sciences Publishers.

On the extrema of a nonconvex functional with double-well potential in 1D

**Authors:**Gao, David , Lu, Xioajun**Date:**2016**Type:**Text , Journal article**Relation:**Zeitschrift fur Angewandte Mathematik und Physik Vol. 67, no. 3 (2016), p. 1-7**Full Text:**false**Reviewed:****Description:**This paper mainly investigates the extrema of a nonconvex functional with double-well potential in 1D through the approach of nonlinear differential equations. Based on the canonical duality method, the corresponding Euler–Lagrange equation with Neumann boundary condition can be converted into a cubic dual algebraic equation, which will help find the local extrema for the primal problem. © 2016, Springer International Publishing.

Complete solutions and triality theory to a nonconvex optimization problem with double-well potential in Rn

- Morales-Silva, Daniel, Gao, David

**Authors:**Morales-Silva, Daniel , Gao, David**Date:**2013**Type:**Text , Journal article**Relation:**Numerical Algebra, Control and Optimization Vol. 3, no. 2 (2013), p. 271-282**Full Text:****Reviewed:****Description:**The main purpose of this research note is to show that the triality theory can always be used to identify both global minimizer and the biggest local maximizer in global optimization. An open problem left on the double- min duality is solved for a nonconvex optimization problem with double-well potential in ℝn, which leads to a complete set of analytical solutions. Also a convergency theorem is proved for linear perturbation canonical dual method, which can be used for solving global optimization problems with multiple so- lutions. The methods and results presented in this note pave the way towards the proof of the triality theory in general cases.

**Authors:**Morales-Silva, Daniel , Gao, David**Date:**2013**Type:**Text , Journal article**Relation:**Numerical Algebra, Control and Optimization Vol. 3, no. 2 (2013), p. 271-282**Full Text:****Reviewed:****Description:**The main purpose of this research note is to show that the triality theory can always be used to identify both global minimizer and the biggest local maximizer in global optimization. An open problem left on the double- min duality is solved for a nonconvex optimization problem with double-well potential in ℝn, which leads to a complete set of analytical solutions. Also a convergency theorem is proved for linear perturbation canonical dual method, which can be used for solving global optimization problems with multiple so- lutions. The methods and results presented in this note pave the way towards the proof of the triality theory in general cases.

Global optimal trajectory in Chaos and NP-Hardness

- Latorre, Vittorio, Gao, David

**Authors:**Latorre, Vittorio , Gao, David**Date:**2016**Type:**Text , Journal article**Relation:**International Journal of Bifurcation and Chaos Vol. 26, no. 8 (2016), p. 1-14**Full Text:****Reviewed:****Description:**This paper presents an unconventional theory and method for solving general nonlinear dynamical systems. Instead of the direct iterative methods, the discretized nonlinear system is first formulated as a global optimization problem via the least squares method. A newly developed canonical duality theory shows that this nonconvex minimization problem can be solved deterministically in polynomial time if a global optimality condition is satisfied. The so-called pseudo-chaos produced by linear iterative methods are mainly due to the intrinsic numerical error accumulations. Otherwise, the global optimization problem could be NP-hard and the nonlinear system can be really chaotic. A conjecture is proposed, which reveals the connection between chaos in nonlinear dynamics and NP-hardness in computer science. The methodology and the conjecture are verified by applications to the well-known logistic equation, a forced memristive circuit and the Lorenz system. Computational results show that the canonical duality theory can be used to identify chaotic systems and to obtain realistic global optimal solutions in nonlinear dynamical systems. The method and results presented in this paper should bring some new insights into nonlinear dynamical systems and NP-hardness in computational complexity theory. © 2016 World Scientific Publishing Company.

**Authors:**Latorre, Vittorio , Gao, David**Date:**2016**Type:**Text , Journal article**Relation:**International Journal of Bifurcation and Chaos Vol. 26, no. 8 (2016), p. 1-14**Full Text:****Reviewed:****Description:**This paper presents an unconventional theory and method for solving general nonlinear dynamical systems. Instead of the direct iterative methods, the discretized nonlinear system is first formulated as a global optimization problem via the least squares method. A newly developed canonical duality theory shows that this nonconvex minimization problem can be solved deterministically in polynomial time if a global optimality condition is satisfied. The so-called pseudo-chaos produced by linear iterative methods are mainly due to the intrinsic numerical error accumulations. Otherwise, the global optimization problem could be NP-hard and the nonlinear system can be really chaotic. A conjecture is proposed, which reveals the connection between chaos in nonlinear dynamics and NP-hardness in computer science. The methodology and the conjecture are verified by applications to the well-known logistic equation, a forced memristive circuit and the Lorenz system. Computational results show that the canonical duality theory can be used to identify chaotic systems and to obtain realistic global optimal solutions in nonlinear dynamical systems. The method and results presented in this paper should bring some new insights into nonlinear dynamical systems and NP-hardness in computational complexity theory. © 2016 World Scientific Publishing Company.

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.

Double well potential function and its optimization in the N-dimensional real space -- Part I

- Fang, Shucherng, Gao, David, Lin, Gang-Xuan, Sheu, Ruey-Lin, Xing, Wenxun

**Authors:**Fang, Shucherng , Gao, David , Lin, Gang-Xuan , Sheu, Ruey-Lin , Xing, Wenxun**Date:**2017**Type:**Text , Journal article**Relation:**Journal of Industrial and Management Optimization Vol. 13, no. 3 (2017), p. 1291-1305**Full Text:**false**Reviewed:****Description:**A special type of multi-variate polynomial of degree 4, called the double well potential function, is studied. It is derived from a discrete approximation of the generalized Ginzburg-Landau functional, and we are interested in understanding its global minimum solution and all local non-global points. The main difficulty for the model is due to its non-convexity. In Part I of the paper, we first characterize the global minimum solution set, whereas the study for local non-global optimal solutions is left for Part II. We show that, the dual of the Lagrange dual of the double well potential problem is a linearly constrained convex minimization problem, which, under a designated nonlinear transformation, can be equivalently mapped to a portion of the original double well potential function containing the global minimum. In other words, solving the global minimum of the double well potential function is essentially a convex minimization problem, despite of its non-convex nature. Numerical examples are provided to illustrate the important features of the problem and the mapping in between.

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.

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.

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.

**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.

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