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14Ruan, Ning
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33Global optimization
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15Canonical duality theory
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7Canonical duality theories
60802 Computation Theory and Mathematics
609 Engineering
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508 Information and Computing Sciences
401 Mathematical Sciences
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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:**

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 approx imation 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 nonlin ear 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.

Canonical finite element method for solving nonconvex variational problems to post buckling beam problem

**Authors:**Ali, Elaf , Gao, David**Date:**2016**Type:**Text , Conference proceedings**Relation:**2nd International Conference on Numerical Computations : Theory and Algorithms, NUMTA 2016; Pizzo Calabro, Italy; 19th-25th June 2016; published in AIP Proceedings of the 2nd International Conference "Numerical Computations: Theory and Algorithms Vol. 1776, p. 1-4**Full Text:****Reviewed:****Description:**The goal of this paper is to solve the post buckling phenomena of a large deformed elastic beam by a canonical dual mixed finite element method (CD-FEM). The total potential energy of this beam is a nonconvex functional which can be used to model both pre-and post-buckling problems. Different types of dual stress interpolations are used in order to verify the triality theory. Applications are illustrated with different boundary conditions and external loads by using semi-definite programming (SDP) algorithm. The results show that the global minimum of the total potential energy is stable buckled configuration, the local maximum solution leads to the unbuckled state, and both of these two solutions are numerically stable. While the local minimum is unstable buckled configuration and very sensitive to both stress interpolations and the external loads.

**Authors:**Ali, Elaf , Gao, David**Date:**2016**Type:**Text , Conference proceedings**Relation:**2nd International Conference on Numerical Computations : Theory and Algorithms, NUMTA 2016; Pizzo Calabro, Italy; 19th-25th June 2016; published in AIP Proceedings of the 2nd International Conference "Numerical Computations: Theory and Algorithms Vol. 1776, p. 1-4**Full Text:****Reviewed:****Description:**The goal of this paper is to solve the post buckling phenomena of a large deformed elastic beam by a canonical dual mixed finite element method (CD-FEM). The total potential energy of this beam is a nonconvex functional which can be used to model both pre-and post-buckling problems. Different types of dual stress interpolations are used in order to verify the triality theory. Applications are illustrated with different boundary conditions and external loads by using semi-definite programming (SDP) algorithm. The results show that the global minimum of the total potential energy is stable buckled configuration, the local maximum solution leads to the unbuckled state, and both of these two solutions are numerically stable. While the local minimum is unstable buckled configuration and very sensitive to both stress interpolations and the external loads.

Canonical duality approach for non-linear dynamical systems

**Authors:**Ruan, Ning , Gao, David**Date:**2014**Type:**Text , Journal article**Relation:**IMA Journal of Applied Mathematics (Institute of Mathematics and Its Applications) Vol. 79, no. 2 (2014), p. 313-325**Full Text:**false**Reviewed:****Description:**This paper presents a canonical dual approach for solving a non-linear population growth problem governed by the well-known logistic equation. Using the finite difference and least squares methods, the non-linear differential equation is first formulated as a non-convex optimization problem with unknown parameters. We then prove that by the canonical duality theory, this non-convex problem is equivalent to a concave maximization problem over a convex feasible space, which can be solved easily to obtain a global optimal solution to this challenging problem. Several illustrative examples are presented.

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:**

Video driven traffic modelling

- Zhou, Hailing, Creighton, Douglas, Wei, Lei, Gao, David, Nahavandi, Saeid

**Authors:**Zhou, Hailing , Creighton, Douglas , Wei, Lei , Gao, David , Nahavandi, Saeid**Date:**2013**Type:**Text , Conference paper**Relation:**2013 IEEE/ASME International Conference on Advanced Intelligent Mechatronics: Mechatronics for Human Wellbeing, AIM 2013 p. 506-511**Full Text:**false**Reviewed:****Description:**We propose Video Driven Traffic Modelling (VDTM) for accurate simulation of real-world traffic behaviours with detailed information and low-cost model development and maintenance. Computer vision techniques are employed to estimate traffic parameters. These parameters are used to build and update a traffic system model. The model is simulated using the Paramics traffic simulation platform. Based on the simulation techniques, effects of traffic interventions can be evaluated in order to achieve better decision makings for traffic management authorities. In this paper, traffic parameters such as vehicle types, times of starting trips and corresponding origin-destinations are extracted from a video. A road network is manually defined according to the traffic composition in the video, and individual vehicles associated with extracted properties are modelled and simulated within the defined road network using Paramics. VDTM has widespread potential applications in supporting traffic decision-makings. To demonstrate the effectiveness, we apply it in optimizing a traffic signal control system, which adaptively adjusts green times of signals at an intersection to reduce traffic congestion.**Description:**E1

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).

