Nondominated Points Research Articles

B&B method for discrete partial order optimization

The paper extends the branch and bound (B&B) method, primarily developed for the solution of global, discrete, and vector programming problems, to finding nondominated points in a partially ordered space/set. The framework of the generalized B&B method is standard, it includes partition, estimation, and pruning steps, but bounds are different, they are set-valued, and, as a particular case, the bounds may be singletons. For bounding, the method uses a set ordering in the following sense. One set is “less or equal” than the other if for any element of the first set there is a “greater or equal” element in the second one. The defined set ordering is a partial order in the space of sets consisting of mutually nondominated elements. In the B&B method, partitioning is applied to the parts of the original space with nondominated upper bounds. Parts with small upper bounds (less than some lower bound) are pruned. Convergence of the method to the set of all nondominated points is established. The acceleration with respect to the enumerative search is achieved through group evaluation of elements of the original space. Further, the developed B&B method is extended to vector partial order optimization. In the latter case, several partial orders are defined in the decision space. Finally, we develop a B&B method for a so-called constrained partial order optimization problem, where the feasible set is defined by a family of partial orders.

Optimizing a linear function over the nondominated set of multiobjective integer programs

AbstractIn this paper, we develop two algorithms to optimize a linear function over the nondominated set of multiobjective integer programs. The algorithms iteratively generate nondominated points and converge to the optimal solution reducing the feasible set. The first algorithm proposes improvements to an existing algorithm employing a decomposition and search procedure in finding a new point. Differently, the second algorithm maximizes one of the criteria throughout the algorithm and generates new points by setting bounds on the linear function value. The decomposition and search procedure in the algorithms is accompanied by problem‐specific mechanisms in order to explore the objective function space efficiently. The algorithms are designed to produce solutions that meet a prespecified accuracy level. We conduct experiments on multiobjective combinatorial optimization problems and show that the algorithms work well.

A biobjective Dijkstra algorithm

We generalize the Dijkstra algorithm to the Biobjective Shortest Path (BSP) problem. The proposed method keeps only one candidate label per node in a priority queue of size n. In this way, we introduce a novel algorithm to solve the one-to-all BSP problem determining all non-dominated points in the outcome space and one efficient path associated with each of them. For the case of the one-to-one BSP problem, we incorporate the classical bidirectional search scheme in the proposed algorithm to reduce the number of iterations in practice. The proposed algorithm also includes pruning strategies to avoid the computation of unnecessary labels. The result is a fast algorithm to solve the one-to-one BSP problem in large networks. A computational experiment comparing the performance of the proposed method and the state-of-the-art methods is included.

A comparative study of two key algorithms in multiple objective linear programming

Multiple objective linear programming problems are solved with a variety of algorithms. While these algorithms vary in philosophy and outlook, most of them fall into two broad categories: those that are decision space-based and those that are objective space-based. This paper reports the outcome of a computational investigation of two key representative algorithms, one of each category, namely the parametric simplex algorithm which is a prominent representative of the former and the primal variant of Bensons Outer-approximation algorithm which is a prominent representative of the latter. The paper includes a procedure to compute the most preferred nondominated point which is an important feature in the implementation of these algorithms and their comparison. Computational and comparative results on problem instances ranging from small to medium and large are provided.

An Economic-Reliability Security-Constrained Optimal Dispatch for Microgrids

This paper presents a new security-constrained multiobjective optimal dispatch (SC-MOOD) framework for an economic and reliable operation of microgrids. The framework is developed based on a computationally effective multiobjective optimization technique, Pareto concavity elimination transformation (PaCcET). The new method considers grid steady-state and dynamic constraints while solving the optimal dispatch in both grid-connected and islanded modes. The constraints consist of power balance, voltage magnitude, line flows, power generation, frequency, and voltage transients. The proposed framework finds the most economic operating solutions to not only minimize the generation cost but also minimize the reliability cost. In this paper, the PaCcET is utilized to solve the SC-MOOD. The PaCcET uses an extraordinary transformation to first transfer all the points from a multiobjective space to a transformed objective space. It then solves a linear combination of transformed objectives using a single-objective optimizer to find all the nondominated points of the original multiobjective space. The performance of the new framework is verified using the simulation results in a microgrid with several distributed energy resources in both grid-connected and islanded modes. The results are then compared with two multiobjective optimization techniques, nondominated sorting genetic algorithm II and multiobjective particle swarm optimization, as two well-known benchmarks.

