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3Fang, Shucherng
3Ruan, Ning
2Latorre, Vittorio
2Lin, Gang-Xuan
2Sheu, Ruey-Lin
2Xing, Wenxun
2Yang, Chunhua
2Zhou, Xiaojun
2Zhu, Jinghao
1Billups, Stephen
1Chen, Yi
1Easterling, David
1Huang, Tingwen
1Jin, Zhong
1Li, Chuandong
1Morales-Silva, Daniel
1Thacker, William
1Wang, Chao
1Watson, Layne

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13Global optimization
60802 Computation Theory and Mathematics
6Canonical duality theory
3Polynomial optimization
2Canonical duality theories
2Chaos
2Computational complexity
2Double well potential
2Dual problem
2Global minimizers
2Global optimal solutions
2Non-convex quadratic programming
2Optimization
10906 Electrical and Electronic Engineering
10913 Mechanical Engineering
1Algorithm VTDIRECT
1Anticipating synchronization
1Augmented Lagrangian

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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.

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.

On modeling and global solutions for d.c. optimization problems by canonical duality theory

**Authors:**Jin, Zhong , Gao, David**Date:**2017**Type:**Text , Journal article**Relation:**Applied Mathematics and Computation Vol. 296, no. (2017), p. 168-181**Full Text:****Reviewed:****Description:**This paper presents a canonical d.c. (difference of canonical and convex functions) programming problem, which can be used to model general global optimization problems in complex systems. It shows that by using the canonical duality theory, a large class of nonconvex minimization problems can be equivalently converted to a unified concave maximization problem over a convex domain, which can be solved easily under certain conditions. Additionally, a detailed proof for triality theory is provided, which can be used to identify local extremal solutions. Applications are illustrated and open problems are presented.**Description:**This paper presents a canonical d.c. (difference of canonical and convex functions) programming problem, which can be used to model general global optimization problems in complex systems. It shows that by using the canonical duality theory, a large class of nonconvex minimization problems can be equivalently converted to a unified concave maximization problem over a convex domain, which can be solved easily under certain conditions. Additionally, a detailed proof for triality theory is provided, which can be used to identify local extremal solutions. Applications are illustrated and open problems are presented. © 2016 Elsevier Inc.

**Authors:**Jin, Zhong , Gao, David**Date:**2017**Type:**Text , Journal article**Relation:**Applied Mathematics and Computation Vol. 296, no. (2017), p. 168-181**Full Text:****Reviewed:****Description:**This paper presents a canonical d.c. (difference of canonical and convex functions) programming problem, which can be used to model general global optimization problems in complex systems. It shows that by using the canonical duality theory, a large class of nonconvex minimization problems can be equivalently converted to a unified concave maximization problem over a convex domain, which can be solved easily under certain conditions. Additionally, a detailed proof for triality theory is provided, which can be used to identify local extremal solutions. Applications are illustrated and open problems are presented.**Description:**This paper presents a canonical d.c. (difference of canonical and convex functions) programming problem, which can be used to model general global optimization problems in complex systems. It shows that by using the canonical duality theory, a large class of nonconvex minimization problems can be equivalently converted to a unified concave maximization problem over a convex domain, which can be solved easily under certain conditions. Additionally, a detailed proof for triality theory is provided, which can be used to identify local extremal solutions. Applications are illustrated and open problems are presented. © 2016 Elsevier Inc.

Canonical duality for solving general nonconvex constrained problems

- Latorre, Vittorio, Gao, David

**Authors:**Latorre, Vittorio , Gao, David**Date:**2016**Type:**Text , Journal article**Relation:**Optimization Letters Vol. 10, no. 8 (2016), p. 1763-1779**Full Text:****Reviewed:****Description:**This paper presents a canonical duality theory for solving a general nonconvex constrained optimization problem within a unified framework to cover Lagrange multiplier method and KKT theory. It is proved that if both target function and constraints possess certain patterns necessary for modeling real systems, a perfect dual problem (without duality gap) can be obtained in a unified form with global optimality conditions provided.While the popular augmented Lagrangian method may produce more difficult nonconvex problems due to the nonlinearity of constraints. Some fundamental concepts such as the objectivity and Lagrangian in nonlinear programming are addressed.

**Authors:**Latorre, Vittorio , Gao, David**Date:**2016**Type:**Text , Journal article**Relation:**Optimization Letters Vol. 10, no. 8 (2016), p. 1763-1779**Full Text:****Reviewed:****Description:**This paper presents a canonical duality theory for solving a general nonconvex constrained optimization problem within a unified framework to cover Lagrange multiplier method and KKT theory. It is proved that if both target function and constraints possess certain patterns necessary for modeling real systems, a perfect dual problem (without duality gap) can be obtained in a unified form with global optimality conditions provided.While the popular augmented Lagrangian method may produce more difficult nonconvex problems due to the nonlinearity of constraints. Some fundamental concepts such as the objectivity and Lagrangian in nonlinear programming are addressed.

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.

Global solutions to a class of CEC benchmark constrained optimization problems

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

**Authors:**Zhou, Xiaojun , Gao, David , Yang, Chunhua**Date:**2016**Type:**Text , Journal article**Relation:**Optimization Letters Vol. 10, no. 3 (2016), p. 457-472**Full Text:****Reviewed:****Description:**This paper aims to solve a class of CEC benchmark constrained optimization problems that have been widely studied by nature-inspired optimization algorithms. Based on canonical duality theory, these challenging problems can be reformulated as a unified canonical dual problem over a convex set, which can be solved deterministically to obtain global optimal solutions in polynomial time. Applications are illustrated by some well-known CEC benchmark problems, and comparisons with other methods have demonstrated the effectiveness of the proposed approach. © 2014, Springer-Verlag Berlin Heidelberg.

