A Toolkit for Generating Scalable Stochastic Multiobjective Test Problems

Real-world optimization problems typically include uncertainties over various aspects of the problem formulation. Some existing algorithms are designed to cope with stochastic multiobjective optimization problems, but in order to benchmark them, a proper framework still needs to be established. This paper presents a novel toolkit that generates scalable, stochastic, multiobjective optimization problems. A stochastic problem is generated by transforming the objective vectors of a given deterministic test problem into random vectors. All random objective vectors are bounded by the feasible objective space, defined by the deterministic problem. Therefore, the global solution for the deterministic problem can also serve as a reference for the stochastic problem. A simple parametric distribution for the random objective vector is defined in a radial coordinate system, allowing for direct control over the dual challenges of convergence towards the true Pareto front and diversity across the front. An example for a stochastic test problem, generated by the toolkit, is provided

Approximating Pareto frontier using a hybrid line search approach

This is the post-print version of the final paper published in Information Sciences. The published article is available from the link below. Changes resulting from the publishing process, such as peer review, editing, corrections, structural formatting, and other quality control mechanisms may not be reflected in this document. Changes may have been made to this work since it was submitted for publication. Copyright @ 2010 Elsevier B.V.The aggregation of objectives in multiple criteria programming is one of the simplest and widely used approach. But it is well known that this technique sometimes fail in different aspects for determining the Pareto frontier. This paper proposes a new approach for multicriteria optimization, which aggregates the objective functions and uses a line search method in order to locate an approximate efficient point. Once the first Pareto solution is obtained, a simplified version of the former one is used in the context of Pareto dominance to obtain a set of efficient points, which will assure a thorough distribution of solutions on the Pareto frontier. In the current form, the proposed technique is well suitable for problems having multiple objectives (it is not limited to bi-objective problems) and require the functions to be continuous twice differentiable. In order to assess the effectiveness of this approach, some experiments were performed and compared with two recent well known population-based metaheuristics namely ParEGO and NSGA II. When compared to ParEGO and NSGA II, the proposed approach not only assures a better convergence to the Pareto frontier but also illustrates a good distribution of solutions. From a computational point of view, both stages of the line search converge within a short time (average about 150 ms for the first stage and about 20 ms for the second stage). Apart from this, the proposed technique is very simple, easy to implement and use to solve multiobjective problems.CNCSIS IDEI 2412, Romani

A benchmark test problem toolkit for multi-objective path optimization

Due to the complexity of multi-objective optimization problems (MOOPs) in general, it is crucial to test MOOP methods on some benchmark test problems. Many benchmark test problem toolkits have been developed for continuous parameter/numerical optimization, but fewer toolkits reported for discrete combinational optimization. This paper reports a benchmark test problem toolkit especially for multi-objective path optimization problem (MOPOP), which is a typical category of discrete combinational optimization. With the reported toolkit, the complete Pareto front of a generated test problem of MOPOP can be deduced and found out manually, and the problem scale and complexity are controllable and adjustable. Many methods for discrete combinational MOOPs often only output a partial or approximated Pareto front. With the reported benchmark test problem toolkit for MOPOP, we can now precisely tell how many true Pareto points are missed by a partial Pareto front, or how large the gap is between an approximated Pareto front and the complete one

Scalable and customizable benchmark problems for many-objective optimization

Solving many-objective problems (MaOPs) is still a significant challenge in the multi-objective optimization (MOO) field. One way to measure algorithm performance is through the use of benchmark functions (also called test functions or test suites), which are artificial problems with a well-defined mathematical formulation, known solutions and a variety of features and difficulties. In this paper we propose a parameterized generator of scalable and customizable benchmark problems for MaOPs. It is able to generate problems that reproduce features present in other benchmarks and also problems with some new features. We propose here the concept of generative benchmarking, in which one can generate an infinite number of MOO problems, by varying parameters that control specific features that the problem should have: scalability in the number of variables and objectives, bias, deceptiveness, multimodality, robust and non-robust solutions, shape of the Pareto front, and constraints. The proposed Generalized Position-Distance (GPD) tunable benchmark generator uses the position-distance paradigm, a basic approach to building test functions, used in other benchmarks such as Deb, Thiele, Laumanns and Zitzler (DTLZ), Walking Fish Group (WFG) and others. It includes scalable problems in any number of variables and objectives and it presents Pareto fronts with different characteristics. The resulting functions are easy to understand and visualize, easy to implement, fast to compute and their Pareto optimal solutions are known.This work has been supported by the Brazilian agencies (i) National Council for Scientific and Technological Development (CNPq); (ii) Coordination for the Improvement of Higher Education (CAPES) and (iii) Foundation for Research of the State of Minas Gerais (FAPEMIG, in Portuguese)

