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Engineering optimization problems often need to solve constrained optimization, and many methods have been proposed to solve constrained optimization problems over the past few years. Constraint-handling is one of the major concerns when solving constrained optimization problems. This paper presents a new constraint-handling method, which reform inequation constraint as equation constraint without any parameter. In addition, we adopt the penalty function method to balance objective and constraints, and solved unconstrained optimization by heuristic method named novel global harmony search algorithm (NGHS). The NGHS algorithm utilizes position updating and mutation strategy with low probability. After solving two engineering optimization problems, the results demonstrated that the constraint-handing technique is effective, and it can be an efficient alternative on solving constrained optimization problems.

The main contribution of present work is to validate the performance of Richardson iteration method for linear equations with symmetric positive definite (SPD) matrix. Meanwhile, Richardson iterati...

Nonnegative linear least square (NLLS) is an important optimization problem, which often appears in processing the physical data. In this paper, we use a recently proposed algorithm named harmony s...

Sparse solution is an active topic with a range of applications. This paper demonstrated some sparse solutions of linear complementarity problem (LCP) and absolute value equation (AVE). Possible future work is that how to compute sparse solution of LCP and AVE with the relations between LCP and AVE.

Biogeography-based optimization (BBO) is a competitive population optimization algorithm based on biogeography theory with inherently insufficient exploration capability and slow convergence speed. To overcome limitations, we propose an improved variant of BBO, named PRBBO, for solving global optimization problems. In PRBBO, a hybrid migration operator with random ring topology, a modified mutation operator, and a self-adaptive Powell's method are rational integrated together. The hybrid migration operator with random ring topology, denoted as RMO, is created by using local ring topology to replace global topology, which can avoid the asymmetrical migration operation and enhance potential population diversity. The self-adaptive Powell's method is amended by using self-adaptive parameters for suiting evolution process to enhance solution precision quickly. Extensive experimental tests are carried out on 24 benchmark functions to show effectiveness of the proposed algorithm. Simulation results were compared with original BBO, ABC, DE, other variants of the BBO, and other state-of-the-art evolutionary algorithms. Finally, the effectiveness of operators on the performance of PRBBO is also discussed.

0-1 optimization problem plays an important role in operational research. In this paper, we use a recently proposed algorithm named harmony search with teaching-learning (HSTL) strategy which derived from Teaching-Learning-Based Optimization (TLBO) for solving. Four strategies (Harmony memory consideration, teaching-learning strategy, local pitch adjusting and random mutation) are employed to improve the performance of HS algorithm. Numerical results demonstrated very good computational performance.

This paper investigate the NP-hard absolute value equation (AVE) Ax - |x| = b, where A is an arbitrary square matrix whose singular values exceed one. The significance of the absolute value equation arises from the fact that linear programs, quadratic programs, bimatrix games and other problems can all be reduced to the linear complementarity problem that in turn is equivalent to the absolute value equation. This paper present a new smoothing function to the AVE. Based on this function, a smoothing Newton method is proposed for solving the AVE under the less stringent condition that the singular values of A exceed 1. The global convergence of the method is established under appropriate conditions. Preliminary numerical results indicate that this method is promising.

We investigate the NP-hard absolute value equation (AVE) Ax − |x|=b, where A is an arbitrary square matrix whose singular values exceed one. The significance of the absolute value equations arises from the fact that linear programs, quadratic programs, bimatrix games and other problems can all be reduced to the linear complementarity problem that in turn is equivalent to the absolute value equation. In this paper, after a lower uniform smooth approximation function of absolute value function is established, absolute value equation is transformed into smooth optimization problem, and solved by smoothing Newton method. The convergence of method is shown, and the preliminary numerical experiments are given.

An iterative method for absolute value equation (AVE) Ax – ∣x∣ = b, where A is an arbitrary square matrix whose singular values exceed one, is presented. The method converges to solution of AVE after finite iterations. Finally, this method is applied to solve two-point boundary value problem. Numerical experiments show that this method has advantages of less number of iterations and high precision.

We study a general holographic superconductor model in four dimensional AdS back hole background. We explore the properties of the phase transitions in the scalar condensation and conclude the correspondence of a lower critical temperature with a deeper condensation gap holds with various scalar field mass above zero. We also examine the effects of the model parameters on the phase transition and find that this model allows first order phase transitions to occur with non-negative scalar mass when the model parameter is above a threshold value. Finally, we study the effects of the backreaction and scalar mass on the threshold model parameters.

In this paper a new method is proposed for computing Nash Equilibrium of bimatrix games. Firstly, bimatrix games were reformed as non-monotonic linear complementarity problem (LCP). Then, the non-monotonic LCP is formulated as absolute value equation (AVE), and solved by recently proposed algorithm named novel global harmony search (NGHS). Numerical experiment shows that the NGHS algorithm can achieve as many as possible equilibrium pair of strategies.

