Fast Pareto set generation for nonlinear optimal control problems with multiple objectives

Many practical engineering problems involve the determination of optimal control trajectories for given multiple and conflicting objectives. These conflicting objectives typically give rise to a set of Pareto optimal solutions. To enhance real-time decision making efficient approaches are required for determining the Pareto set in a fast and accurate way. Hereto, the current paper integrates efficient multiple objective scalarisation strategies (e.g., Normal Boundary Intersection and Normalised Normal Constraint) with fast deterministic approaches for dynamic optimisation (e.g., Single and Multiple Shooting). All techniques have been implemented as an easy-to-use add-on module of the automatic control and dynamic optimisation toolkit ACADO (both freely available at ). Several algorithmic synergies (e.g., hot-start initialisation strategies) are exploited for an additional speed-up. The features of ACADO Multi-Objective are discussed and its use is illustrated on different multiple objective optimal control problems arising in several engineering disciplines.     

Adaptive weighted sum method for multiobjective optimization: a new method for Pareto front generation

This paper presents an adaptive weighted sum (AWS) method for multiobjective optimization problems. The method extends the previously developed biobjective AWS method to problems with more than two objective functions. In the first phase, the usual weighted sum method is performed to approximate the Pareto surface quickly, and a mesh of Pareto front patches is identified. Each Pareto front patch is then refined by imposing additional equality constraints that connect the pseudonadir point and the expected Pareto optimal solutions on a piecewise planar hypersurface in the -dimensional objective space. It is demonstrated that the method produces a well-distributed Pareto front mesh for effective visualization, and that it finds solutions in nonconvex regions. Two numerical examples and a simple structural optimization problem are solved as case studies. Presented as paper AIAA-2004-4322 at the 10th AIAA-ISSMO Multidisciplinary Analysis and Optimization Conference, Albany, New York, August 30–September 1, 2004     

Pareto set analysis: local measures of objective coupling in multiobjective design optimization

Multiobjective optimization focuses on the explicit trade-offs between competing criteria. A particular case is the study of combined optimal design and optimal control, or co-design, of smart artifacts where the artifact design and controller design objectives compete. In the system-level co-design problem, the objective is often the weighted sum of these two objectives. A frequently referenced practice is to solve co-design problems in a sequential manner: design first, control next. The success of this approach depends on the form of coupling between the two subproblems. In this paper, the coupling vector derived for a system problem with unidirectional coupling is shown to be related to the alignment of competing objectives, as measured by the polar cone of objective gradients, in the bi-objective programming formulation. Further, it is shown that a measure describing the case where a range of objective weighting values for the system objective result in identical design solutions can be normalized when the system problem is considered as a bi-objective one. Changes to the mathematical structure and input parameter values of a bi-objective programming problem can lead to changes in the shape of the attainable set and its Pareto boundary. We illustrate the link between the coupling and alignment measures and the outcomes of the Pareto set. Systematically studying changes to coupling and alignment measures due to changes to the multiobjective formulation can yield deeper insights into the system-level design problem. Two examples illustrate these results.     

多目标动态优化中Pareto随机合作博弈研究综述

随着经济全球化的不断深入,“合作共赢”的发展战略越来越被人们接受,进而合作博弈也被合理地应用到多个领域.与静态合作博弈相比,动态博弈的约束条件为动态方程,其具有优化行为、多个玩家共同存在、决策结果的持久性以及对环境变化的鲁棒性等特点.由于动态系统总是受到某些随机波动的干扰,将这些内部随机波动和外部随机扰动考虑到系统模型中更为实际.随机动态合作博弈同时考虑策略行为、动态演化与随机因素之间的相互作用,其可能是最复杂的决策形式之一.鉴于此,对多目标动态优化中随机合作博弈的进展进行综述:首先,回顾多目标合作博弈的研究背景,给出Pareto最优性的定义和基本性质;其次,综述确定性的合作博弈;再次,分别论述随机合作博弈和平均场随机合作博弈;最后,提出随机合作博弈几个未来研究方向.     

Automatic generation of controllers for embodied legged organisms: a Pareto evolutionary multi-objective approach

In this paper, we investigate the use of a self-adaptive Pareto evolutionary multi-objective optimization (EMO) approach for evolving the controllers of virtual embodied organisms. The objective of this paper is to demonstrate the trade-off between quality of solutions and computational cost. We show empirically that evolving controllers using the proposed algorithm incurs significantly less computational cost when compared to a self-adaptive weighted sum EMO algorithm, a self-adaptive single-objective evolutionary algorithm (EA) and a hand-tuned Pareto EMO algorithm. The main contribution of the self-adaptive Pareto EMO approach is its ability to produce sufficiently good controllers with different locomotion capabilities in a single run, thereby reducing the evolutionary computational cost and allowing the designer to explore the space of good solutions simultaneously. Our results also show that self-adaptation was found to be highly beneficial in reducing redundancy when compared against the other algorithms. Moreover, it was also shown that genetic diversity was being maintained naturally by virtue of the system's inherent multi-objectivity.     

