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Global minimizer of large scale stochastic rosenbrock function : canonical duality approach

**Authors:**Li, Chaojie , Gao, David**Date:**2012**Type:**Text , Conference paper**Relation:**19th International Conference on Neural Information Processing, ICONIP 2012 Vol. 7666 LNCS, p. 677-682**Full Text:****Reviewed:****Description:**Canonical duality theory for solving the well-known benchmark test problem of stochastic Rosenbrock function is explored by two canonical transformations. Global optimality criterion is analytically obtained, which shows that the stochastic disturbance of these parameters could be eliminated by a proper canonical dual transformation. Numerical simulations illustrate the canonical duality theory is potentially powerful for solving this benchmark test problem and many other challenging problems in global optimization and complex network systems. © 2012 Springer-Verlag.**Description:**2003010651

**Authors:**Li, Chaojie , Gao, David**Date:**2012**Type:**Text , Conference paper**Relation:**19th International Conference on Neural Information Processing, ICONIP 2012 Vol. 7666 LNCS, p. 677-682**Full Text:****Reviewed:****Description:**Canonical duality theory for solving the well-known benchmark test problem of stochastic Rosenbrock function is explored by two canonical transformations. Global optimality criterion is analytically obtained, which shows that the stochastic disturbance of these parameters could be eliminated by a proper canonical dual transformation. Numerical simulations illustrate the canonical duality theory is potentially powerful for solving this benchmark test problem and many other challenging problems in global optimization and complex network systems. © 2012 Springer-Verlag.**Description:**2003010651

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.

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.

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

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.

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

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.

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

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.

Impulsive synchronization of state delayed discrete complex networks with switching topology

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

**Authors:**Li, Chaojie , Gao, David , Liu, Chao**Date:**2012**Type:**Text , Conference paper**Relation:**19th International Conference on Neural Information Processing, ICONIP 2012 Vol. 7665 LNCS, p. 50-57**Full Text:****Reviewed:****Description:**In this paper, global exponential synchronization of a class of discrete delayed complex networks with switching topology is 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. © 2012 Springer-Verlag.**Description:**2003010652

**Authors:**Li, Chaojie , Gao, David , Liu, Chao**Date:**2012**Type:**Text , Conference paper**Relation:**19th International Conference on Neural Information Processing, ICONIP 2012 Vol. 7665 LNCS, p. 50-57**Full Text:****Reviewed:****Description:**In this paper, global exponential synchronization of a class of discrete delayed complex networks with switching topology is 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. © 2012 Springer-Verlag.**Description:**2003010652

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.

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

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.

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.

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:**

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 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.

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

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.

Optimization of matrix semirings for classification systems

- Gao, David, Kelarev, Andrei, Yearwood, John

**Authors:**Gao, David , Kelarev, Andrei , Yearwood, John**Date:**2011**Type:**Text , Journal article**Relation:**Bulletin of the Australian Mathematical Society Vol. 84, no. 3 (2011), p. 492-503**Full Text:****Reviewed:****Description:**The max-plus algebra is well known and has useful applications in the investigation of discrete event systems and affine equations. Structural matrix rings have been considered by many authors too. This article introduces more general structural matrix semirings, which include all matrix semirings over the max-plus algebra. We investigate properties of ideals in this construction motivated by applications to the design of centroid-based classification systems, or classifiers, as well as multiple classifiers combining several initial classifiers. The first main theorem of this paper shows that structural matrix semirings possess convenient visible generating sets for ideals. Our second main theorem uses two special sets to determine the weights of all ideals and describe all matrix ideals with the largest possible weight, which are optimal for the design of classification systems. Â© Copyright Australian Mathematical Publishing Association Inc. 2011.**Description:**2003009498

**Authors:**Gao, David , Kelarev, Andrei , Yearwood, John**Date:**2011**Type:**Text , Journal article**Relation:**Bulletin of the Australian Mathematical Society Vol. 84, no. 3 (2011), p. 492-503**Full Text:****Reviewed:****Description:**The max-plus algebra is well known and has useful applications in the investigation of discrete event systems and affine equations. Structural matrix rings have been considered by many authors too. This article introduces more general structural matrix semirings, which include all matrix semirings over the max-plus algebra. We investigate properties of ideals in this construction motivated by applications to the design of centroid-based classification systems, or classifiers, as well as multiple classifiers combining several initial classifiers. The first main theorem of this paper shows that structural matrix semirings possess convenient visible generating sets for ideals. Our second main theorem uses two special sets to determine the weights of all ideals and describe all matrix ideals with the largest possible weight, which are optimal for the design of classification systems. Â© Copyright Australian Mathematical Publishing Association Inc. 2011.**Description:**2003009498

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