Generalized decomposition and cross entropy methods for many-objective optimization

Decomposition-based algorithms for multi-objective optimization problems have increased in popularity in the past decade. Although their convergence to the Pareto optimal front (PF) is in several instances superior to that of Pareto-based algorithms, the problem of selecting a way to distribute or guide these solutions in a high-dimensional space has not been explored. In this work, we introduce a novel concept which we call generalized decomposition. Generalized decomposition provides a framework with which the decision maker (DM) can guide the underlying evolutionary algorithm toward specific regions of interest or the entire Pareto front with the desired distribution of Pareto optimal solutions. Additionally, it is shown that generalized decomposition simplifies many-objective problems by unifying the three performance objectives of multi-objective evolutionary algorithms – convergence to the PF, evenly distributed Pareto optimal solutions and coverage of the entire front – to only one, that of convergence. A framework, established on generalized decomposition, and an estimation of distribution algorithm (EDA) based on low-order statistics, namely the cross-entropy method (CE), is created to illustrate the benefits of the proposed concept for many objective problems. This choice of EDA also enables the test of the hypothesis that low-order statistics based EDAs can have comparable performance to more elaborate EDAs

MONEDA: scalable multi-objective optimization with a neural network-based estimation of distribution algorithm

The Extension Of Estimation Of Distribution Algorithms (Edas) To The Multiobjective Domain Has Led To Multi-Objective Optimization Edas (Moedas). Most Moedas Have Limited Themselves To Porting Single-Objective Edas To The Multi-Objective Domain. Although Moedas Have Proved To Be A Valid Approach, The Last Point Is An Obstacle To The Achievement Of A Significant Improvement Regarding "Standard" Multi-Objective Optimization Evolutionary Algorithms. Adapting The Model-Building Algorithm Is One Way To Achieve A Substantial Advance. Most Model-Building Schemes Used So Far By Edas Employ Off-The-Shelf Machine Learning Methods. However, The Model-Building Problem Has Particular Requirements That Those Methods Do Not Meet And Even Evade. The Focus Of This Paper Is On The Model- Building Issue And How It Has Not Been Properly Understood And Addressed By Most Moedas. We Delve Down Into The Roots Of This Matter And Hypothesize About Its Causes. To Gain A Deeper Understanding Of The Subject We Propose A Novel Algorithm Intended To Overcome The Draw-Backs Of Current Moedas. This New Algorithm Is The Multi-Objective Neural Estimation Of Distribution Algorithm (Moneda). Moneda Uses A Modified Growing Neural Gas Network For Model-Building (Mb-Gng). Mb-Gng Is A Custom-Made Clustering Algorithm That Meets The Above Demands. Thanks To Its Custom-Made Model-Building Algorithm, The Preservation Of Elite Individuals And Its Individual Replacement Scheme, Moneda Is Capable Of Scalably Solving Continuous Multi-Objective Optimization Problems. It Performs Better Than Similar Algorithms In Terms Of A Set Of Quality Indicators And Computational Resource Requirements.This work has been funded in part by projects CNPq BJT 407851/2012-7, FAPERJ APQ1 211.451/2015, MINECO TEC2014-57022-C2-2-R and TEC2012-37832-C02-01

Improving Performance Insensitivity of Large-scale Multiobjective Optimization via Monte Carlo Tree Search

