Subset simulation for multi-objective optimization

Subset simulation is an efficient Monte Carlo technique originally developed for structural reliability problems, and further modified to solve single-objective optimization problems based on the idea that an extreme event (optimization problem) can be considered as a rare event (reliability problem). In this paper subset simulation is extended to solve multi-objective optimization problems by taking advantages of Markov Chain Monte Carlo and a simple evolutionary strategy. In the optimization process, a non-dominated sorting algorithm is introduced to judge the priority of each sample and handle the constraints. To improve the diversification of samples, a reordering strategy is proposed. A Pareto set can be generated after limited iterations by combining the two sorting algorithms together. Eight numerical multi-objective optimization benchmark problems are solved to demonstrate the efficiency and robustness of the proposed algorithm. A parametric study on the sample size in a simulation level and the proportion of seed samples is performed to investigate the performance of the proposed algorithm. Comparisons are made with three existing algorithms. Finally, the proposed algorithm is applied to the conceptual design optimization of a civil jet.     

TSP的量子蚂蚁算法求解

在分析量子算法的基本概念的基础上,提出了一种新的算法——量子蚂蚁算法。量子蚂蚁算法结合了量子计算中量子旋转门的量子信息和蚂蚁寻优的特点,为解决实际问题提供的一种新的优化方法。本文将量子蚂蚁算法应用于TSP问题的研究,通过选取国际通用的TSP实例库中多个实例进行测试,表明了新算法具有很好的精确度和鲁棒性,即使对于大规模问题,也能以很小的种群和不长的时间求得相对误差较小的满意解。     

Multi-objective design optimization of an engine accessory drive system with a robustness analysis

Designing a good engine accessory drive system becomes a hard work with its increasingly complicated configuration and high demands on its dynamic characteristics. In this work, a hybrid mutation particle swarm optimization (HMPSO) algorithm is presented to optimize the key structure parameters of an engine accessory drive system for its vibration control. The superiority of the HMPSO algorithm against several other concerned metaheuristic algorithms in terms of solution quality and stability are verified by non-parametric statistical tests on ten benchmark functions. The design problem of the engine accessory drive system is a multi-objective optimization problem; the weighted sum method and main target method are applied to convert it to a single-objective one. Optimization on an example engine accessory drive system using the HMPSO algorithm demonstrates obvious improvement in system vibration after optimization. A robustness analysis is conducted to identify the robustness of dynamic responses of the engine accessory drive system with respect to small variations of the design variables relative to the optimal design in the design space, and suggestions on design of an engine accessory drive system are given according to it.     

GOSAC: global optimization with surrogate approximation of constraints

We introduce GOSAC, a global optimization algorithm for problems with computationally expensive black-box constraints and computationally cheap objective functions. The variables may be continuous, integer, or mixed-integer. GOSAC uses a two-phase optimization approach. The first phase aims at finding a feasible point by solving a multi-objective optimization problem in which the constraints are minimized simultaneously. The second phase aims at improving the feasible solution. In both phases, we use cubic radial basis function surrogate models to approximate the computationally expensive constraints. We iteratively select sample points by minimizing the computationally cheap objective function subject to the constraint function approximations. We assess GOSAC’s efficiency on computationally cheap test problems with integer, mixed-integer, and continuous variables and two environmental applications. We compare GOSAC to NOMAD and a genetic algorithm (GA). The results of the numerical experiments show that for a given budget of allowed expensive constraint evaluations, GOSAC finds better feasible solutions more efficiently than NOMAD and GA for most benchmark problems and both applications. GOSAC finds feasible solutions with a higher probability than NOMAD and GOSAC.     

Sequential Semidefinite Program for Maximum Robustness Design of Structures under Load Uncertainty

A robust structural optimization scheme as well as an optimization algorithm are presented based on the robustness function. Under the uncertainties of the external forces based on the info-gap model, the maximization of the robustness function is formulated as an optimization problem with infinitely many constraints. By using the quadratic embedding technique of uncertainty and the S-procedure, we reformulate the problem into a nonlinear semidefinite programming problem. A sequential semidefinite programming method is proposed which has a global convergent property. It is shown through numerical examples that optimum designs of various linear elastic structures can be found without difficulty.The authors are grateful to the Associate Editor and two anonymous referees for handling the paper efficiently as well as for helpful comments and suggestions.     