In this paper, we use a recently proposed algorithm named hybrid Differential Evolution with Biogeography-based Optimization (DEBBO) for solving NP-hard Absolute Value Equation (AVE) Ax− |x| = b, where A is an arbitrary square matrix whose singular values exceed one. DEBBO accepts good qualities of DE and BBO, and numerical experiments show that the DEBBO algorithm can avoid precocious phenomena to a certain extent and it has higher rate of convergence than the other BBOvariants algorithms.

Nonnegative Linear Least Squares (NLLS) is an important optimization problems, which often appears in processing the physical data. In this paper, we use a recently proposed algorithm named Novel Global Harmony Search (NGHS) for solving NLLS problems. Numerical results are reported which demonstrate very good computational performance.

Harmony Search (HS) and Teaching-Learning-Based Optimization (TLBO) as new swarm intelligent optimization algorithms have received much attention in recent years. Both of them have shown outstanding performance for solving NP-Hard optimization problems. However, they also suffer dramatic performance degradation for some complex high-dimensional optimization problems. Through a lot of experiments, we find that the HS and TLBO have strong complementarity each other. The HS has strong global exploration power but low convergence speed. Reversely, the TLBO has much fast convergence speed but it is easily trapped into local search. In this work, we propose a hybrid search algorithm named HSTLBO that merges the two algorithms together for synergistically solving complex optimization problems using a self-adaptive selection strategy. In the HSTLBO, both HS and TLBO are modified with the aim of balancing the global exploration and exploitation abilities, where the HS aims mainly to explore the unknown regions and the TLBO aims to rapidly exploit high-precision solutions in the known regions. Our experimental results demonstrate better performance and faster speed than five state-of-the-art HS variants and show better exploration power than five good TLBO variants with similar run time, which illustrates that our method is promising in solving complex high-dimensional optimization problems. The experiment on portfolio optimization problems also demonstrate that the HSTLBO is effective in solving complex read-world application.

This paper presents an improved differential evolution (DE) based on orthogonal crossover to solve complex absolute value equation problems. Absolute value equation problems are NP-hard multimodal problems. Differential evolution (DE) is one of fast and robust, accurate evolutionary algorithms for global optimization problems and solution of absolute value equation problems. In this paper, orthogonal based differential evolution, which is proposed through incorporation differential evolution with orthogonal crossover, is excogitated to solve three absolute value equation problems. Several experiments are conducted on absolute value equation problems, the solution accuracy, convergence rate and robustness of ODE are compared with other state-ofthe-art orthogonal based algorithms. The comparison results show that ODE is a fast, effective, efficient and robust algorithm for solving absolute value equation problems.

Harmony search (HS) algorithm is an emerging population-based metaheuristic algorithm, which is inspired by the music improvisation process. The HS method has been developed rapidly and applied widely during the past decade. In this paper, an improved global harmony search algorithm, named harmony search based on teaching-learning (HSTL), is presented for high dimension complex optimization problems. In HSTL algorithm, four strategies (harmony memory consideration, teaching-learning strategy, local pitch adjusting, and random mutation) are employed to maintain the proper balance between convergence and population diversity, and dynamic strategy is adopted to change the parameters. The proposed HSTL algorithm is investigated and compared with three other state-of-the-art HS optimization algorithms. Furthermore, to demonstrate the robustness and convergence, the success rate and convergence analysis is also studied. The experimental results of 31 complex benchmark functions demonstrate that the HSTL method has strong convergence and robustness and has better balance capacity of space exploration and local exploitation on high dimension complex optimization problems.

A new method is proposed for the linear complementarity problem (LCP). Firstly we formulate the LCP into a multiobjective optimization problem (MOP), and study the relations between the efficient solution of MOP and optimal solution of LCP. Based on the efficient solution of MOP, we define zero-efficient solution. Then we indicate that zero-efficient solution of the MOP is also the solution to the LCP. Finally some standard LCP examples are respectively transformed into MOP and solved by minimax method. Numerical results indicate that the proposed method is effective.

This text studies a kind of obstacle problem. Combining with difference principle, we transform the original problem into monotone linear complementarity problem, and propose a novel method called potential-reduction interior point algorithm for monotone linear complementarity problem. We establish global and finite convergence of the new method. The reliability and efficiency of the algorithm is demonstrated by the numerical experiments of standard linear complementarity problems and the examples of obstacle problem with free boundary.

The construction example of algebra system is to verify the existence of a complex algebra system, and it is a NP-hard problem. In this paper, to solve this kind of problems, firstly, a mathematical optimization model for construction example of algebra system is established. Secondly, an improved harmony search algorithm based on NGHS algorithm (INGHS) is proposed to find as more solutions as possible for the optimization model; in the proposed INGHS algorithm, to achieve the balance between exploration power and exploitation power in the search process, a global best strategy and parameters dynamic adjustment method are present. Finally, nine construction examples of algebra system are used to evaluate the optimization model and performance of INGHS. The experimental results show that the proposed algorithm has strong performance for solving complex construction example problems of algebra system.

