Enhancing Evolutionary Algorithms With Pattern Mining for Sparse Large-Scale Multi-Objective Optimization Problems

Sparse large-scale multi-objective optimization problems(SLMOPs)are common in science and engineering.However,the large-scale problem represents the high dimensionality of the decision space,requiring algorithms to traverse vast expanse with limited computational resources.Furthermore,in the context of sparse,most variables in Pareto optimal solutions are zero,making it difficult for algorithms to identify non-zero variables efficiently.This paper is dedicated to addressing the challenges posed by SLMOPs.To start,we introduce innovative objective functions customized to mine maximum and minimum candidate sets.This substantial enhancement dramatically improves the efficacy of frequent pattern mining.In this way,selecting candidate sets is no longer based on the quantity of nonzero variables they contain but on a higher proportion of nonzero variables within specific dimensions.Additionally,we unveil a novel approach to association rule mining,which delves into the intricate relationships between non-zero variables.This novel methodology aids in identifying sparse distributions that can potentially expedite reductions in the objective function value.We extensively tested our algorithm across eight benchmark problems and four real-world SLMOPs.The results demonstrate that our approach achieves competitive solutions across various challenges.

A Two-Layer Encoding Learning Swarm Optimizer Based on Frequent Itemsets for Sparse Large-Scale Multi-Objective Optimization

Traditional large-scale multi-objective optimization algorithms(LSMOEAs)encounter difficulties when dealing with sparse large-scale multi-objective optimization problems(SLM-OPs)where most decision variables are zero.As a result,many algorithms use a two-layer encoding approach to optimize binary variable Mask and real variable Dec separately.Nevertheless,existing optimizers often focus on locating non-zero variable posi-tions to optimize the binary variables Mask.However,approxi-mating the sparse distribution of real Pareto optimal solutions does not necessarily mean that the objective function is optimized.In data mining,it is common to mine frequent itemsets appear-ing together in a dataset to reveal the correlation between data.Inspired by this,we propose a novel two-layer encoding learning swarm optimizer based on frequent itemsets(TELSO)to address these SLMOPs.TELSO mined the frequent terms of multiple particles with better target values to find mask combinations that can obtain better objective values for fast convergence.Experi-mental results on five real-world problems and eight benchmark sets demonstrate that TELSO outperforms existing state-of-the-art sparse large-scale multi-objective evolutionary algorithms(SLMOEAs)in terms of performance and convergence speed.

A GLOBALLY AND SUPERLINEARLY CONVERGENT TRUST REGION METHOD FOR LC^1 OPTIMIZATION PROBLEMS

A new trust region algorithm for solving convex LC 1 optimization problem is presented.It is proved that the algorithm is globally convergent and the rate of convergence is superlinear under some reasonable assumptions.

Global Optimization for Solving Linear Non-Quadratic Optimal Control Problems

This paper presents a global optimization approach to solving linear non-quadratic optimal control problems. The main work is to construct a differential flow for finding a global minimizer of the Hamiltonian function over a Euclid space. With the Pontryagin principle, the optimal control is characterized by a function of the adjoint variable and is obtained by solving a Hamiltonian differential boundary value problem. For computing an optimal control, an algorithm for numerical practice is given with the description of an example.

GLOBAL OPTIMIZATION OF PUMP CONFIGURATION PROBLEM USING EXTENDED CROWDING GENETIC ALGORITHM

An extended crowding genetic algorithm (ECGA) is introduced for solvingoptimal pump configuration problem, which was presented by T. Westerlund in 1994. This problem hasbeen found to be non-convex, and the objective function contained several local optima and globaloptimality could not be ensured by all the traditional MINLP optimization method. The concepts ofspecies conserving and composite encoding are introduced to crowding genetic algorithm (CGA) formaintain the diversity of population more effectively and coping with the continuous and/or discretevariables in MINLP problem. The solution of three-levels pump configuration got from DICOPT++software (OA algorithm) is also given. By comparing with the solutions obtained from DICOPT++, ECPmethod, and MIN-MIN method, the ECGA algorithm proved to be very effective in finding the globaloptimal solution of multi-levels pump configuration via using the problem-specific information.

RECURRENT NEURAL NETWORK MODEL BASED ON PROJECTIVE OPERATOR AND ITS APPLICATION TO OPTIMIZATION PROBLEMS

The recurrent neural network （RNN） model based on projective operator was studied. Different from the former study, the value region of projective operator in the neural network in this paper is a general closed convex subset of n-dimensional Euclidean space and it is not a compact convex set in general, that is, the value region of projective operator is probably unbounded. It was proved that the network has a global solution and its solution trajectory converges to some equilibrium set whenever objective function satisfies some conditions. After that, the model was applied to continuously differentiable optimization and nonlinear or implicit complementarity problems. In addition, simulation experiments confirm the efficiency of the RNN.