A closer look at drawbacks of minimizing weighted sums of objectives for Pareto set generation in multicriteria optimization problems

A standard technique for generating the Pareto set in multicriteria optimization problems is to minimize (convex) weighted sums of the different objectives for various different settings of the weights. However, it is well-known that this method succeeds in getting points from all parts of the Pareto set only when the Pareto curve is convex. This article provides a geometrical argument as to why this is the case.Secondly, it is a frequent observation that even for convex Pareto curves, an evenly distributed set of weights fails to produce an even distribution of points from all parts of the Pareto set. This article aims to identify the mechanism behind this observation. Roughly, the weight is related to the slope of the Pareto curve in the objective space in a way such that an even spread of Pareto points actually corresponds to often very uneven distributions of weights. Several examples are provided showing assumed shapes of Pareto curves and the distribution of weights corresponding to an even spread of points on those Pareto curves.     

MOSubdue: a Pareto dominance-based multiobjective Subdue algorithm for frequent subgraph mining

Graph-based data mining approaches have been mainly proposed to the task popularly known as frequent subgraph mining subject to a single user preference, like frequency, size, etc. In this work, we propose to deal with the frequent subgraph mining problem from multiobjective optimization viewpoint, where a subgraph (or solution) is defined by several user-defined preferences (or objectives), which are conflicting in nature. For example, mined subgraphs with high frequency are often of small size, and vice-versa. Use of such objectives in the multiobjective subgraph mining process generates Pareto-optimal subgraphs, where no subgraph is better than another subgraph in all objectives. We have applied a Pareto dominance approach for the evaluation and search subgraphs regarding to both proximity and diversity in multiobjective sense, which has incorporated in the framework of Subdue algorithm for subgraph mining. The method is called multiobjective subgraph mining by Subdue (MOSubdue) and has several advantages: (i) generation of Pareto-optimal subgraphs in a single run (ii) selection of subgraph-seeds from the candidate subgraphs based on all objectives (iii) search in the multiobjective subgraphs lattice space, and (iv) capability to deal with different multiobjective frequent subgraph mining tasks by customizing the tackled objectives. The good performance of MOSubdue is shown by performing multiobjective subgraph mining defined by two and three objectives on two real-life datasets.     

Leak localization in water distribution networks using a mixed model-based/data-driven approach

This paper proposes a new method for leak localization in water distribution networks (WDNs). In a first stage, residuals are obtained by comparing pressure measurements with the estimations provided by a WDN model. In a second stage, a classifier is applied to the residuals with the aim of determining the leak location. The classifier is trained with data generated by simulation of the WDN under different leak scenarios and uncertainty conditions. The proposed method is tested both by using synthetic and experimental data with real WDNs of different sizes. The comparison with the current existing approaches shows a performance improvement.     

Regularization networks for inverse problems: A state-space approach

Linear inverse problems with discrete data are equivalent to the estimation of the continuous-time input of a linear dynamical system from samples of its output. The solution obtained by means of regularization theory has the structure of a neural network similar to classical RBF networks. However, the basis functions depend in a nontrivial way on the specific linear operator to be inverted and the adopted regularization strategy. By resorting to the Bayesian interpretation of regularization, we show that such networks can be implemented rigorously and efficiently whenever the linear operator admits a state-space representation. An analytic expression is provided for the basis functions as well as for the entries of the matrix of the linear system used to compute the weights. The results are illustrated through a deconvolution problem where the spontaneous secretory rate of luteinizing hormone (LH) of the hypophisis is reconstructed from measurements of plasma LH concentrations.     

