### On modeling and complete solutions to general fixpoint problems in multi-scale systems with applications

**Authors:**Ruan, Ning , Gao, David**Date:**2018**Type:**Text , Journal article**Relation:**Fixed Point Theory and Applications Vol. 2018, no. 1 (2018), p. 1-19**Full Text:****Reviewed:****Description:**This paper revisits the well-studied fixed point problem from a unified viewpoint of mathematical modeling and canonical duality theory, i.e., the general fixed point problem is first reformulated as a nonconvex optimization problem, its well-posedness is discussed based on the objectivity principle in continuum physics; then the canonical duality theory is applied for solving this challenging problem to obtain not only all fixed points, but also their stability properties. Applications are illustrated by problems governed by nonconvex polynomial, exponential, and logarithmic operators. This paper shows that within the framework of the canonical duality theory, there is no difference between the fixed point problems and nonconvex analysis/optimization in multidisciplinary studies.

### On SPD method for solving canonical dual problem in post buckling of large deformed elastic beam

**Authors:**Ali, Elaf , Gao, David**Date:**2018**Type:**Text , Journal article**Relation:**Communications in Mathematical Sciences Vol. 16, no. 5 (2018), p. 1225-1240**Full Text:****Reviewed:****Description:**This paper presents a new methodology and algorithm for solving post buckling problems of a large deformed elastic beam. The total potential energy of this beam is a nonconvex functional, which can be used to model both pre- and post-buckling phenomena. By using a canonical dual finite element method, a new primal-dual semi-definite programming (PD-SDP) algorithm is presented, which can be used to obtain all possible post-buckled solutions. Applications are illustrated by several numerical examples with different boundary conditions. We find that the global minimum solution of the nonconvex potential leads to a stable configuration of the buckled beam, the local maximum solution leads to the unbuckled state, and both of these two solutions are numerically stable. However, the local minimum solution leads to an unstable buckled state, which is very sensitive to axial compressive forces, thickness of beam, numerical precision, and the size of finite elements. The method and algorithm proposed in this paper can be used for solving general nonconvex variational problems in engineering and sciences.

### On topology optimization and canonical duality method

**Authors:**Gao, David**Date:**2018**Type:**Text , Journal article**Relation:**Computer Methods in Applied Mechanics and Engineering Vol. 341, no. (2018), p. 249-277**Full Text:**false**Reviewed:****Description:**Topology optimization for general materials is correctly formulated as a bi-level knapsack problem, which is considered to be NP-hard in global optimization and computer science. By using canonical duality theory (CDT) developed by the author, the linear knapsack problem can be solved analytically to obtain global optimal solution at each design iteration. Both uniqueness, existence, and NP-hardness are discussed. The novel CDT method for general topology optimization is refined and tested by both 2-D and 3-D benchmark problems. Numerical results show that without using filter and any other artificial technique, the CDT method can produce exactly 0-1 optimal density distribution with almost no checkerboard pattern. Its performance and novelty are compared with the popular SIMP and BESO approaches. Additionally, some mathematical and conceptual mistakes in literature are explicitly addressed. A brief review on the canonical duality theory for modeling multi-scale complex systems and for solving general nonconvex/discrete problems are given in Appendix. This paper demonstrates a simple truth: elegant designs come from correct model and theory. © 2018

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

**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

**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.

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

**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.

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

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

### Canonical duality for solving general nonconvex constrained problems

**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.

### 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.

### Discrete state transition algorithm for unconstrained integer optimization problems

**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 optimal trajectory in Chaos and NP-Hardness

**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

**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.

### 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.

### On unified modeling, theory, and method for solving multi-scale global optimization problems

**Authors:**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 Conference Proceedings Vol. 1776, p. 1-8**Full Text:****Reviewed:****Description:**A unified model is proposed for general optimization problems in multi-scale complex systems. Based on this model and necessary assumptions in physics, the canonical duality theory is presented in a precise way to include traditional duality theories and popular methods as special applications. Two conjectures on NP-hardness are proposed, which should play important roles for correctly understanding and efficiently solving challenging real-world problems. Applications are illustrated for both nonconvex continuous optimization and mixed integer nonlinear programming.

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

**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.

### A multiobjective state transition algorithm for single machine scheduling

**Authors:**Zhou, Xiaojun , Hanoun, Samer , Gao, David , Nahavandi, Saeid**Date:**2015**Type:**Text , Conference paper**Relation:**3rd World Congress on Global Optimization in Engineering and Science, WCGO 2013; Anhui, China; 8th-12th July 2013 Vol. 95, p. 79-88**Full Text:**false**Reviewed:****Description:**In this paper, a discrete state transition algorithm is introduced to solve a multiobjective single machine job shop scheduling problem. In the proposed approach, a non-dominated sort technique is used to select the best from a candidate state set, and a Pareto archived strategy is adopted to keep all the non-dominated solutions. Compared with the enumeration and other heuristics, experimental results have demonstrated the effectiveness of the multiobjective state transition algorithm. © Springer International Publishing Switzerland 2015.

### Advances in Global Optimization

**Authors:**Gao, David , Ruan, Ning , Xing, Wenxun**Date:**2015**Type:**Text , Conference paper**Relation:**3rd World Congress on Global Optimization in Engineering and Science, WCGO 2013; Anhui, China; 8th-12th July 2013 Vol. 95**Full Text:**false**Reviewed:****Description:**This proceedings volume addresses advances in global optimization - a multidisciplinary research field that deals with the analysis, characterization, and computation of global minima and/or maxima of nonlinear, non-convex, and nonsmooth functions in continuous or discrete forms. The volume contains selected papers from the third biannual World Congress on Global Optimization in Engineering & Science (WCGO), held in the Yellow Mountains, Anhui, China on July 8-12, 2013. The papers fall into eight topical sections: mathematical programming; combinatorial optimization; duality theory; topology optimization; variational inequalities and complementarity problems; numerical optimization; stochastic models and simulation; and complex simulation and supply chain analysis.

### Application of canonical duality theory to fixed point problem

**Authors:**Ruan, Ning , Gao, David**Date:**2015**Type:**Text , Conference paper**Relation:**3rd World Congress on Global Optimization in Engineering and Science, WCGO 2013; Anhui, China; 8th-12th July 2013 Vol. 95, p. 157-163**Full Text:**false**Reviewed:****Description:**In this paper, we study general fixed point problem. We first rewrite the original problem in the canonical framework. Then, we proposed a canonical transformation of this problem, which leads to a convex differentiable dual problem and new iteration method. An illustrative example is presented. © Springer International Publishing Switzerland 2015.