A SPARSE SUBSPACE TRUNCATED NEWTON METHOD FOR LARGE-SCALE BOUND CONSTRAINED NONLINEAR OPTIMIZATION

In this paper we report a sparse truncated Newton algorithm for handling large-scale simple bound nonlinear constrained minimixation problem. The truncated Newton method is used to update the variables with indices outside of the active set, while the projected gradient method is used to update the active variables. At each iterative level, the search direction consists of three parts, one of which is a subspace truncated Newton direction, the other two are subspace gradient and modified gradient directions. The subspace truncated Newton direction is obtained by solving a sparse system of linear equations. The global convergence and quadratic convergence rate of the algorithm are proved and some numerical tests are given.

Hybridizing grey wolf optimization with differential evolution for global optimization and test scheduling for 3D stacked SoC

A new meta-heuristic method is proposed to enhance current meta-heuristic methods for global optimization and test scheduling for three-dimensional （3D） stacked system-on-chip （SoC） by hybridizing grey wolf optimization with differential evo- lution （HGWO）. Because basic grey wolf optimization （GWO） is easy to fall into stagnation when it carries out the operation of at- tacking prey, and differential evolution （DE） is integrated into GWO to update the previous best position of grey wolf Alpha, Beta and Delta, in order to force GWO to jump out of the stagnation with DE＇s strong searching ability. The proposed algorithm can accele- rate the convergence speed of GWO and improve its performance. Twenty-three well-known benchmark functions and an NP hard problem of test scheduling for 3D SoC are employed to verify the performance of the proposed algorithm. Experimental results show the superior performance of the proposed algorithm for exploiting the optimum and it has advantages in terms of exploration.

Homotopy Continuous Method for Weak Efficient Solution of Multiobjective Optimization Problem with Feasible Set Unbounded Condition

In this paper, we propose a homotopy continuous method (HCM) for solving a weak efficient solution of multiobjective optimization problem (MOP) with feasible set unbounded condition, which is arising in Economical Distributions, Engineering Decisions, Resource Allocations and other field of mathematical economics and engineering problems. Under the suitable assumption, it is proved to globally converge to a weak efficient solution of (MOP), if its x-branch has no weak infinite solution.

A Filled Function with Adjustable Parameters for Unconstrained Global Optimization

A filled function with adjustable parameters is suggested in this paper for finding a global minimum point of a general class of nonlinear programming problems with a bounded and closed domain. This function has two adjustable parameters. We will discuss the properties of the proposed filled function. Conditions on this function and on the values of parameters are given so that the constructed function has the desired properties of traditional filled function.

GLOBAL CONVERGENCE OF TRUST REGION ALGORITHM FOR EQUALITY AND BOUND CONSTRAINED NONLINEAR OPTIMIZATION

This paper presents a trust region two phase model algorithm for solving the equality and bound constrained nonlinear optimization problem. A concept of substationary point is given. Under suitable assumptions,the global convergence of this algorithm is proved without assuming the linear independence of the gradient of active constraints. A numerical example is also presented.

Numerical Approach of Network Problems in Optimal Mass Transportation

In this paper, we focus on the theoretical and numerical aspects of network problems. For an illustration, we consider the urban traffic problems. And our effort is concentrated on the numerical questions to locate the optimal network in a given domain (for example a town). Mainly, our aim is to find the network so as the distance between the population position and the network is minimized. Another problem that we are interested is to give an numerical approach of the Monge and Kantorovitch problems. In the literature, many formulations (see for example [1-4]) have not yet practical applications which deal with the permutation of points. Let us mention interesting numerical works due to E. Oudet begun since at least in 2002. He used genetic algorithms to identify optimal network (see [5]). In this paper we introduce a new reformulation of the problem by introducing permutations . And some examples, based on realistic scenarios, are solved.

A New Searching Strategy for the Lost Plane Based on RBF Neural Network Model and Global Optimization Model

In this paper, we construct two models for the searching task for a lost plane. Model 1 determines the searching area. We predict the trajectory of floats generated after the disintegration of the plane by using RBF neural network model, and then determine the searching area according to the trajectory. With the pass of time, the searching area will also be constantly moving along the trajectory. Model 2 develops a maritime search plan to achieve the purpose of completing the search in the shortest time. We optimize the searching time and transform the problem into the 0-1 knapsack problem. Solving this problem by improved genetic algorithm, we can get the shortest searching time and the best choice for the search power.

Orthogonal genetic algorithm for solving quadratic bilevel programming problems

A quadratic bilevel programming problem is transformed into a single level complementarity slackness problem by applying Karush-Kuhn-Tucker（KKT） conditions.To cope with the complementarity constraints,a binary encoding scheme is adopted for KKT multipliers,and then the complementarity slackness problem is simplified to successive quadratic programming problems,which can be solved by many algorithms available.Based on 0-1 binary encoding,an orthogonal genetic algorithm,in which the orthogonal experimental design with both two-level orthogonal array and factor analysis is used as crossover operator,is proposed.Numerical experiments on 10 benchmark examples show that the orthogonal genetic algorithm can find global optimal solutions of quadratic bilevel programming problems with high accuracy in a small number of iterations.