New polynomial time algorithms to compute a set of Pareto optimal paths for multi-objective shortest path problems

Multi-objective shortest path problem (MOSPP) is an active area of research because of its application in a large number of systems such as transportation systems, communication systems, power transmission systems, pipeline distribution systems of water, gas, blood and drainage, neural decision systems, production planning and project planning. In these networks it becomes necessary to find the best path from one node to a specified or all other nodes. The computational complexity of the existing algorithms in the literature to compute all Pareto minimum paths from a specified source node to all other nodes in an MOSPP is of exponential order in the worst case. Instead of generating all the values of the Pareto minimum paths in exponential time, we propose an algorithm to find a set of values of the Pareto minimum paths in polynomial time, which is very significant in many contexts. If an MOSPP of a network is having negative cycle, all the existing algorithm only indicate the existence of the negative cycle, that too after exponential number of operations. However, applying the proposed algorithm, we can find a set of Pareto minimum paths of any MOSPP of a network even if it contains negative cycles. The proposed algorithm is illustrated with examples.     

Computational experimentations in market and supply-chain co-design: a mixed agent approach

The synthetic environment for analysis and simulations (SEAS) is a computational experimentation environment that mimics real life economies, with multiple interlinked markets, multiple goods and services, multiple firms and channels and multiple consumers, all built from the ground up. It is populated with human agents who make strategically complex decisions and artificial agents who make simple but detail intensive decisions. These agents can be calibrated with real data and allowed to make the same decisions in this synthetic economy as their real life counterparts. The resulting outcomes can be surprisingly accurate. This paper discusses the research in this area and goes on to detail the architecture of SEAS. It also presents a detailed case study of market and supply-chain co-design for business-to-business e-commerce in the PC industry.     

Space exploration and global optimization for computationally intensive design problems: a rough set based approach

Modern engineering design problems often involve computation-intensive analysis and simulation processes. Design optimization based on such processes is desired to be efficient, informative and transparent. This work proposes a rough set based approach that can identify multiple sub-regions in a design space, within which all of the design points are expected to have a performance value equal to or less than a given level. The rough set method is applied iteratively on a growing sample set. A novel termination criterion is also developed to ensure a modest number of total expensive function evaluations to identify these sub-regions and search for the global optimum. The significance of the proposed method is twofold. First, it provides an intuitive method to establish the mapping from the performance space to the design space, i.e. given a performance level, its corresponding design region(s) can be identified. Such a mapping could be potentially used to explore and visualize the entire design space. Second, it can be naturally extended to a global optimization method. It also bears potential for more broad application to problems such as metamodeling-based design and robust design optimization. The proposed method was tested with a number of test problems and compared with a few well-known global optimization algorithms.     

Information mediation in cyberspace: Scalable methods for declarative information networks

An end-to-end discussion, from logical architecture to implementation, of issues and design decisions in declarative information networks is presented. A declarative information network is defined to be a dynamic and decentralized structure where value-added services are declared and applied as mediators in a scalable and controlled manner. A primary result is the need to adopt dynamically linked ontologies as the semantic basis for knowledge sharing in scalable networks. It is shown that data mining techniques provide a promising basis upon which to explore and develop this result. Our prototype system, entitled Mystique, is described in terms of KQML, distributed object management, and distributed agent execution. An example shows how we map our architecture into the World Wide Web (WWW) and transform the appearance of the WWW into an intelligently integrated and multi-subject distributed information network.     

U-shaped energy loss curves utilization for distributed generation optimization in distribution networks

We propose novel techniques to find the optimal achieve the maximum loss reduction for distribution networks location, size, and power factor of distributed generation （DG） to Determining the optimal DG location and size is achieved simultaneously using the energy loss curves technique for a pre-selected power factor that gives the best DG operation. Based on the network＇s total load demand, four DG sizes are selected. They are used to form energy loss curves for each bus and then for determining the optimal DG options. The study shows that by defining the energy loss minimization as the objective function, the time-varying load demand significantly affects the sizing of DG resources in distribution networks, whereas consideration of power loss as the objective function leads to inconsistent interpretation of loss reduction and other calculations. The devised technique was tested on two test distribution systems of varying size and complexity and validated by comparison with the exhaustive iterative method （EIM） and recently published results. Results showed that the proposed technique can provide an optimal solution with less computation.     

On solving constrained optimization problems with neural networks:a penalty method approach

Deals with the use of neural networks to solve linear and nonlinear programming problems. The dynamics of these networks are analyzed. In particular, the dynamics of the canonical nonlinear programming circuit are analyzed. The circuit is shown to be a gradient system that seeks to minimize an unconstrained energy function that can be viewed as a penalty method approximation of the original problem. Next, the implementations that correspond to the dynamical canonical nonlinear programming circuit are examined. It is shown that the energy function that the system seeks to minimize is different than that of the canonical circuit, due to the saturation limits of op-amps in the circuit. It is also noted that this difference can cause the circuit to converge to a different state than the dynamical canonical circuit. To remedy this problem, a new circuit implementation is proposed.     