Risk-averse feasible policies for large-scale multistage stochastic linear programs

We develop tractable semidefinite programming based approximations for distributionally robust individual and joint chance constraints, assuming that only the first- and second-order moments as well as the support of the uncertain parameters are given. It is known that robust chance constraints can be conservatively approximated by Worst-Case Conditional Value-at-Risk (CVaR) constraints. We first prove that this approximation is exact for robust individual chance constraints with concave or (not necessarily concave) quadratic constraint functions, and we demonstrate that the Worst-Case CVaR can be computed efficiently for these classes of constraint functions. Next, we study the Worst-Case CVaR approximation for joint chance constraints. This approximation affords intuitive dual interpretations and is provably tighter than two popular benchmark approximations. The tightness depends on a set of scaling parameters, which can be tuned via a sequential convex optimization algorithm. We show that the approximation becomes essentially exact when the scaling parameters are chosen optimally and that the Worst-Case CVaR can be evaluated efficiently if the scaling parameters are kept constant. We evaluate our joint chance constraint approximation in the context of a dynamic water reservoir control problem and numerically demonstrate its superiority over the two benchmark approximations.     

Convergence Criteria for Hierarchical Overlapping Coordination of Linearly Constrained Convex Design Problems

Decomposition of multidisciplinary engineering system design problems into smaller subproblems is desirable because it enhances robustness and understanding of the numerical results. Moreover, subproblems can be solved in parallel using the optimization technique most suitable for the underlying mathematical form of the subproblem. Hierarchical overlapping coordination (HOC) is an interesting strategy for solving decomposed problems. It simultaneously uses two or more design problem decompositions, each of them associated with different partitions of the design variables and constraints. Coordination is achieved by the exchange of information between decompositions. This article presents the HOC algorithm and several new sufficient conditions for convergence of the algorithm to the optimum in the case of convex problems with linear constraints. One of these equivalent conditions involves the rank of the constraint matrix that is computationally efficient to verify. Computational results obtained by applying the HOC algorithm to quadratic programming problems of various sizes are included for illustration.     

模糊蚁群算法及其在TSP中的应用

在传统蚁群算法的基础上加入了使用模糊规则表更新信息素的策略,提出了一种新的算法——模糊蚁群算法.算法结合了模糊控制中输入输出的模糊化处理和蚁群寻优的特点,为实际问题提供了新的解决手段.文中将模糊蚁群算法应用于TSP问题,通过对中国31个省会城市等实例数据进行的测试,验证表明了新算法具有良好的有效性和鲁棒性.     

An artificial neural network meta-model for constrained simulation optimization

This paper presents artificial neural network (ANN) meta-models for expensive continuous simulation optimization (SO) with stochastic constraints. These meta-models are used within a sequential experimental design to approximate the objective function and the stochastic constraints. To capture the non-linear nature of the ANN, the SO problem is iteratively approximated via non-linear programming problems whose (near) optimal solutions obtain estimates of the global optima. Following the optimization step, a cutting plane-relaxation scheme is invoked to drop uninformative estimates of the global optima from the experimental design. This approximation is iterated until a terminating condition is met. To study the robustness and efficiency of the proposed algorithm, a realistic inventory model is used; the results are compared with those of the OptQuest optimization package. These numerical results indicate that the proposed meta-model-based algorithm performs quite competitively while requiring slightly fewer simulation observations.     