Enhanced Butterfly Optimization Algorithm for Large-Scale Optimization Problems

To solve large-scale optimization problems,Fragrance coefficient and variant Particle Swarm local search Butterfly Optimization Algorithm(FPSBOA)is proposed.In the position update stage of Butterfly Optimization Algorithm(BOA),the fragrance coefficient is designed to balance the exploration and exploitation of BOA.The variant particle swarm local search strategy is proposed to improve the local search ability of the current optimal butterfly and prevent the algorithm from falling into local optimality.192000-dimensional functions and 201000-dimensional CEC 2010 large-scale functions are used to verify FPSBOA for complex large-scale optimization problems.The experimental results are statistically analyzed by Friedman test and Wilcoxon rank-sum test.All attained results demonstrated that FPSBOA can better solve more challenging scientific and industrial real-world problems with thousands of variables.Finally,four mechanical engineering problems and one ten-dimensional process synthesis and design problem are applied to FPSBOA,which shows FPSBOA has the feasibility and effectiveness in real-world application problems.

Discrete Differential Evolution for Mixed Discrete Non-Linear Problems

Differential evolution （DE） is a global optimizer for continuous design variables. To enhance DE, it is necessary to handle discrete design variables. In this paper, a discrete differential evolution （DDE） algorithm is proposed to handle discrete design variables The proposed DDE is based on the DE/l/rand/bin method. In the proposed DDE, the mutation ratio is regarded as the exchange probability, and thus, no modifications of DE/l/rand/bin are required. In addition, in order to maintain diversity through the search process, we initialize all search points. By introducing the initialization of all search points, global or quasi-optimum solution can be found. We validate the proposed DDE by applying it to several benchmark problems.

CONVEXIFICATION AND CONCAVIFICATION METHODS FOR SOME GLOBAL OPTIMIZATION PROBLEMS

In this paper, firstly, we propose several convexification and concavification transformations to convert a strictly monotone function into a convex or concave function, then we propose several convexification and concavification transformations to convert a non-convex and non-concave objective function into a convex or concave function in the programming problems with convex or concave constraint functions, and propose several convexification and concavification transformations to convert a non-monotone objective function into a convex or concave function in some programming problems with strictly monotone constraint functions. Finally, we prove that the original programming problem can be converted into an equivalent concave minimization problem, or reverse convex programming problem or canonical D.C. programming problem. Then the global optimal solution of the original problem can be obtained by solving the converted concave minimization problem, or reverse convex programming problem or canonical D.C. programming problem using the existing algorithms about them.

Dual-Stage Hybrid Learning Particle Swarm Optimization Algorithm for Global Optimization Problems

Particle swarm optimization(PSO)is a type of swarm intelligence algorithm that is frequently used to resolve specific global optimization problems due to its rapid convergence and ease of operation.However,PSO still has certain deficiencies,such as a poor trade-off between exploration and exploitation and premature convergence.Hence,this paper proposes a dual-stage hybrid learning particle swarm optimization(DHLPSO).In the algorithm,the iterative process is partitioned into two stages.The learning strategy used at each stage emphasizes exploration and exploitation,respectively.In the first stage,to increase population variety,a Manhattan distance based learning strategy is proposed.In this strategy,each particle chooses the furthest Manhattan distance particle and a better particle for learning.In the second stage,an excellent example learning strategy is adopted to perform local optimization operations on the population,in which each particle learns from the global optimal particle and a better particle.Utilizing the Gaussian mutation strategy,the algorithm’s searchability in particular multimodal functions is significantly enhanced.On benchmark functions from CEC 2013,DHLPSO is evaluated alongside other PSO variants already in existence.The comparison results clearly demonstrate that,compared to other cutting-edge PSO variations,DHLPSO implements highly competitive performance in handling global optimization problems.

具约束向量拟均衡问题的Global有效解的最优条件

向量均衡问题是运筹学的重要组成部分,其研究的主要内容包含各种解的存在性、稳定性、连续性、连通性、适定性、最优条件。向量均衡问题的解主要有有效解、弱有效解、强有效解、Global有效解、Henig有效解、超有效解。研究向量均衡问题各种有效解的最优条件是向量均衡问题的一个重要课题。首先,在实Hausdorff拓扑线性空间中引入具约束条件的向量拟均衡问题及其Global有效解的概念;其次,在实拓扑线性空间中分析了锥-凸、几乎锥-类凸与几乎锥-次类凸3种广义凸性的内在关系;最后,在3种广义凸性条件下借助于凸集分离定理给出了具约束条件的向量拟均衡问题Global有效解的充要条件。

GLOBAL CONVERGENCE OF QPFTH METHOD FOR LARGE-SCALE NONLINEAR SPARSE CONSTRAINED OPTIMIZATION

A QP-free, truncated hybrid (QPFTH) method was proposed and developed in [6] forsolving sparse large-scale nonlinear programming problems. In the hybrid method, a truncatedNewton method is combined with the method of multiplier. In every iteration level, either atruncated solution for a symmetric system of linear equations is determined by CG algorithmor an unconstrained subproblem is solved by the limited memory BFGS algorithm such thatthe hybrid algorithm is suitable to large-scale problems. In this paper, the consistency in thehybrid method and a steplength procedure are discussed and developed. The global convergenceof QPFTH method is proved and the two-step Q-quadratic convergence rate is further analyzed.