Disjunctive Programming for Multiobjective Discrete Optimisation

In this paper, I view and present the multiobjective discrete optimisation problem as a particular case of disjunctive programming where one seeks to identify efficient solutions from within a disjunction formed by a set of systems. The proposed approach lends itself to a simple yet effective iterative algorithm that is able to yield the set of all nondominated points, both supported and nonsupported, for a multiobjective discrete optimisation problem. Each iteration of the algorithm is a series of feasibility checks and requires only one formulation to be solved to optimality that has the same number of integer variables as that of the single objective formulation of the problem. The application of the algorithm shows that it is particularly effective when solving constrained multiobjective discrete optimisation problem instances. The online supplement is available at https://doi.org/10.1287/ijoc.2017.0804 .

On fuzzy approaches for enlarging skyline query results

In the last decade, skyline queries have gained much attention and are proved to be valuable for multi-criteria decisions. Based on the concept of Pareto dominance, they return the non-dominated points, called the skyline points. In practice, it may happen that the skyline only contains a small number of points which could be insufficient for the user needs. In this paper, we discuss two fuzzy-set-based approaches to enriching the small skyline with particular points that could serve the decision makers’ needs. The basic idea consists in identifying the most interesting points among the non-skyline ones. On the one hand, we introduce a novel fuzzy dominance relationship which makes more demanding the dominance between the points of interest. So, much points would be considered as incomparable and then as elements of the new relaxed skyline. On the other hand, we leverage an appropriate fuzzy closeness relation to retrieve non skyline points that are fuzzily close to some skyline points. Furthermore, we develop efficient algorithms to compute the relaxed variants of skyline. Extensive experiments are conducted to demonstrate the effectiveness of our approaches and analyze the performance of the proposed algorithms. A comparative study between the approaches presented is made as well.

ND-Tree-Based Update: A Fast Algorithm for the Dynamic Nondominance Problem

In this paper, we propose a new method called ND-Tree-based update (ND-Tree) for the dynamic nondominance problem, i.e., the problem of online update of a Pareto archive composed of mutually nondominated points. It uses a new ND-Tree data structure in which each node represents a subset of points contained in a hyperrectangle defined by its local approximate ideal and nadir points. By building subsets containing points located close in the objective space and using basic properties of the local ideal and nadir points we can efficiently avoid searching many branches in the tree. ND-Tree may be used in multiobjective evolutionary algorithms and other multiobjective metaheuristics to update an archive of potentially nondominated points. We prove that the proposed algorithm has sublinear time complexity under mild assumptions. We experimentally compare ND-Tree to the simple list, Quad-tree, and M-Front methods using artificial and realistic benchmarks with up to ten objectives and show that with this new method substantial reduction of the number of point comparisons and computational time can be obtained. Furthermore, we apply the method to the nondominated sorting problem showing that it is highly competitive to some recently proposed algorithms dedicated to this problem.

Multi-objective optimization of WEDM process parameters for taper cutting of AISI D2 tool steel

Wire electrical discharge machining (WEDM) process is one of the most popular method for the producing complex geometries in hard and wear resistant materials such as those used in tooling industry. This taper cutting involves in making of inclined surface, and it is particularly significant in the manufacturing of tooling that requires draft angles. In the present paper, experiments were conducted in order to find the effect of process parame- ters such as taper angle, geometrical thickness and servo voltage on response variables such as angular error and cutting speed on AISI D2 tool steel using statistical design of experiments. The experiments were planned using Central Composite design (CCD) which is part of Response Surface Methodology (RSM) – involving three variables with five levels. Multi objective optimization was conducted for maximizing the Cutting speed and minimizing the angular error using genetic algorithm. The optimization procedure leads to creation of non-dominated optimal points which gave an insight regarding the optimal operating conditions of the process.

GoNDEF: an exact method to generate all non-dominated points of multi-objective mixed-integer linear programs

Most real-world problems involve multiple conflicting criteria. These problems are called multi-criteria/multi-objective optimization problems (MOOP). The main task in solving MOOPs is to find the non-dominated (ND) points in the objective space or efficient solutions in the decision space. A ND point is a point in the objective space with objective function values that cannot be improved without worsening another objective function. In this paper, we present a new method that generates the set of ND points for a multi-objective mixed-integer linear program (MOMILP). The Generator of ND and Efficient Frontier (GoNDEF) for MOMILPs finds that the ND points represented as points, line segments, and facets consist of every type of ND point. First, the GoNDEF sets integer variables to the values that result in ND points. Fixing integer variables to specific values results in a multi-objective linear program (MOLP). This MOLP has its own set of ND points. A subset of this set establishes a subset of the ND points set of the MOMILP. In this paper, we present an extensive theoretical analysis of the GoNDEF and illustrate its effectiveness on a set of instance problems.