一种求解柔性资源约束前摄性项目调度问题的启发式算法

在不确定环境中,一个具有较高鲁棒性的进度计划可以保证项目的稳定实施。考虑到现实中资源可能具有多种技能,会对制定鲁棒性较高进度计划的过程产生影响,因此本文研究了柔性资源约束下前摄性项目调度优化问题。首先界定研究问题;然后从鲁棒性最大化的视角出发,构建了研究问题的优化模型,在对模型进行分析的基础上将其分解为经典鲁棒优化和资源技能分配两个子模型;随后设计了求解问题的基于削峰算法的启发式算法;最后用一个实际案例验证了算法有效性,并分析了关键参数对进度计划鲁棒性的影响,得到如下结论:项目进度计划鲁棒性随着项目工期的延长、资源可用量的增加或资源柔性的提高而增大。     

An effective multiagent evolutionary algorithm integrating a novel roulette inversion operator for engineering optimization

Multiagent systems have been studied and widely used in the field of artificial intelligence and computer science to catalyze computation intelligence. In this paper, a multiagent evolutionary algorithm called RAER based on the ERA multiagent modeling pattern is proposed, where ERA has the same architecture as Swarm including three parts of Environment, Reactive rules and Agents. RAER integrates a novel roulette inversion operator (RIO) proposed in this paper and theoretically proved to conquer the irrationality of the inversion operator (IO) designed by John Holland when used for real code stochastic optimization algorithms. Experiments for numerical optimization of 4 benchmark functions show that the RIO operator bears better functioning than IO operator. And experiments for numerical optimization of 12 benchmark functions are used to examine the performance and scalability of RAER along the problem dimensions ranging 20-10 000, results indicate that RAER outperforms other comparative algorithms significantly. Also, two engineering optimization problems of a stable linear system approximation and a welded beam design are used to examine the applicability of RAER. Results show that RAER has better search ability and faster convergence speed. Especially for the approximation problem, REAR can find the proper optima belonging to different fixed search areas, which is significantly better than other algorithms and shows that RAER can search the problem domains more thoroughly than other algorithms. Hence, RAER is efficient and practical.     

A GRASP based on DE to solve single machine scheduling problem with SDST

When handling combinatorial optimization problems, we try to get the optimal arrangement of discrete entities so that the requirements and the constraints are satisfied. These problems become more and more important in various industrial and academic fields. So, over the past years, several techniques have been proposed to solve them. In this paper, we are interested in the single machine scheduling problem with Sequence-Dependent Setup Times, which can be solved through different approaches. We present a hybrid algorithm which combines Greedy Randomized Adaptive Search Procedure and Differential Evolution for tackling this problem. Our algorithm is tested on benchmark instances from the literature. The computational experiments prove the efficiency of this algorithm.     

Firefly penalty-based algorithm for bound constrained mixed-integer nonlinear programming

In this article, we aim to extend the firefly algorithm (FA) to solve bound constrained mixed-integer nonlinear programming (MINLP) problems. An exact penalty continuous formulation of the MINLP problem is used. The continuous penalty problem comes out by relaxing the integrality constraints and by adding a penalty term to the objective function that aims to penalize integrality constraint violation. Two penalty terms are proposed, one is based on the hyperbolic tangent function and the other on the inverse hyperbolic sine function. We prove that both penalties can be used to define the continuous penalty problem, in the sense that it is equivalent to the MINLP problem. The solutions of the penalty problem are obtained using a variant of the metaheuristic FA for global optimization. Numerical experiments are given on a set of benchmark problems aiming to analyze the quality of the obtained solutions and the convergence speed. We show that the firefly penalty-based algorithm compares favourably with the penalty algorithm when the deterministic DIRECT or the simulated annealing solvers are invoked, in terms of convergence speed.     

Asymptotic Results for Random Multidimensional Assignment Problems

The multidimensional assignment problem (MAP) is an NP-hard combinatorial optimization problem occurring in applications such as data association and target tracking. In this paper, we investigate characteristics of the mean optimal solution values for random MAPs with axial constraints. Throughout the study, we consider cost coefficients taken from three different random distributions: uniform, exponential and standard normal. In the cases of uniform and exponential costs, experimental data indicates that the mean optimal value converges to zero when the problem size increases. We give a short proof of this result for the case of exponentially distributed costs when the number of elements in each dimension is restricted to two. In the case of standard normal costs, experimental data indicates the mean optimal value goes to negative infinity with increasing problem size. Using curve fitting techniques, we develop numerical estimates of the mean optimal value for various sized problems. The experiments indicate that numerical estimates are quite accurate in predicting the optimal solution value of a random instance of the MAP.     