Low-rank matrix recovery (LRMR) has been becoming an increasingly popular technique for analyzing data with missing entries, gross corruptions, and outliers. As a significant component of LRMR, the model of low-rank representation (LRR) seeks the lowest-rank representation among all samples and it is robust for recovering subspace structures. This paper attempts to solve the problem of LRR with partially observed entries. Firstly, we construct a nonconvex minimization by taking the low rankness, robustness, and incompletion into consideration. Then we employ the technique of augmented Lagrange multipliers to solve the proposed program. Finally, experimental results on synthetic and real-world datasets validate the feasibility and effectiveness of the proposed method.

Biogeography-based optimization (BBO) is a new biogeography inspired, population-based algorithm, which mainly uses migration operator to share information among solutions. Similar to crossover operator in genetic algorithm, migration operator is a probabilistic operator and only generates the vertex of a hyperrectangle defined by the emigration and immigration vectors. Therefore, the exploration ability of BBO may be limited. Orthogonal crossover operator with quantization technique (QOX) is based on orthogonal design and can generate representative solution in solution space. In this paper, a BBO variant is presented through embedding the QOX operator in BBO algorithm. Additionally, a modified migration equation is used to improve the population diversity. Several experiments are conducted on 23 benchmark functions. Experimental results show that the proposed algorithm is capable of locating the optimal or closed-to-optimal solution. Comparisons with other variants of BBO algorithms and state-of-the-art orthogonal-based evolutionary algorithms demonstrate that our proposed algorithm possesses faster global convergence rate, high-precision solution, and stronger robustness. Finally, the analysis result of the performance of QOX indicates that QOX plays a key role in the proposed algorithm.

Two classical first order iteration methods, Richardson iteration and HSS iteration for linear systems with positive definite matrix, are demonstrated. Theoretical analyses and computational results show that the HSS iteration has the advantages of fast convergence speed, high computation efficiency, and without requirement of symmetry.

In this paper, an upper uniform smooth approximation function of absolute value function is proposed, and some properties of uniform smooth approximation function are studied. Then, absolute value equation (AVE), Ax - |x| = b, where A is a square matrix whose singular values exceed one, is transformed into smooth optimization problem by using the upper uniform smooth approximation function, and solved by quasi-Newton method. Numerical results in solving given AVE problems demonstrated that our algorithm is valid and superior to lower uniform smooth approximation function.

Aiming at the existing problems of support vector machine ensemble, such as strong randomicity, larger scale of training subsets size and high complexity of ensemble classifier, this paper put forward a novel SVM ensemble construction method based on clustering analysis. Firstly, the samples are clustered into several clusters according to their distribution with rival penalty competitive learning algorithm(RPCL). Then a small quantity of representative instances are chosen as training sets and training SVM that adopt self-perturbation in population convergence speed. Finally Ensemble improvement SVM is constructed by relative majority voting. Man-made data are used to test C_PSOSVM. Experiment result illustrate that the algorithm can improve ensemble SVM classification precision, reducing time-space complexity compared with Bagging, Adaboost.

In last decades, there has been much effort on the solution and the analysis of the nonlinear complementarity problem (NCP) by reformulating NCP as an unconstrained minimization. In this paper, we propose a new method for the NCP under the condition that the solution exists. We formulate the NCP into a multiobjective optimization problem (MOP), and define zero-efficient solution of MOP. Then we indicate that zero-efficient solution of the MOP is also the solution to the NCP. Finally we use minimax method for the MOP. Numerical results indicate that the proposed method is promising.

A feasible interior point method is proposed for solving the NP-hard absolute value equation (AVE) when the singular values of A exceed one. We formulate the NP-hard AVE as linear complementary problem, and prove that the solution to AVE is existent and unique under suitable assumptions. Then we present a feasible interior point algorithm for AVE based on the Newton direction and centering direction. We show that this algorithm has the polynomial complexity. Preliminary numerical results show that this method is promising.

Modern interior point methods originated from an algorithm introduced by Karmarkar in 1984 for linear programming. In the years since then, algorithms and software for linear programming have become quite popular, while extensions to more general classes of problems, such as convex quadratic programming, linear complementarity problem, semi-definite programming, second order cone programming and nonconvex and nonlinear problems, have reached varying levels of maturity. In this paper we review the interior point algorithms and applications in some optimization problems, such as linear programming, linear complementarity problem, semi-definite programming and some convex programming. Combining with the current studies, we conclude that " applications of interior point algorithms and kernel function-based interior point algorithms" will be the research focuses in the future.

An iterative method for solving a class of nonnegative linear least squares problems is presented. Firstly, nonnegative least squares problem is transformed into monotone linear complementarity problem. Then we present an iterative algorithm for monotone linear complementarity problem based on the fixed-point principle. We prove that this method converges to optimal solution of original problem after finite iterations. At last, we give some numerical examples to indicate that the method is feasible and effective.

We present a new method for solving bimatrix games problems. Firstly, bimatrix games problem is transformed into general linear complementarity problem. Then we apply mixed integer linear programming method to linear complementarity problem. At last, we give some numerical examples to indicate that the method is feasible and effective to bimatrix games problems.