Optimal design of water distribution networks by a discrete state transition algorithm

- Zhou, Xiaojun, Gao, David, Simpson, Angus

**Authors:**Zhou, Xiaojun , Gao, David , Simpson, Angus**Date:**2016**Type:**Text , Journal article**Relation:**Engineering Optimization Vol. 48, no. 4 (2016), p. 603-628**Full Text:**false**Reviewed:****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.

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.

Video driven traffic modelling in paramics

- Zhou, Hailing, Creighton, Douglas, Lim, Cheepeng, Wei, Lei, Gao, David

**Authors:**Zhou, Hailing , Creighton, Douglas , Lim, Cheepeng , Wei, Lei , Gao, David**Date:**2013**Type:**Text , Conference paper**Relation:**Proceedings - UKSim 15th International Conference on Computer Modelling and Simulation, UKSim 2013 p. 525-530**Full Text:**false**Reviewed:****Description:**With urbanization and vehicle availability, there exist many traffic problems including congestion, environmental impact and safety. In order to address these problems, we propose a video driven traffic modelling system in this paper. The system can simulate real-world traffic activities in a computer, based on traffic data recorded in videos. Video processing is employed to estimate metrics such as traffic volumes. These metrics are used to update the traffic system model, which is then simulated using the ParamicsTM traffic simulation platform. Video driven traffic modelling has widespread potential application in traffic systems, due to the convenience and reduced costs of model development and maintenance. Experiments are conducted in this paper to demonstrate the effectiveness of the proposed system. Â© 2013 IEEE.**Description:**2003011214

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.

Discrete state transition algorithm for unconstrained integer optimization problems

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

**Authors:**Zhou, Xiaojun , Gao, David , Yang, Chunhua , Gui, Weihua**Date:**2016**Type:**Text , Journal article**Relation:**Neurocomputing Vol. 173, no. (2016), p. 864-874**Full Text:****Reviewed:****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.

**Authors:**Zhou, Xiaojun , Gao, David , Yang, Chunhua , Gui, Weihua**Date:**2016**Type:**Text , Journal article**Relation:**Neurocomputing Vol. 173, no. (2016), p. 864-874**Full Text:****Reviewed:****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.

Global solutions to fractional programming problem with ratio of nonconvex functions

**Authors:**Ruan, Ning , Gao, David**Date:**2015**Type:**Text , Journal article**Relation:**Applied Mathematics and Computation Vol. 255, no. (2015), p. 66-72**Full Text:**false**Reviewed:****Description:**This paper presents a canonical dual approach for minimizing a sum of quadratic function and a ratio of nonconvex functions in Rn. By introducing a parameter, the problem is first equivalently reformed as a nonconvex polynomial minimization with elliptic constraint. It is proved that under certain conditions, the canonical dual is a concave maximization problem in R2 that exhibits no duality gap. Therefore, the global optimal solution of the primal problem can be obtained by solving the canonical dual problem. © 2014 Elsevier Inc. All rights reserved.

Mixed finite element solutions to contact problems of nonlinear Gao beam on elastic foundation

- Gao, David, Machalova, Jitka, Netuka, Horymir

**Authors:**Gao, David , Machalova, Jitka , Netuka, Horymir**Date:**2015**Type:**Text , Journal article**Relation:**Nonlinear Analysis: Real World Applications Vol. 22, no. (2015), p. 537-550**Full Text:**false**Reviewed:****Description:**This paper analyzes nonlinear contact problems of a large deformed beam on an elastic foundation. The beam model is governed by a nonlinear fourth-order differential equation developed by Gao (1996); while the elastic foundation model is assumed as Winkler's type. Based on a decomposition method, the nonlinear variational inequality problem is able to be reformed as a min-max problem of a saddle Lagrangian. Therefore, by using mixed finite element method with independent discretization-interpolations for foundation and beam elements, the nonlinear contact problem in continuous space is eventually converted as a nonlinear mixed complementarity problem, which can be solved by combination of interior-point and Newton methods. Applications are illustrated by different boundary conditions. Results show that the nonlinear Gao beam is more stiffer than the Euler-Bernoulli beam.

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.

Global optimal solutions to general sensor network localization problem

**Authors:**Ruan, Ning , Gao, David**Date:**2014**Type:**Text , Journal article**Relation:**Performance Evaluation Vol. 75-76, no. (2014), p. 1-16**Full Text:**false**Reviewed:****Description:**Sensor network localization problem is to determine the position of the sensor nodes in a network given pairwise distance measurements. Such problem can be formulated as a quartic polynomial minimization via the least squares method. This paper presents a canonical duality theory for solving this challenging problem. It is shown that the nonconvex minimization problem can be reformulated as a concave maximization dual problem over a convex set in a symmetrical matrix space, and hence can be solved efficiently by combining a general (linear or quadratic) perturbation technique with existing optimization techniques. Applications are illustrated by solving some relatively large-scale problems. Our results show that the general sensor network localization problem is not NP-hard unless its canonical dual problem has no solution in its positive definite domain. Fundamental ideas for solving general NP-hard problems are discussed. (C) 2014 Elsevier B.V. All rights reserved.

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