Fast computation of equispaced Pareto manifolds and Pareto fronts for multiobjective optimization problems

In this paper, we consider the problem of generating a well sampled discrete representation of the Pareto manifold or the Pareto front corresponding to the equilibrium points of a multi-objective optimization problem. We show how the introduction of simple additional constraints into a continuation procedure produces equispaced points in either of those two sets. Moreover, we describe in detail a novel algorithm for global continuation that requires two orders of magnitude less function evaluations than evolutionary algorithms commonly used to solve this problem. The performance of the methods is demonstrated on problems from the current literature.     

Multiobjective model predictive control

This paper proposes a novel model predictive control (MPC) scheme based on multiobjective optimization. At each sampling time, the MPC control action is chosen among the set of Pareto optimal solutions based on a time-varying, state-dependent decision criterion. Compared to standard single-objective MPC formulations, such a criterion allows one to take into account several, often irreconcilable, control specifications, such as high bandwidth (closed-loop promptness) when the state vector is far away from the equilibrium and low bandwidth (good noise rejection properties) near the equilibrium. After recasting the optimization problem associated with the multiobjective MPC controller as a multiparametric multiobjective linear or quadratic program, we show that it is possible to compute each Pareto optimal solution as an explicit piecewise affine function of the state vector and of the vector of weights to be assigned to the different objectives in order to get that particular Pareto optimal solution. Furthermore, we provide conditions for selecting Pareto optimal solutions so that the MPC control loop is asymptotically stable, and show the effectiveness of the approach in simulation examples.     

A multiobjective approach for maximizing the reach or GRP of different brands in TV advertising

This paper focuses on a multiobjective optimization problem in TV advertising from an advertising agency's perspective, which involves deciding on which commercial breaks to air the ads of various brands to jointly maximize reach or gross rating point (GRP) for the different brands subject to budget constraints, brand competition constraints, and other scheduling constraints. We present a multiobjective integer programming formulation of this problem and develop and implement algorithms for generating provably Pareto‐optimal solutions. We also develop reduction and visualization procedures to aid a decision maker in choosing suitable subsets of the Pareto‐optimal solutions obtained. Numerical experiments on five TV advertising problems involving 20–40 objective functions and thousands of decision variables and constraints demonstrate the effectiveness of the proposed formulation and solution methods in generating Pareto‐optimal objective vectors that reflect brand priorities and that are well distributed along the Pareto front.     

The augmented weighted Tchebychev norm for optimizing a linear function over an integer efficient set of a multicriteria linear program

In this paper, we propose a new exact algorithm, using an augmented weighted Tchebychev norm, for optimizing a linear function on the efficient set of a multiple objective integer linear programming problem. This norm is optimized progressively by improving the value of the linear criteria and going through some efficient solutions. The method produced not only the best efficient solution of the linear objective function but also a subset of nondominated solutions that can help decision makers to select the best decision among a large set of Pareto solutions.     

Decision-making based on approximate and smoothed Pareto curves

We consider bicriteria optimization problems and investigate the relationship between two standard approaches to solving them: (i) computing the Pareto curve and (ii) the so-called decision maker’s approach in which both criteria are combined into a single (usually nonlinear) objective function. Previous work by Papadimitriou and Yannakakis showed how to efficiently approximate the Pareto curve for problems like Shortest Path, Spanning Tree, and Perfect Matching. We wish to determine for which classes of combined objective functions the approximate Pareto curve also yields an approximate solution to the decision maker’s problem. We show that an FPTAS for the Pareto curve also gives an FPTAS for the decision-maker’s problem if the combined objective function is growth bounded like a quasi-polynomial function. If the objective function, however, shows exponential growth then the decision-maker’s problem is NP-hard to approximate within any polynomial factor. In order to bypass these limitations of approximate decision making, we turn our attention to Pareto curves in the probabilistic framework of smoothed analysis. We show that in a smoothed model, we can efficiently generate the (complete and exact) Pareto curve with a small failure probability if there exists an algorithm for generating the Pareto curve whose worst-case running time is pseudopolynomial. This way, we can solve the decision-maker’s problem w.r.t. any non-decreasing objective function for randomly perturbed instances of, e.g. Shortest Path, Spanning Tree, and Perfect Matching.     

The multiobjective multidimensional knapsack problem: a survey and a new approach

The knapsack problem (KP) and its multidimensional version (MKP) are basic problems in combinatorial optimization. In this paper, we consider their multiobjective extension (MOKP and MOMKP), for which the aim is to obtain or approximate the set of efficient solutions. In the first step, we classify and briefly describe the existing works that are essentially based on the use of metaheuristics. In the second step, we propose the adaptation of the two‐phase Pareto local search (2PPLS) to the resolution of the MOMKP. With this aim, we use a very large scale neighborhood in the second phase of the method, that is the PLS. We compare our results with state‐of‐the‐art results and show that the results we obtained were never reached before by heuristics for biobjective instances. Finally, we consider the extension to three‐objective instances.     

