A trust-region algorithm combining line search filter technique for nonlinear constrained optimization

In this paper, we propose a trust-region algorithm in association with line search filter technique for solving nonlinear equality constrained programming. At current iteration, a trial step is formed as the sum of a normal step and a tangential step which is generated by trust-region subproblem and the step size is decided by interior backtracking line search together with filter methods. Then, the next iteration is determined. This is different from general trust-region methods in which the next iteration is determined by the ratio of the actual reduction to the predicted reduction. The global convergence analysis for this algorithm is presented under some reasonable assumptions and the preliminary numerical results are reported.     

An affine scaling interior trust-region method combining with nonmonotone line search filter technique for linear inequality constrained minimization

This paper proposes an affine scaling interior trust-region method in association with nonmonotone line search filter technique for solving nonlinear optimization problems subject to linear inequality constraints. Based on a Newton step which is derived from the complementarity conditions of linear inequality constrained optimization, a trust-region subproblem subject only to an ellipsoidal constraint is defined by minimizing a quadratic model with an appropriate quadratic function and scaling matrix. The nonmonotone schemes combining with trust-region strategy and line search filter technique can bring about speeding up the convergence progress in the case of high nonlinear. A new backtracking relevance condition is given which assures global convergence without using the switching condition used in the traditional line search filter technique. The fast local convergence rate of the proposed algorithm is achieved which is not depending on any external restoration procedure. The preliminary numerical experiments are reported to show effectiveness of the proposed algorithm.     

A new cubic convergent method for solving a system of nonlinear equations

We present a new cubic convergent method for solving a system of nonlinear equations. The new method can be viewed as a modified Chebyshev's method in which the difference of Jacobian matrixes replaces three order tensor. Therefore, the new method reduces the storage and computational cost. The new method possesses the local cubic convergence as well as Chebyshev's method. A rule is deduced to ensure the descent property of the search direction, and a nonmonotone line search technique is used to guarantee the global convergence. Numerical results indicate that the new method is competitive and efficient for some classical test problems.     

A spectral conjugate gradient method for solving large-scale unconstrained optimization

This paper establishes a spectral conjugate gradient method for solving unconstrained optimization problems, where the conjugate parameter and the spectral parameter satisfy a restrictive relationship. The search direction is sufficient descent without restarts in per-iteration. Moreover, this feature is independent of any line searches. Under the standard Wolfe line searches, the global convergence of the proposed method is proved when $\left|{\beta }_k\right|\le {\beta }_k^{FR}$ holds. The preliminary numerical results are presented to show effectiveness of the proposed method.     

一种新的遗传算法求解有等式约束的优化问题

针对有等式约束的优化问题,提出了一种新的遗传算法.该算法是在种群初始化、交叉、变异操作过程中使用求解参数方程的方法处理等式约束,违反不等式约束的个体用死亡罚函数进行惩罚设计出的实数编码遗传算法.数值实验结果表明,新算法性能优于现有其它算法;它不仅可以处理线性等式约束,而且还可以处理非线性等式约束,同时提高了收敛速度和解的精度,是一种通用强、高效稳健的智能算法.     

A hybrid steepest descent method for constrained convex optimization

This paper describes a hybrid steepest descent method to decrease over time any given convex cost function while keeping the optimization variables in any given convex set. The method takes advantage of the properties of hybrid systems to avoid the computation of projections or of a dual optimum. The convergence to a global optimum is analyzed using Lyapunov stability arguments. A discretized implementation and simulation results are presented and analyzed. This method is of practical interest to integrate real-time convex optimization into embedded controllers thanks to its implementation as a dynamical system, its simplicity, and its low computation cost.     

Sufficient descent conjugate gradient methods for large-scale optimization problems

Sufficient descent condition is very crucial in establishing the global convergence of nonlinear conjugate gradient method. In this paper, we modified two conjugate gradient methods such that both methods satisfy this property. Under suitable conditions, we prove the global convergence of the proposed methods. Numerical results show that the proposed methods are efficient for the given test problems.     

