Fast computation of equispaced Pareto manifolds and Pareto fronts for multiobjective optimization problems

In this paper, we consider the problem of generating a well sampled discrete representation of the Pareto manifold or the Pareto front corresponding to the equilibrium points of a multi-objective optimization problem. We show how the introduction of simple additional constraints into a continuation procedure produces equispaced points in either of those two sets. Moreover, we describe in detail a novel algorithm for global continuation that requires two orders of magnitude less function evaluations than evolutionary algorithms commonly used to solve this problem. The performance of the methods is demonstrated on problems from the current literature.     

Pareto optima of isostatic trusses

Multicriterion optimization of elastic stress limited isostatic trusses is considered and a numerical method for determining the Pareto optimal set of the problem is developed. The weight of the structure and some chosen nodal displacements are taken as design criteria, and member areas are used as design variables. The corresponding bicriterion problem with weight and one displacement in the objective function is solved exactly, and this result is used as a basis of the proposed method. By introducing certain parameters, each joined with one displacement criterion, the Pareto optimal solutions of the general problem may be obtained with any accuracy. Two examples are given to illustrate the method.     

Image thresholding based on Pareto multiobjective optimization

A new image thresholding method based on multiobjective optimization following the Pareto approach is presented. This method allows to optimize several segmentation criteria simultaneously, in order to improve the quality of the segmentation. To obtain the Pareto front and then the optimal Pareto solution, we adapted the evolutionary algorithm NSGA-II (Deb et al., 2002). The final solution or Pareto solution corresponds to that allowing a compromise between the different segmentation criteria, without favouring any one. The proposed method was evaluated on various types of images. The obtained results show the robustness of the method, and its non dependence towards the kind of the image to be segmented.     

Scalable Pareto set generation for multiobjective co-design problems in water distribution networks: a continuous relaxation approach

In this paper, we study the multiobjective co-design problem of optimal valve placement and operation in water distribution networks, addressing the minimization of average pressure and pressure variability indices. The presented formulation considers nodal pressures, pipe flows and valve locations as decision variables, where binary variables are used to model the placement of control valves. The resulting optimization problem is a multiobjective mixed integer nonlinear optimization problem. As conflicting objectives, average zone pressure and pressure variability can not be simultaneously optimized. Therefore, we present the concept of Pareto optima sets to investigate the trade-offs between the two conflicting objectives and evaluate the best compromise. We focus on the approximation of the Pareto front, the image of the Pareto optima set through the objective functions, using the weighted sum, normal boundary intersection and normalized normal constraint scalarization techniques. Each of the three methods relies on the solution of a series of single-objective optimization problems, which are mixed integer nonlinear programs (MINLPs) in our case. For the solution of each single-objective optimization problem, we implement a relaxation method that solves a sequence of nonlinear programs (NLPs) whose stationary points converge to a stationary point of the original MINLP. The relaxed NLPs have a sparse structure that come from the sparse water network graph constraints. In solving the large number of relaxed NLPs, sparsity is exploited by tailored techniques to improve the performance of the algorithms further and render the approaches scalable for large scale networks. The features of the proposed scalarization approaches are evaluated using a published benchmarking network model.     

Note on singular optima in laminate design problems

This paper studies the design of laminates subject to restrictions on the ply strength. The minimum weight design is considered. It is shown that this formulation includes singular optima, which are similar to the ones observed in topology optimization including local stress constraints. In laminate design, these singular optima are linked to the removal of ‘zero thickness’ plies from the stacking sequence. It is shown how the fiber orientation variables can circumvent the singularity by relaxing the strength constraints related to such vanishing plies. This demonstrates the key role of fiber orientations in the optimization of laminates and the need for their efficient treatment as design variables.     

Uncertainty management in multiobjective hydro-thermal self-scheduling under emission considerations

In this paper, a stochastic multiobjective framework is proposed for a day-ahead short-term Hydro Thermal Self-Scheduling (HTSS) problem for joint energy and reserve markets. An efficient linear formulations are introduced in this paper to deal with the nonlinearity of original problem due to the dynamic ramp rate limits, prohibited operating zones, operating services of thermal plants, multi-head power discharge characteristics of hydro generating units and spillage of reservoirs. Besides, system uncertainties including the generating units’ contingencies and price uncertainty are explicitly considered in the stochastic market clearing scheme. For the stochastic modeling of probable multiobjective optimization scenarios, a lattice Monte Carlo simulation has been adopted to have a better coverage of the system uncertainty spectrum. Consequently, the resulting multiobjective optimization scenarios should concurrently optimize competing objective functions including GENeration COmpany's (GENCO's) profit maximization and thermal units’ emission minimization. Accordingly, the ɛ-constraint method is used to solve the multiobjective optimization problem and generate the Pareto set. Then, a fuzzy satisfying method is employed to choose the most preferred solution among all Pareto optimal solutions. The performance of the presented method is verified in different case studies. The results obtained from ɛ-constraint method is compared with those reported by weighted sum method, evolutionary programming-based interactive Fuzzy satisfying method, differential evolution, quantum-behaved particle swarm optimization and hybrid multi-objective cultural algorithm, verifying the superiority of the proposed approach.     

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.     

Decomposition methods in multiobjective discrete-time dynamic problems

Decomposition methods for multicriteria dynamic (discrete-time) problems are derived. In these methods, the original problem is reduced to a series of multicriteria subproblems related to individual stages. Hence, the dimensionality of decision variables in each subproblem is smaller than in the original problem. The following decomposition procedures for such problems are developed: (1) a dynamic programming method, (2) a two-point boundary value problem method, (3) multilevel methods, and (4) the formulation of a temporal hierarchy. For completeness, methods for multicriteria dynamic problems are reviewed that, at the outset, transform a problem into a series of single-objective problems. Formulation of the multiobjective problem in the context of a multilayer temporal hierarchy is also presented. The temporal structure motivates problem simplification by decomposing the overall decision-making problem according to relative time scales.     