**Authors:**Zhou, Xiaojun , Gao, David , Yang, Chunhua**Date:**2016**Type:**Text , Journal article**Relation:**Optimization Letters Vol. 10, no. 3 (2016), p. 457-472**Full Text:****Reviewed:****Description:**This paper aims to solve a class of CEC benchmark constrained optimization problems that have been widely studied by nature-inspired optimization algorithms. Based on canonical duality theory, these challenging problems can be reformulated as a unified canonical dual problem over a convex set, which can be solved deterministically to obtain global optimal solutions in polynomial time. Applications are illustrated by some well-known CEC benchmark problems, and comparisons with other methods have demonstrated the effectiveness of the proposed approach. © 2014, Springer-Verlag Berlin Heidelberg.

Global solutions to nonconvex optimization of 4th-order polynomial and log-sum-exp functions

**Authors:**Chen, Yi , Gao, David**Date:**2016**Type:**Text , Journal article**Relation:**Journal of Global Optimization Vol. 64, no. 3 (2016), p. 417-431**Full Text:****Reviewed:****Description:**This paper presents a canonical dual approach for solving a nonconvex global optimization problem governed by a sum of 4th-order polynomial and a log-sum-exp function. Such a problem arises extensively in engineering and sciences. Based on the canonical dualityâ€“triality theory, this nonconvex problem is transformed to an equivalent dual problem, which can be solved easily under certain conditions. We proved that both global minimizer and the biggest local extrema of the primal problem can be obtained analytically from the canonical dual solutions. As two special cases, a quartic polynomial minimization and a minimax problem are discussed. Existence conditions are derived, which can be used to classify easy and relative hard instances. Applications are illustrated by several nonconvex and nonsmooth examples. © 2014, Springer Science+Business Media New York.

**Authors:**Chen, Yi , Gao, David**Date:**2016**Type:**Text , Journal article**Relation:**Journal of Global Optimization Vol. 64, no. 3 (2016), p. 417-431**Full Text:****Reviewed:****Description:**This paper presents a canonical dual approach for solving a nonconvex global optimization problem governed by a sum of 4th-order polynomial and a log-sum-exp function. Such a problem arises extensively in engineering and sciences. Based on the canonical dualityâ€“triality theory, this nonconvex problem is transformed to an equivalent dual problem, which can be solved easily under certain conditions. We proved that both global minimizer and the biggest local extrema of the primal problem can be obtained analytically from the canonical dual solutions. As two special cases, a quartic polynomial minimization and a minimax problem are discussed. Existence conditions are derived, which can be used to classify easy and relative hard instances. Applications are illustrated by several nonconvex and nonsmooth examples. © 2014, Springer Science+Business Media New York.

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.

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.

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.

- Gao, David, Watson, Layne, Easterling, David, Thacker, William, Billups, Stephen

**Authors:**Gao, David , Watson, Layne , Easterling, David , Thacker, William , Billups, Stephen**Date:**2013**Type:**Text , Journal article**Relation:**Optimization Methods and Software Vol. 28, no. 2 (2013), p. 313-326**Full Text:**false**Reviewed:****Description:**This paper presents a massively parallel global deterministic direct search method (VTDIRECT) for solving nonconvex quadratic minimization problems with either box or1 integer constraints. Using the canonical dual transformation, these well-known NP-hard problems can be reformulated as perfect dual stationary problems (with zero duality gap). Under certain conditions, these dual problems are equivalent to smooth concave maximization over a convex feasible space. Based on a perturbation method proposed by Gao, the integer programming problem is shown to be equivalent to a continuous unconstrained Lipschitzian global optimization problem. The parallel algorithm VTDIRECT is then applied to solve these dual problems to obtain global minimizers. Parallel performance results for several nonconvex quadratic integer programming problems are reported. © 2013 Copyright Taylor and Francis Group, LLC.**Description:**2003010580

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.

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.

- Yuan, Y. B., Fang, Shucherng, Gao, David

**Authors:**Yuan, Y. B. , Fang, Shucherng , Gao, David**Date:**2012**Type:**Text , Journal article**Relation:**Journal of Global Optimization Vol. 52, no. 2 (2012), p. 195-209**Full Text:**false**Reviewed:****Description:**This paper studies the canonical duality theory for solving a class of quadri- nomial minimization problems subject to one general quadratic constraint. It is shown that the nonconvex primal problem in Rn can be converted into a concave maximization dual problem over a convex set in R2 , such that the problem can be solved more efficiently. The existence and uniqueness theorems of global minimizers are provided using the triality theory. Examples are given to illustrate the results obtained. © 2011 Springer Science+Business Media, LLC.

Global optimization over a box via canonical dual function

- Zhu, Jinghao, Wang, Chao, Gao, David

**Authors:**Zhu, Jinghao , Wang, Chao , Gao, David**Date:**2011**Type:**Text , Journal article**Relation:**Journal of Computational and Applied Mathematics Vol. 235, no. 5 (January 2011), p. 1141-1147**Full Text:**false**Reviewed:****Description:**In this paper, we study global concave optimization by the canonical dual function. A differential flow on the dual feasible space is introduced. We show that the flow reaches a global minimizer of the concave function over a box. An example is illustrated.

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

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:**

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