单纯形法的布谷鸟搜索算法及其在约束优化问题中的应用

为求解约束优化问题,针对布谷鸟搜索算法(CS)后期收敛速度慢,求解精度不高等不足,利用单纯形法局部搜索能力强的特点,提出了基于单纯形法的布谷鸟搜索算法(SMCS)。算法首先用CS算法进行全局搜索,再用单纯形法进行局部搜索。10个标准测试函数的实验结果表明,SMCS算法相对于CS算法有更强的寻优能力,再将算法用于求解减速器设计、伸缩绳设计、焊接条设计等约束优化问题。实验结果表明,CS算法和SMCS算法均能求出比其他文献更优的解,且SMCS算法求出的解更优、稳定性更强。     

A new subspace minimization conjugate gradient method based on tensor model for unconstrained optimization

ABSTRACT

A new subspace minimization conjugate gradient method based on tensor model is proposed and analysed. If the objective function is close to a quadratic, we construct a quadratic approximation model in a two-dimensional subspace to generate the search direction; otherwise, we construct a tensor model. It is remarkable that the search direction satisfies the sufficient descent property. We prove the global convergence of the proposed method under mild assumptions. Numerical comparisons are given with well-known CGOPT and CG_DESCENT and show that the proposed algorithm is very promising.     

A non-monotone inexact non-interior continuation method based on a parametric smoothing function for LWCP

This paper considers the linear weighted complementarity problem (denoted by LWCP). We introduce a parametric smoothing function which is a broad class of smoothing functions for the LWCP and enjoys some favourable properties. Based on this function, we propose a new non-interior continuation method for solving the LWCP. In general, the non-interior continuation method consists of finding an exact solution of a system of equations at each iteration, which may be cumbersome if one is solving a large-scale problem. To overcome this difficulty, our method uses an inexact Newton method to solve the corresponding linear system approximately and adopts a non-monotone line search to obtain a step size. Under suitable assumptions, we show that the proposed method is globally and locally quadratically convergent. Preliminary numerical results are also reported.     

Modified harmony search optimization for constrained design problems

Constrained optimization is a major real-world problem. Constrained optimization problems consist of an objective function subjected to both linear and nonlinear constraints. Here a constraint handling procedure based on the fitness priority-based ranking method (FPBRM) is proposed. It is embedded into a harmony search (HS) algorithm that allows it to satisfy constraints. The HS algorithm is conceptualized using the musical process of searching for a perfect state of harmony. Here, the original heuristic HS was improved by combining both improved and global-best methods along with the FPBRM. The resulting modified harmony search (MHS) was then compared with the original HS technique and other optimization methods for several test problems.     

An improved infeasible SSLE method for constrained optimization without strict complementarity

In this paper, using “working set” technique for determining the active set, a new SSLE-Type algorithm with arbitrary initial point for constrained optimization is presented. At each iteration, we first introduce a new working set based on a multiplier function, then an improved direction is obtained by three systems of linear equations with the same coefficient matrix and possess small scale, and to avoid the Maratos effect, another correction direction is yielded by a simple explicit formula. Under some mild conditions, the algorithm is proved to be globally and strongly convergent, and the superlinear or even quadratic convergence can be obtained without the strict complementarity. Finally, some interesting numerical results are reported.     

Inexact quasi-Newton global convergent method for solving constrained nonsmooth equations

In this work we introduce a new method for solving nonsmooth equations with simple constraints. The method is based on the inexact and quasi-Newton approaches with backtracking strategy. Some conditions are given that ensure global superlinear convergence to a solution of the equation. We also propose a nonmonotone algorithm scheme. Both versions of the algorithm were constructed for Lipschitz continuous equations.     