解非线性规划的多目标遗传算法及其收敛性

给出非线性约束规划问题的一种新解法。它既不需用传统的惩罚函数,又不需区分可行解和不可行解,新方法把带约束的非线性规划问题转化成为两个目标函数优化问题,其中一个是原约束问题的目标函数,另一个是违反约束的度函数,并利用多目标优化中的Pareto优劣关系设计了一种新的选择算子,通过对搜索操作和参数的合理设计给出了一种新型遗传算法,且给出了算法的收敛性证明,最后数据实验表明该算法对带约束的非线性规划问题求解是非常有效的。     

Multiobjective optimization using an adaptive weighting scheme

A new Pareto front approximation method is proposed for multiobjective optimization problems (MOPs) with bound constraints. The method employs a hybrid optimization approach using two derivative-free direct search techniques, and intends to solve black box simulation-based MOPs where the analytical form of the objectives is not known and/or the evaluation of the objective function(s) is very expensive. A new adaptive weighting scheme is proposed to convert a multiobjective optimization problem to a single objective optimization problem. Another contribution of this paper is the generalization of the star discrepancy-based performance measure for problems with more than two objectives. The method is evaluated using five test problems from the literature, and a realistic engineering problem. Results show that the method achieves an arbitrarily close approximation to the Pareto front with a good collection of well-distributed nondominated points for all six test problems.     

Bicriteria scheduling of a material handling robot in an m-machine cell to minimize the energy consumption of the robot and the cycle time

This study considers a flowshop type production system consisting of m machines. A material handling robot transports the parts between the machines and loads and unloads the machines. We consider the sequencing of the robot moves and determining the speeds of these moves simultaneously. These decisions affect both the robot’s energy consumption and the production speed of the system. In this study, these two objectives are considered simultaneously. We propose a second order cone programming formulation to find Pareto efficient solutions. We also develop a heuristic algorithm that finds a set of approximate Pareto efficient solutions. The conic formulation can find robot schedules for small cells with less number of machines in reasonable computation times. Our heuristic algorithm can generate a large set of approximate Pareto efficient solutions in a very short computational time. Proposed solution approaches help the decision-maker to achieve the best trade-off between the throughput of a cell and the energy efficiency of a material handling robot.     

On some multiobjective optimization problems arising in biology

In this paper, we discuss modelling and solving some multiobjective optimization problems arising in biology. A class of comparison problems for string selection in molecular biology and a relocation problem in conservation biology are modelled as multiobjective optimization programmes. Some discussions about applications, solvability and different variants of the obtained models are given, as well. A crucial part of the study is based upon the Pareto optimization which refers to the Pareto solutions of multiobjective optimization problems. For such solution, improvement of some objective function can only be obtained at the expense of the deterioration of at least one other objective function.     

基于动态Pareto解集的微粒群优化算法及其在多目标规划中的应用

在传统的微粒群优化算法的基础上,提出了一种基于动态Pareto解集的求解多目标规划问题的方法。Pareto解集在每次迭代过程中进行动态更新和信息共享,在加入新产生的Pareto近似最优解同时去除解集中已经不是Pareto解的数据,每个个体随机地与Pareto解集中的结果进行信息交换,从而保证在快速找到Pareto解的同时保持多样性。并通过三个标准的测试函数证明了算法的有效性。     

基于Pareto支配的双目标优化求解非线性双层规划问题

双层规划问题是一类具有双层递阶结构的系统优化问题。采用Pareto支配的双目标优化策略求解非线性双层规划问题。利用K-T条件把双层规划问题等价转化单层规划问题,进而结合约束部分建立可行性度量目标形成双目标规划问题。在基本的差分进化算法框架中融入非负的最小二乘曲线拟合判断候选解的可行性,构造基于动态概率的Pareto支配选择策略挑选下一代个体,解决种群容易陷入局部最优的缺陷。15个标准函数的测试结果对比显示,该算法在求解非线性双层规划问题中具有较好的全局寻优能力、较低的计算复杂度、较强的稳定性和适用性,可以获得全局最优解。     

MOTGA: A multiobjective Tchebycheff based genetic algorithm for the multidimensional knapsack problem

This paper presents a new multiobjective genetic algorithm based on the Tchebycheff scalarizing function, which aims to generate a good approximation of the nondominated solution set of the multiobjective problem. The algorithm performs several stages, each one intended for searching potentially nondominated solutions in a different part of the Pareto front. Pre-defined weight vectors act as pivots to define the weighted-Tchebycheff scalarizing functions used in each stage. Therefore, each stage focuses the search on a specific region, leading to an iterative approximation of the entire nondominated set.     