A cutting plane method for solving harvest scheduling models with area restrictions

We describe a cutting plane algorithm for an integer programming problem that arises in forest harvest scheduling. Spatial harvest scheduling models optimize the binary decisions of cutting or not cutting forest management units in different time period subject to logistical, economic and environmental restrictions. One of the most common constraints requires that the contiguous size of harvest openings (i.e., clear-cuts) cannot exceed an area threshold in any given time period or over a set of periods called green-up. These so-called adjacency or green-up constraints make the harvest scheduling problem combinatorial in nature and very hard to solve. Our proposed cutting plane algorithm starts with a model without area restrictions and adds constraints only if a violation occurs during optimization. Since violations are less likely if the threshold area is large, the number of constraints is kept to a minimum. The utility of the approach is illustrated by an application, where the landowner needs to assess the cost of forest certification that involves clear-cut size restrictions stricter than what is required by law. We run empirical tests and find that the new method performs best when existing models fail: when the number of units is high or the allowable clear-cut size is large relative to average unit size. Since this scenario is the norm rather than the exception in forestry, we suggest that timber industries would greatly benefit from the method. In conclusion, we describe a series of potential applications beyond forestry.     

Reducing the bandwidth of a sparse matrix with a genetic algorithm

Abstract

The matrix bandwidth minimization problem (MBMP) consists in finding a permutation of the lines and columns of a given sparse matrix in order to keep the non-zero elements in a band that is as close as possible to the main diagonal. Equivalently in terms of graph theory, MBMP is defined as the problem of finding a labelling of the vertices of a given graph G such that its bandwidth is minimized. In this paper, we propose an improved genetic algorithm (GA)-based heuristic for solving the matrix bandwidth minimization problem, motivated by its robustness and efficiency in a wide area of optimization problems. Extensively computational results are reported for an often used set of benchmark instances. The obtained results on the different instances investigated show improvement of the quality of the solutions and demonstrate the efficiency of our GA compared to the existing methods in the literature.     

An Algorithm for Solving Optimization Problems with One Linear Objective Function and Finitely Many Constraints of Fuzzy Relation Inequalities

An optimization model with one linear objective function and fuzzy relation equation constraints was presented by Fang and Li (1999) as well as an efficient solution procedure was designed by them for solving such a problem. A more general case of the problem, an optimization model with one linear objective function and finitely many constraints of fuzzy relation inequalities, is investigated in this paper. A new approach for solving this problem is proposed based on a necessary condition of optimality given in the paper. Compared with the known methods, the proposed algorithm shrinks the searching region and hence obtains an optimal solution fast. For some special cases, the proposed algorithm reaches an optimal solution very fast since there is only one minimum solution in the shrunk searching region. At the end of the paper, two numerical examples are given to illustrate this difference between the proposed algorithm and the known ones.     

Maximally diverse grouping: an iterated tabu search approach

The maximally diverse grouping problem (MDGP) consists of finding a partition of a set of elements into a given number of mutually disjoint groups, while respecting the requirements of group size constraints and diversity. In this paper, we propose an iterated tabu search (ITS) algorithm for solving this problem. We report computational results on three sets of benchmark MDGP instances of size up to 960 elements and provide comparisons of ITS to five state-of-the-art heuristic methods from the literature. The results demonstrate the superiority of the ITS algorithm over alternative approaches. The source code of the algorithm is available for free download via the internet.     

Using conical partition to globally maximizing the nonlinear sum of ratios

This article presents a global optimization algorithm for globally maximizing the sum of concave–convex ratios problem with a convex feasible region. The algorithm uses a branch and bound scheme where a concave envelope of the objective function is constructed to obtain an upper bound of the optimal value by using conical partition. As a result, the upper-bound subproblems during the algorithm search are all ordinary convex programs with less variables and constraints and do not grow in size from iterations to iterations in the computation procedure, and furthermore a new bounding tightening strategy is proposed such that the upper-bound convex relaxation subproblems are closer to the original nonconvex problem to enhance solution procedure. At last, some numerical examples are given to vindicate our conclusions.