A Pareto-based hybrid multiobjective evolutionary approach for constrained multipath traffic engineering optimization in MPLS/GMPLS networks

This paper proposes a hybrid evolutionary algorithm for solving the constrained multipath traffic engineering problem in MPLS (Multi-Protocol Label Switching) network and its extended architecture GMPLS (Generalized MPLS). Multipath traffic engineering is gaining more importance in contemporary networks. It aims to satisfy the requirements of emerging network applications while optimizing the network performance and the utilization of the available resources within the network. A formulation of this problem as a multiobjective constrained mixed-integer program, which is known to be NP-hard, is first extended. Then, we develop a hybrid heuristic algorithm based on combining linear programming with a devised Pareto-based genetic algorithm for approximating the optimal Pareto curve. A numerical example is adopted from the literature to evaluate and compare the performance of six variations of the proposed heuristic. We study the statistical significance of the results using Kruskal–Wallis nonparametric test. We also compare the results of the heuristic approach with the lexicographic weighted Chebyshev method using a variety of performance metrics.     

A multiobjective approach for optimal placement and sizing of distributed generators and capacitors in distribution network

Both active and reactive power play important roles in power system transmission and distribution networks. While active power does the useful work, reactive power supports the voltage that necessitates control from system reliability aspect as deviation of voltage from nominal range may lead to inadvertent operation and premature failure of system components. Reactive power flow must also be controlled in the system to maximize the amount of real power that can be transferred across the power transmitting media. This paper proposes an approach to simultaneously minimize the real power loss and the net reactive power flow in the system when reinforced with distributed generators (DGs) and shunt capacitors (SCs). With the suggested method, the system performance, reliability and loading capacity can be increased by reduction of losses. A multiobjective evolutionary algorithm based on decomposition (MOEA/D) is adopted to select optimal sizes and locations of DGs and SCs in large scale distribution networks with objectives being minimizing system real and reactive power losses. MOEA/D is the process of decomposition of a multiobjective optimization problem into a number of scalar optimization subproblems and optimizing those concurrently. Case studies with standard IEEE 33-bus, 69-bus, 119-bus distribution networks and a practical 83-bus distribution network are performed. Output results of MOEA/D method are compared with similar past studies and notable improvement is observed.     

Integrated approach of network reconfiguration with distributed generation and shunt capacitors placement for power loss minimization in radial distribution networks

This article presents the significance of efficient hybrid heuristic search algorithm(HS-PABC) based on Harmony search algorithm (HSA) and particle artificial bee colony algorithm (PABC) in the context of performance enhancement of distribution network through simultaneous network reconfiguration along with optimal allocation and sizing of distributed generators and shunt capacitors. The premature and slow convergence over multi model fitness landscape is the main limitation in standard HSA. In the proposed hybrid algorithm the harmony memory vector of HSA is intelligently enhanced through PABC algorithm during the optimization process to reach the optimal solution within the search space. In hybrid approach, the exploration ability of HSA and the exploitation ability of PABC algorithm are integrated to blend the potency of both algorithms. The box plot and Wilcoxon rank sum tests are used to show the quality of the solution obtained by hybrid HS-PABC with respect to HSA.The computational results prove the integrated approach of the network reconfiguration problem along with optimal placement and sizing of DG units and shunt capacitors as an efficient approach with respect to power loss reduction and voltage profile enhancement. The results obtained on 69 and 118 node network by hybrid HS-PABC method and the standard HSA reveals the effeciency of the proposed approach which guarantees to achieve global optimal solution with less iteration.     

ParEGO: a hybrid algorithm with on-line landscape approximation for expensive multiobjective optimization problems

This paper concerns multiobjective optimization in scenarios where each solution evaluation is financially and/or temporally expensive. We make use of nine relatively low-dimensional, nonpathological, real-valued functions, such as arise in many applications, and assess the performance of two algorithms after just 100 and 250 (or 260) function evaluations. The results show that NSGA-II, a popular multiobjective evolutionary algorithm, performs well compared with random search, even within the restricted number of evaluations used. A significantly better performance (particularly, in the worst case) is, however, achieved on our test set by an algorithm proposed herein-ParEGO-which is an extension of the single-objective efficient global optimization (EGO) algorithm of Jones et al. ParEGO uses a design-of-experiments inspired initialization procedure and learns a Gaussian processes model of the search landscape, which is updated after every function evaluation. Overall, ParEGO exhibits a promising performance for multiobjective optimization problems where evaluations are expensive or otherwise restricted in number.