Efficient implementation of an active set algorithm for large-scale portfolio selection

This paper deals with the efficient implementation of parametric quadratic programming that is specialized for large-scale mean-variance portfolio selection with a dense covariance matrix. The aim is to calculate the whole Pareto front of solutions that represent the trade-off between maximizing expected return and minimizing variance of return.We describe and compare in a uniform framework several techniques to speed up the necessary matrix operations, namely the initial matrix decomposition, the solution process in each iteration, and the matrix updates. Techniques considered include appropriate ordering of the matrix rows and columns, reducing the size of the system of linear equations, and dividing the system into two parts. Regarding implementation, we suggest to simultaneously use two different matrix representations that are specifically adapted to certain parts of the algorithm and propose a technique that prevents algorithm stalling due to numerical errors. Finally, we analyse and compare the runtime of these algorithm variants on a set of benchmark problems. As we demonstrate, the most sophisticated variant is several orders of magnitude faster than the standard implementation on all tested problem instances.     

Designing pareto optimal nonlinear filters for image processing

We consider a method of designing multidimensional polynomial filters characterized by an interval of a discrete functional Volterra series. This method allows designing Pareto optimal nonlinear filters by several criteria. We prove that finding a set of Pareto optimal alternatives is equivalent to minimizing a weighted target function. The designed nonlinear filter is characterized by the plane tangent to a convex optimal solving function, whose curvature is determined by how contradictory the given criteria are. We show examples of polynomial filter design for image processing and compare them with known filters of the same class.     

Modelling and prediction of complex non-linear processes by using Pareto multi-objective genetic programming

In this paper, a new multi-objective genetic programming (GP) with a diversity preserving mechanism and a real number alteration operator is presented and successfully used for Pareto optimal modelling of some complex non-linear systems using some input–output data. In this study, two different input–output data-sets of a non-linear mathematical model and of an explosive cutting process are considered separately in three-objective optimisation processes. The pertinent conflicting objective functions that have been considered for such Pareto optimisations are namely, training error (TE), prediction error (PE), and the length of tree (complexity of the network) (TL) of the GP models. Such three-objective optimisation implementations leads to some non-dominated choices of GP-type models for both cases representing the trade-offs among those objective functions. Therefore, optimal Pareto fronts of such GP models exhibit the trade-off among the corresponding conflicting objectives and, thus, provide different non-dominated optimal choices of GP-type models. Moreover, the results show that no significant optimality in TE and PE may occur when the TL of the corresponding GP model exceeds some values.     

迁移工作流系统中基于Pareto的服务主体优选

建立了服务主体优选的数学模型，采用Pareto遗传算法对多目标问题进行优化，给出了适用于该模型的操作算子，并提出了在最优解集中选取决策方案的算法。实验结果表明，该方案效果明显优于文献[3]中给出的解决方案。     

Automatic calibration of a rapid flood spreading model using multiobjective optimisations

In order to successfully calibrate a numerical model, multiple criteria should be considered. Multi-objective differential evolution (MODE) and multi-objective particle swarm optimisation (MOPSO) have proved effective in numerous such applications, where most of the techniques relying on the condition of Pareto efficiency to compare different solutions. We describe the performance of two population based search algorithms [nondominated sorting particle swarm optimisation (NSPSO), and nondominated sorting differential evolution (NSDE)] when applied to calibration of a rapid flood spreading model (RFSM). Formulation of an automatic calibration strategy for the RFSM is outline. The simulations show that the both methods possess the ability to find the optimal Pareto front.     

一种结合多目标免疫算法和线性规划的双行设备布局方法

设备布局对于提高生产效率和降低运营成本具有重要意义. 本文针对半导体加工制造中常见的双行设备布局问题, 提出了一种结合多目标免疫算法和线性规划的双行设备布局方法来同时优化物料流成本和布局面积两个目标. 首先, 建立了问题的混合整数规划模型;其次, 针对问题既含有组合方面(机器排序)又含有连续方面(机器精确位置)的特点, 分别设计了一种多目标免疫算法来获取非支配的机器排序集合, 提出了一种基于线性规划的方法来构造任一非支配机器排序对应的连续的非支配解集;最后, 由所有连续的非支配解来构造最后Pareto解. 实验结果表明, 该方法对于小规模问题能获得最优Pareto解, 对于大规模问题能够获得具有良好分布性的Pareto解且其质量远好于NSGA-II和精确算法获得的解.     

A new approach for online multiobjective optimization of mechatronic systems

We present a new concept for online multiobjective optimization and its application to the optimization of the operating point assignment for a doubly-fed linear motor. This problem leads to a time-dependent multiobjective optimization problem. In contrast to classical optimization where the aim is to find the (global) minimum of a single function, we want to simultaneously minimize k objective functions. The solution to this problem is given by the set of optimal compromises, the so-called Pareto set. In the case of the linear motor, there are two conflicting aims which both have to be maximized: the degree of efficiency and the inverter utilization factor. The objective functions depend on velocity, force and power, which can be modeled as time-dependent parameters. For a fixed point of time, the entire corresponding Pareto set can be computed by means of a recently developed set-oriented numerical method. An online computation of the time-dependent Pareto sets is not possible, because the computation itself is too complex. Therefore, we combine the computation of the Pareto set with numerical path following techniques. Under certain smoothness assumptions the set of Pareto points can be characterized as the set of zeros of a certain function. Here, path following allows to track the evolution of a given solution point through time.