The large-scale multiobjective optimization problem (LSMOP) is characterized by simultaneously optimizing multiple conflicting objectives and involving hundreds of decision variables. {Many real-world applications in engineering fields can be modeled as LSMOPs; simultaneously, engineering applications require insensitivity in performance.} This requirement usually means that the results from the algorithm runs should not only be good for every run in terms of performance but also that the performance of multiple runs should not fluctuate too much, i.e., the algorithm shows good insensitivity. Considering that substantial computational resources are requested for each run, it is essential to improve upon the performance of the large-scale multiobjective optimization algorithm, as well as the insensitivity of the algorithm. However, existing large-scale multiobjective optimization algorithms solely focus on improving the performance of the algorithms, leaving the insensitivity characteristics unattended. {In this work, we propose an evolutionary algorithm for solving LSMOPs based on Monte Carlo tree search, the so-called LMMOCTS, which aims to improve the performance and insensitivity for large-scale multiobjective optimization problems.} The proposed method samples the decision variables to construct new nodes on the Monte Carlo tree for optimization and evaluation. {It selects nodes with good evaluation for further search to reduce the performance sensitivity caused by large-scale decision variables.} We compare the proposed algorithm with several state-of-the-art designs on different benchmark functions. We also propose two metrics to measure the sensitivity of the algorithm. The experimental results confirm the effectiveness and performance insensitivity of the proposed design for solving large-scale multiobjective optimization problems.Comment: 12 pages, 11 figure

sParEGO – A hybrid optimization algorithm for expensive uncertain multi-objective optimization problems

Evaluations of candidate solutions to real-world problems are often expensive to compute, are characterised by uncertainties arising from multiple sources, and involve simultaneous consideration of multiple conflicting objectives. Here, the task of an optimizer is to find a set of solutions that offer alternative robust trade-offs between objectives, where robustness comprises some user-defined measure of the ability of a solution to retain high performance in the presence of uncertainties. Typically, understanding the robustness of a solution requires multiple evaluations of performance under different uncertain conditions – but such an approach is infeasible for expensive problems with a limited evaluation budget. To overcome this issue, a new hybrid optimization algorithm for expensive uncertain multi-objective optimization problems is proposed. The algorithm – sParEGO – uses a novel uncertainty quantification approach to assess the robustness of a candidate design without having to rely on expensive sampling techniques. Hypotheses on the relative performance of the algorithm compared to an existing method for deterministic problems are tested using two benchmark problems, and provide preliminary indication that sParEGO is an effective technique for identifying robust trade-off surfaces

Neural Architecture Search as Multiobjective Optimization Benchmarks: Problem Formulation and Performance Assessment

The ongoing advancements in network architecture design have led to remarkable achievements in deep learning across various challenging computer vision tasks. Meanwhile, the development of neural architecture search (NAS) has provided promising approaches to automating the design of network architectures for lower prediction error. Recently, the emerging application scenarios of deep learning have raised higher demands for network architectures considering multiple design criteria: number of parameters/floating-point operations, and inference latency, among others. From an optimization point of view, the NAS tasks involving multiple design criteria are intrinsically multiobjective optimization problems; hence, it is reasonable to adopt evolutionary multiobjective optimization (EMO) algorithms for tackling them. Nonetheless, there is still a clear gap confining the related research along this pathway: on the one hand, there is a lack of a general problem formulation of NAS tasks from an optimization point of view; on the other hand, there are challenges in conducting benchmark assessments of EMO algorithms on NAS tasks. To bridge the gap: (i) we formulate NAS tasks into general multi-objective optimization problems and analyze the complex characteristics from an optimization point of view; (ii) we present an end-to-end pipeline, dubbed $\texttt{EvoXBench}$, to generate benchmark test problems for EMO algorithms to run efficiently -- without the requirement of GPUs or Pytorch/Tensorflow; (iii) we instantiate two test suites comprehensively covering two datasets, seven search spaces, and three hardware devices, involving up to eight objectives. Based on the above, we validate the proposed test suites using six representative EMO algorithms and provide some empirical analyses. The code of $\texttt{EvoXBench}$ is available from $\href{https://github.com/EMI-Group/EvoXBench}{\rm{here}}$