A review of Nadir point estimation procedures using evolutionary approaches: a tale of dimensionality reduction

Estimation of the nadir objective vector is an important task, particularly for multi-objective optimization problems having more than two conflicting objectives. Along with the ideal point, nadir point can be used to normalize the objectives so that multi-objective optimization algorithms can be used more reliably. The knowledge of the nadir point is also a pre-requisite to many multiple criteria decision making methodologies.Moreover, nadir point is useful for an aid in interactive methodologies and visualization softwares catered for multi-objective optimization. However, the computation of exact nadir point formore than two objectives is not an easy matter, simply because nadir point demands the knowledge of extreme Paretooptimal solutions. In the past few years, researchers have proposed several nadir point estimation procedures using evolutionary optimization methodologies. In this paper, we review the past studies and reveal an interesting chronicle of events in this direction. To make the estimation procedure computationally faster and more accurate, the methodologies were refined one after the other by mainly focusing on increasingly lower dimensional subset of Pareto-optimal solutions. Simulation results on a number of numerical test problems demonstrate better efficacy of the approach which aims to find only the extreme Pareto-optimal points compared to its higher-dimensional counterparts

06501 Abstracts Collection -- Practical Approaches to Multi-Objective Optimization

From 10.12.06 to 15.12.06, the Dagstuhl Seminar 06501 ``Practical Approaches to Multi-Objective Optimization\u27\u27 was held in the International Conference and Research Center (IBFI), Schloss Dagstuhl. During the seminar, several participants presented their current research, and ongoing work and open problems were discussed. Abstracts of the presentations given during the seminar as well as abstracts of seminar results and ideas are put together in this paper. The first section describes the seminar topics and goals in general. Links to extended abstracts or full papers are provided, if available

A hybrid integrated multi-objective optimization procedure for estimating nadir point

A nadir point is constructed by the worst objective values of the solutions of the entire Pareto-optimal set. Along with the ideal point, the nadir point provides the range of objective values within which all Pareto-optimal solutions must lie. Thus, a nadir point is an important point to researchers and practitioners interested in multi-objective optimization. Besides, if the nadir point can be computed relatively quickly, it can be used to normalize objectives in many multi-criterion decision making tasks. Importantly, estimating the nadir point is a challenging and unsolved computing problem in case of more than two objectives. In this paper, we revise a previously proposed serial application of an EMO and a local search method and suggest an integrated approach for finding the nadir point. A local search procedure based on the solution of a bi-level achievement scalarizing function is employed to extreme solutions in stabilized populations in an EMO procedure. Simulation results on a number of problems demonstrate the viability and working of the proposed procedure

04461 Abstracts Collection -- Practical Approaches to Multi-Objective Optimization

From 07.11.04 to 12.11.04, the Dagstuhl Seminar 04461 ``Practical Approaches to Multi-Objective Optimization\u27\u27 was held in the International Conference and Research Center (IBFI), Schloss Dagstuhl. During the seminar, several participants presented their current research, and ongoing work and open problems were discussed. Abstracts of the presentations given during the seminar as well as abstracts of seminar results and ideas are put together in this paper. The first section describes the seminar topics and goals in general. Links to extended abstracts or full papers are provided, if available

Comparison of Interactive Evolutionary Multiobjective Optimization Methods Using an Artificial Decision Maker

In interactive evolutionary multiobjective optimization methods, preferences of a decision maker (DM), a domain expert, are iteratively incorporated to generate solutions that reflect the DM’s interests. When comparing these methods, we need means to capture features inherent in the nature of the solution processes. Namely, the DM’s preferences evolve while (s)he learns about the problem’s trade-offs and the feasibility of her/his own preferences. In this work, we implement an artificial decision maker (ADM) to evaluate reference point-based interactive evolutionary methods. A reference point consists of desirable values for the objectives. To simulate several iterations with an interactive method, the ADM generates reference points differently depending on two phases that can be distinguished in the solution process. In the learning phase, reference points simulate exploration to examine various Pareto optimal solutions to find a potential region of interest. Then, reference points of the decision phase mimic a progressive convergence towards the most preferred solution in this region. Each reference point is used to assess the methods’ performances per iteration. The ADM’s performance is demonstrated by comparing several interactive evolutionary methods on benchmark problems with up to 9 objectives. Future work includes consideration of other types of preference information and incorporation of a procedure to automatically switch from the learning to the decision phase

Treed Gaussian Process Regression for Solving Offline Data-Driven Continuous Multiobjective Optimization Problems