Classification of ADHD with bi-objective optimization

Attention Deficit Hyperactive Disorder (ADHD) is one of the most common diseases in school aged children. In this paper, we consider using fMRI data with classification techniques to aid the diagnosis of ADHD and propose a bi-objective ADHD classification scheme based on L1-norm support vector machine (SVM). In our classification model, two objectives, namely, the margin of separation and the empirical error are considered at the same time. Then the normal boundary intersection (NBI) method of Das and Dennis is used to solve the bi-objective optimization problem. A representative nondominated set which reflects the entire trade-off information between the two objectives is obtained. Each representative nondominated point in the set corresponds to an efficient classifier. Finally a decision maker can choose a final efficient classifier from the set according to the performance of each classifier. Our scheme avoids the trial and error process for regularization hyper-parameter selection. Experimental results show that our bi-objective optimization classification scheme for ADHD diagnosis performs considerably better than some traditional classification methods.

Implicit enumeration strategies for the hypervolume subset selection problem

The hypervolume subset selection problem arises within selection procedures of multiobjective evolutionary algorithms as well as for extracting a succinct subset of optimal solutions of a multiobjective optimization problem. Although efficient algorithms are known for two dimensions, this problem becomes NP-hard for more dimensions. In this article, we introduce an integer linear programming formulation for this problem for more than two dimensions that is based on the decomposition of the dominated region of the set of nondominated points. Moreover, we propose a branch and bound algorithm that uses combinatorial arguments and discuss bounding strategies based on the application of three upper bounds. We analyze the performance of the two solution approaches on a wide range of instances. The results indicate that the branch and bound algorithm has better performance for several orders of magnitude.

Finding a representative nondominated set for multi-objective mixed integer programs

In this paper, we develop algorithms to find small representative sets of nondominated points that are well spread over the nondominated frontiers for multi-objective mixed integer programs. We evaluate the quality of representations of the sets by a Tchebycheff distance-based coverage gap measure. The first algorithm aims to substantially improve the computational efficiency of an existing algorithm that is designed to continue generating new points until the decision maker (DM) finds the generated set satisfactory. The algorithm improves the coverage gap value in each iteration by including the worst represented point into the set. The second algorithm, on the other hand, guarantees to achieve a desired coverage gap value imposed by the DM at the outset. In generating a new point, the algorithm constructs territories around the previously generated points that are inadmissible for the new point based on the desired coverage gap value. The third algorithm brings a holistic approach considering the solution space and the number of representative points that will be generated together. The algorithm first approximates the nondominated set by a hypersurface and uses it to plan the locations of the representative points. We conduct computational experiments on randomly generated instances of multi-objective knapsack, assignment, and mixed integer knapsack problems and show that the algorithms work well.

Design and analysis of diversity-based parent selection schemes for speeding up evolutionary multi-objective optimisation

Parent selection in evolutionary algorithms for multi-objective optimisation is usually performed by dominance mechanisms or indicator functions that prefer non-dominated points. We propose to refine the parent selection on evolutionary multi-objective optimisation with diversity-based metrics. The aim is to focus on individuals with a high diversity contribution located in poorly explored areas of the search space, so the chances of creating new non-dominated individuals are better than in highly populated areas. We show by means of rigorous runtime analysis that the use of diversity-based parent selection mechanisms in the Simple Evolutionary Multi-objective Optimiser (SEMO) and Global SEMO for the well known bi-objective functions OneMinMax and LOTZ can significantly improve their performance. Our theoretical results are accompanied by experimental studies that show a correspondence between theory and empirical results and motivate further theoretical investigations in terms of stagnation. We show that stagnation might occur when favouring individuals with a high diversity contribution in the parent selection step and provide a discussion on which scheme to use for more complex problems based on our theoretical and experimental results.