求解约束优化问题改进的水波优化算法

为提高约束优化模型的求解精度,提出一种改进的水波优化算法。设计主-从异构种群,结合ε约束处理技术使主群实现探索可行解,从群利用可行解搜寻全局最优解。为加快收敛速度和增强信息交互,主群中个体可以依概率进行个体间学习,设计水波波长函数,使其随着水波的适应度值和违反约束度及时调整。为避免早期收敛,从群采用自适应学习策略以平衡群体的探索和利用。设计随迭代次数变化的放松约束度,提高算法收敛精度。对比实验结果表明,该算法可以获得高质量的可行解。     

A poly-hybrid PSO optimization method with intelligent parameter adjustment

This paper presents results on a new hybrid optimization method which combines the best features of four traditional optimization methods together with an intelligent adjustment algorithm to speed convergence on unconstrained and constrained optimization problems. It is believed that this is the first time that such a broad array of methods has been employed to facilitate synergistic enhancement of convergence. Particle swarm optimization is based on swarm intelligence inspired by the social behavior and movement dynamics of bird flocking, fish schooling, and swarming theory. This method has been applied for structural damage identification, neural network training, and reactive power optimization. It is also believed that this is the first time an intelligent parameter adjustment algorithm has been applied to maximize the effectiveness of individual component algorithms within the hybrid method. A comprehensive sensitivity analysis of the traditional optimization methods within the hybrid group is used to demonstrate how the relationship among the design variables in a given problem can be used to adjust algorithm parameters. The new method is benchmarked using 11 classical test functions and the results show that the new method outperforms eight of the most recently published search methodologies.     

A sharp augmented Lagrangian-based method in constrained non-convex optimization

In this paper, a novel sharp Augmented Lagrangian-based global optimization method is developed for solving constrained non-convex optimization problems. The algorithm consists of outer and inner loops. At each inner iteration, the discrete gradient method is applied to minimize the sharp augmented Lagrangian function. Depending on the solution found the algorithm stops or updates the dual variables in the inner loop, or updates the upper or lower bounds by going to the outer loop. The convergence results for the proposed method are presented. The performance of the method is demonstrated using a wide range of nonlinear smooth and non-smooth constrained optimization test problems from the literature.     

An improved vector particle swarm optimization for constrained optimization problems

Increasing attention is being paid to solve constrained optimization problems (COP) frequently encountered in real-world applications. In this paper, an improved vector particle swarm optimization (IVPSO) algorithm is proposed to solve COPs. The constraint-handling technique is based on the simple constraint-preserving method. Velocity and position of each particle, as well as the corresponding changes, are all expressed as vectors in order to present the optimization procedure in a more intuitively comprehensible manner. The NVPSO algorithm [30], which uses one-dimensional search approaches to find a new feasible position on the flying trajectory of the particle when it escapes from the feasible region, has been proposed to solve COP. Experimental results showed that searching only on the flying trajectory for a feasible position influenced the diversity of the swarm and thus reduced the global search capability of the NVPSO algorithm. In order to avoid neglecting any worthy position in the feasible region and improve the optimization efficiency, a multi-dimensional search algorithm is proposed to search within a local region for a new feasible position. The local region is composed of all dimensions of the escaped particle’s parent and the current positions. Obviously, the flying trajectory of the particle is also included in this local region. The new position is not only present in the feasible region but also has a better fitness value in this local region. The performance of IVPSO is tested on 13 well-known benchmark functions. Experimental results prove that the proposed IVPSO algorithm is simple, competitive and stable.     

BSTBGA: A hybrid genetic algorithm for constrained multi-objective optimization problems

Most of the existing multi-objective genetic algorithms were developed for unconstrained problems, even though most real-world problems are constrained. Based on the boundary simulation method and trie-tree data structure, this paper proposes a hybrid genetic algorithm to solve constrained multi-objective optimization problems (CMOPs). To validate our approach, a series of constrained multi-objective optimization problems are examined, and we compare the test results with those of the well-known NSGA-II algorithm, which is representative of the state of the art in this area. The numerical experiments indicate that the proposed method can clearly simulate the Pareto front for the problems under consideration.