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

The Kalai-Smorodinski solution for many-objective Bayesian optimization

An ongoing aim of research in multiobjective Bayesian optimization is to extend its applicability to a large number of objectives. While coping with a limited budget of evaluations, recovering the set of optimal compromise solutions generally requires numerous observations and is less interpretable since this set tends to grow larger with the number of objectives. We thus propose to focus on a specific solution originating from game theory, the Kalai-Smorodinsky solution, which possesses attractive properties. In particular, it ensures equal marginal gains over all objectives. We further make it insensitive to a monotonic transformation of the objectives by considering the objectives in the copula space. A novel tailored algorithm is proposed to search for the solution, in the form of a Bayesian optimization algorithm: sequential sampling decisions are made based on acquisition functions that derive from an instrumental Gaussian process prior. Our approach is tested on four problems with respectively four, six, eight, and nine objectives. The method is available in the Rpackage GPGame available on CRAN at https://cran.r-project.org/package=GPGame

SAP- Modular Tool for Specification and Analysis of User Preferences in Multiple-Criteria Model Analysis

Model based Decision Support Systems (DSS) often use multiple-criteria optimization for selecting Pareto-efficient solutions. Such a selection is based on interactive specification of user preferences. This can be done by specification of aspiration and reservation levels for criteria. Diverse graphical user interface could be used for specification of these levels as well as for interpretation of results. In the approach presented in this paper the specified aspiration and reservation levels are used for generation of component achievement functions for corresponding criteria. Such functions can be interpreted as fuzzy membership functions or as functions, which reflect the degree of satisfaction with given values of criteria. The paper outlines the methodological background and modular structure of a DSS shell for multiple-criteria analysis of decision problems that can be represented as Linear Programming (LP) or Mixed Integer Programming (MIP) problems. The DSS shell has been used at IIASA for analysis of decision problems in water quality management and land use for sustainable development planning. The pilot implementation of one component of that DSS, namely the modular software tool for interactive specification of user preferences is described in more detail. The tool has been also used as in a DSS for analysis of non-linear problems in several engineering applications

Exact And Representative Algorithms For Multi Objective Optimization

In most real-life problems, the decision alternatives are evaluated with multiple conflicting criteria. The entire set of non-dominated solutions for practical problems is impossible to obtain with reasonable computational effort. Decision maker generally needs only a representative set of solutions from the actual Pareto front. First algorithm we present is for efficiently generating a well dispersed non-dominated solution set representative of the Pareto front which can be used for general multi objective optimization problem. The algorithm first partitions the criteria space into grids to generate reference points and then searches for non-dominated solutions in each grid. This grid-based search utilizes achievement scalarization function and guarantees Pareto optimality. The results of our experimental results demonstrate that the proposed method is very competitive with other algorithms in literature when representativeness quality is considered; and advantageous from the computational efficiency point of view. Although generating the whole Pareto front does not seem very practical for many real life cases, sometimes it is required for verification purposes or where DM wants to run his decision making structures on the full set of Pareto solutions. For this purpose we present another novel algorithm. This algorithm attempts to adapt the standard branch and bound approach to the multi objective context by proposing to branch on solution points on objective space. This algorithm is proposed for multi objective integer optimization type of problems. Various properties of branch and bound concept has been investigated and explained within the multi objective optimization context such as fathoming, node selection, heuristics, as well as some multi objective optimization specific concepts like filtering, non-domination probability, running in parallel. Potential of this approach for being used both as a full Pareto generation or an approximation approach has been shown with experimental studies

Optimal Forest Strategies for Addressing Tradeoffs and Uncertainty in Economic Development under Old-Growth Constraints

In Canada, governments have historically promoted economic development in rural regions by promoting exploitation of natural resources, particularly forests. Forest resources are an economic development driver in many of the more than 80% of native communities located in forest regions. But forests also provide aboriginal people with cultural and spiritual values, and non-timber forest amenities (e.g., biodiversity, wildlife harvests for meat and fur, etc.), that are incompatible with timber exploitation. Some cultural and other amenities can only be satisfied by maintaining a certain amount of timber in an old-growth state. In that case, resource constraints might be too onerous to satisfy development needs. We employ compromise programming and fuzzy programming to identify forest management strategies that best compromise between development and other objectives, applying our models to an aboriginal community in northern Alberta. In addition to describing how mathematical programming techniques can be applied to regional development and forest management, we conclude from the analysis that no management strategy is able to satisfy all of the technical, environmental and social/cultural constraints and, at the same time, offer aboriginal peoples forest-based economic development. Nonetheless, we demonstrate that extant forest management policies can be improved upon.forest-dependent aboriginal communities, boreal forest, compromise and fuzzy programming, sustainability and uncertainty, International Development, Resource /Energy Economics and Policy, R11, Q23, Q01, C61,