Computing and Updating Hypervolume Contributions in Up to Four Dimensions

Arising in connection with multiobjective selection and archiving, the hypervolume subset selection problem (HSSP) consists in finding a subset of size k ≤ n of a set X ⊂ ℝ <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">d</sup> of n nondominated points in d-dimensional space that maximizes the hypervolume indicator. The incremental greedy approximation to the HSSP has an approximation guarantee of 1 - 1/e, and is polynomial in n and k, while no polynomial exact algorithms are known for d ≥ 3. The decremental greedy counterpart has no known approximation guarantee, but is potentially faster for large k, and still leads to good approximations in practice. The computation and update of individual hypervolume contributions are at the core of the implementation of this greedy strategy. In this paper, new algorithms for the computation and update of hypervolume contributions are developed. In three dimensions, updating the total hypervolume and all individual contributions under single-point changes is performed in linear time, while in the 4-D case all contributions are computed in O(n <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> ) time. As a consequence, the decremental greedy approximation to the HSSP can now be obtained in O(n(n - k) +n log n) and O(n <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> (n - k)) time for d = 3 and d = 4, respectively. Experimental results show that the proposed algorithms significantly outperform existing ones.

Parallel Processing Strategies for Skyline Queries Tolerant to Outliers

This paper deals with the problem of efficiently computing the most preferred objects, in the sense of Pareto ordering, in a set of multidimensional objects involving outliers. These outliers, for instance, odd data from ad sales websites, can hide interesting points, so it is important to limit their impact. To do this, we propose a graded Skyline based on the typicality of each object. Such an approach allows to supply an α-cut of the skyline containing both nondominated points, and almost nondominated but more representative points. The computational cost of a naive algorithm performing this task is very high, especially in high-dimensional spaces, due to the large number of comparisons between objects. Since common skyline optimizations cannot be used in our approach, we propose an implementation on parallel architectures, CUDA and OpenMP. Experimental results on both real-world and synthetic data sets show that this processing strategy makes the approach scalable.

A hybrid approach for biobjective optimization

A large number of the real world planning problems which are today solved using Operations Research methods are actually multiobjective planning problems, but most of them are solved using singleobjective methods. The reason for converting, i.e. simplifying, multiobjective problems to singleobjective problems is that no standard multiobjective solvers exist and specialized algorithms need to be programmed from scratch.In this article we will present a hybrid approach, which operates both in decision space and in objective space. The approach enables massive efficient parallelization and can be used to a wide variety of biobjective Mixed Integer Programming models. We test the approach on the biobjective extension of the classic traveling salesman problem, on the standard datasets, and determine the full set of nondominated points. This has only been done once before (Florios and Mavrotas, 2014), and in our approach we do it in a fraction of the time.

Variants of the $$ \varepsilon $$ ε -constraint method for biobjective integer programming problems: application to p-median-cover problems

We conduct an in-depth analysis of the \(\varepsilon \)-constraint method (ECM) for finding the exact Pareto front for biobjective integer programming problems. We have found up to six possible different variants of the ECM. We first discuss the complexity of each of these variants and their relationship with other exact methods for solving biobjective integer programming problems. By extending some results of Neumayer and Schweigert (OR Spektrum 16:267–276, 1994), we develop two variants of the ECM, both including an augmentation term and requiring \(N+1\) integer programs to be solved, where N is the number of nondominated points. In addition, we present another variant of the ECM, based on the use of elastic constraints and also including an augmentation term. This variant has the same complexity, namely \(N+1\), which is the minimum reached for any exact method. A comparison of the different variants is carried out on a set of biobjective location problems which we call p-median-cover problems; these include the objectives of the p-median and the maximal covering problems. As computational results show, for this class of problems, the augmented ECM with elastic constraint is the most effective variant for finding the Pareto front in an exact manner.

Shortest Paths with Shortest Detours

This paper is concerned with a biobjective routing problem, called the shortest path with shortest detour problem, in which the length of a route is minimized as one criterion and, as second, the maximal length of a detour route if the chosen route is blocked is minimized. Furthermore, the relation to robust optimization is pointed out, and we present a new polynomial time algorithm, which computes a minimal complete set of efficient paths for the shortest path with shortest detour problem. Moreover, we show that the number of nondominated points is bounded by the number of arcs in the graph.

A ranking algorithm for bi-objective quadratic fractional integer programming problems

An algorithm to solve bi-objective quadratic fractional integer programming problems is presented in this paper. The algorithm uses -scalarization technique and a ranking approach of the integer feasible solution to find all nondominated points. In order to avoid solving non-linear integer programming problems during this ranking scheme, the existence of a linear or a linear fractional function is established, which acts as a lower bound on the values of first objective function of the bi-objective problem over the entire feasible set. Numerical examples are also presented in support of the theory.