This is the final version. Available from MIT Press via the DOI in this recordFor offline data-driven multiobjective optimization problems (MOPs), no new data is available during the optimization process. Approximation models (or surrogates) are first built using the provided offline data and an optimizer, e.g. a multiobjective evolutionary algorithm, can then be utilized to find Pareto optimal solutions to the problem with surrogates as objective functions. In contrast to online data-driven MOPs, these surrogates cannot be updated with new data and, hence, the approximation accuracy cannot be improved by considering new data during the optimization process. Gaussian process regression (GPR) models are widely used as surrogates because of their ability to provide uncertainty information. However, building GPRs becomes computationally expensive when the size of the dataset is large. Using sparse GPRs reduces the computational cost of building the surrogates. However, sparse GPRs are not tailored to solve offline data-driven MOPs, where good accuracy of the surrogates is needed near Pareto optimal solutions. Treed GPR (TGPR-MO) surrogates for offline data-driven MOPs with continuous decision variables are proposed in this paper. The proposed surrogates first split the decision space into subregions using regression trees and build GPRs sequentially in regions close to Pareto optimal solutions in the decision space to accurately approximate tradeoffs between the objective functions. TGPR-MO surrogates are computationally inexpensive because GPRs are built only in a smaller region of the decision space utilizing a subset of the data. The TGPR-MO surrogates were tested on distance-based visualizable problems with various data sizes, sampling strategies, numbers of objective functions, and decision variables. Experimental results showed that the TGPR-MO surrogates are computationally cheaper and can handle datasets of large size. Furthermore, TGPR-MO surrogates produced solutions closer to Pareto optimal solutions compared to full GPRs and sparse GPRs.Academy of Finlan

A Hybrid Integrated Multi-Objective Optimization Procedure for Estimating Nadir Point

Abstract. A nadir point is constructed by the worst objective values of the solutions of the entire Pareto-optimal set. Along with the ideal point, the nadir point provides the range of objective values within which all Pareto-optimal solutions must lie. Thus, a nadir point is an important point to researchers and practitioners interested in multi-objective optimization. Besides, if the nadir point can be computed relatively quickly, it can be used to normalize objectives in many multi-criterion decision making tasks. Importantly, estimating the nadir point is a challenging and unsolved computing problem in case of more than two objectives. In this paper, we revise a previously proposed serial application of an EMO and a local search method and suggest an integrated approach for finding the nadir point. A local search procedure based on the solution of a bi-level achievement scalarizing function is employed to extreme solutions in stabilized populations in an EMO procedure. Simulation results on a number of problems demonstrate the viability and working of the proposed procedure

Quantifying and resolving conservation conflicts in forest landscapes via multiobjective optimization

Environmental planning for of the maintenance of different conservation objectives should take into account multiple contrasting criteria based on alternative uses of the landscape. We develop new concepts and approaches to describe and measure conflicts among conservation objectives and for resolving them via multiobjective optimization. To measure conflicts we introduce a compatibility index that quantifies how much targeting a certain conservation objective affects the capacity of the landscape for providing another objective. To resolve such conflicts we find compromise solutions defined in terms of minimax regret, i.e. minimizing the maximum percentage of deterioration among conservation objectives. Finally, we apply our approach for a case study of management for biodiversity conservation and development in a forest landscape. We study conflicts between six different forest species, and we identify management solutions for simultaneously maintaining multiple species’ habitat while obtaining timber harvest revenues. We employ the method for resolving conflicts at a large landscape level across a long 50-years forest planning horizon. Our multiobjective approach can be an instrument for guiding hard choices in the conservation-development nexus with a perspective of developing decision support tools for land use planning. In our case study multiple use management and careful landscape level planning using our approach can reduce conflicts among biodiversity objectives and offer room for synergies in forest ecosystems.peerReviewe

A Hybrid Integrated Multi-Objective Optimization Procedure for Estimating Nadir Point

Abstract. A nadir point is constructed by the worst objective values of the solutions of the entire Pareto-optimal set. Along with the ideal point, the nadir point provides the range of objective values within which all Pareto-optimal solutions must lie. Thus, a nadir point is an important point to researchers and practitioners interested in multi-objective optimization. Besides, if the nadir point can be computed relatively quickly, it can be used to normalize objectives in many multi-criterion decision making tasks. Importantly, estimating the nadir point is a challenging and unsolved computing problem in case of more than two objectives. In this paper, we revise a previously proposed serial application of an EMO and a local search method and suggest an integrated approach for finding the nadir point. A local search procedure based on the solution of a bi-level achievement scalarizing function is employed to extreme solutions in stabilized populations in an EMO procedure. Simulation results on a number of problems demonstrate the viability and working of the proposed procedure