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.     

Numerical treatment of multiobjective optimal control problems

We present a numerical procedure for solving optimal control problems with both linear terminal constraints and multiple criteria. Using a Chebyshev spectral procedure, the problem reduces to a constrained optimization problem which can be solved using hybrid penalty partial quadratic interpolation (HPPQI) technique. The proposed procedure compares quite favorably with other methods on a sample of well-known examples.     

Pareto frontier exploration in multiobjective topology optimization using adaptive weighting and point selection schemes

Topology optimization has been used in many industries and applied to a variety of design problems. In real-world engineering design problems, topology optimization problems often include a number of conflicting objective functions, such to achieve maximum stiffness and minimum mass of a design target. The existence of conflicting objective functions causes the results of the topology optimization problem to appear as a set of non-dominated solutions, called a Pareto-optimal solution set. Within such a solution set, a design engineer can easily choose the particular solution that best meets the needs of the design problem at hand. Pareto-optimal solution sets can provide useful insights that enable the structural features corresponding to a certain objective function to be isolated and explored. This paper proposes a new Pareto frontier exploration methodology for multiobjective topology optimization problems. In our methodology, a level set-based topology optimization method for a single-objective function is extended for use in multiobjective problems, using a population-based approach in which multiple points in the objective space are updated and moved to the Pareto frontier. The following two schemes are introduced so that Pareto-optimal solution sets can be efficiently obtained. First, weighting coefficients are adaptively determined considering the relative position of each point. Second, points in sparsely populated areas are selected and their neighborhoods are explored. Several numerical examples are provided to illustrate the effectiveness of the proposed method.     

On some multiobjective optimization problems arising in biology

In this paper, we discuss modelling and solving some multiobjective optimization problems arising in biology. A class of comparison problems for string selection in molecular biology and a relocation problem in conservation biology are modelled as multiobjective optimization programmes. Some discussions about applications, solvability and different variants of the obtained models are given, as well. A crucial part of the study is based upon the Pareto optimization which refers to the Pareto solutions of multiobjective optimization problems. For such solution, improvement of some objective function can only be obtained at the expense of the deterioration of at least one other objective function.     

A new graphical visualization of n-dimensional Pareto front for decision-making in multiobjective optimization

New challenges in engineering design lead to multiobjective (multicriteria) problems. In this context, the Pareto front supplies a set of solutions where the designer (decision-maker) has to look for the best choice according to his preferences. Visualization techniques often play a key role in helping decision-makers, but they have important restrictions for more than two-dimensional Pareto fronts. In this work, a new graphical representation, called Level Diagrams, for n-dimensional Pareto front analysis is proposed. Level Diagrams consists of representing each objective and design parameter on separate diagrams. This new technique is based on two key points: classification of Pareto front points according to their proximity to ideal points measured with a specific norm of normalized objectives (several norms can be used); and synchronization of objective and parameter diagrams. Some of the new possibilities for analyzing Pareto fronts are shown. Additionally, in order to introduce designer preferences, Level Diagrams can be coloured, so establishing a visual representation of preferences that can help the decision-maker. Finally, an example of a robust control design is presented - a benchmark proposed at the American Control Conference. This design is set as a six-dimensional multiobjective problem.     

Infine functions and nonsmooth multiobjective optimization problems

In this paper, we consider notion of infine functions and we establish necessary and sufficient optimality conditions for a feasible solution of a multiobjective optimization problem involving mixed constraints (equality and inequality) to be an efficient or properly efficient solution. We also obtain duality theorems for Wolf type and Mond-Weir type duals under the generalized invexity assumptions.     

Solving multiobjective problems using cat swarm optimization

This paper proposes a new multiobjective evolutionary algorithm (MOEA) by extending the existing cat swarm optimization (CSO). It finds the nondominated solutions along the search process using the concept of Pareto dominance and uses an external archive for storing them. The performance of our proposed approach is demonstrated using standard test functions. A quantitative assessment of the proposed approach and the sensitivity test of different parameters is carried out using several performance metrics. The simulation results reveal that the proposed approach can be a better candidate for solving multiobjective problems (MOPs).     

Asymptotic convergence of metaheuristics for multiobjective optimization problems

This paper analyzes the convergence of metaheuristics used for multiobjective optimization problems in which the transition probabilities use a uniform mutation rule. We prove that these algorithms converge only if elitism is used.     

Uncertainty in bipolar preference problems

Preferences and uncertainty are common in many real-life problems. In this article, we focus on bipolar preferences and uncertainty modelled via uncontrollable variables, and we assume that uncontrollable variables are specified by possibility distributions over their domains. To tackle such problems, we concentrate on uncertain bipolar problems with totally ordered preferences, and we eliminate the uncertain part of the problem, while making sure that some desirable properties hold about the robustness of the problem and its relationship with the preference of the optimal solutions. We also consider several semantics to order the solutions according to different attitudes with respect to the notions of preference and robustness.     

Exact algorithms for OWA-optimization in multiobjective spanning tree problems

This paper deals with the multiobjective version of the optimal spanning tree problem. More precisely, we are interested in determining the optimal spanning tree according to an Ordered Weighted Average (OWA) of its objective values. We first show that the problem is weakly NP-hard. We then propose different mixed integer programming formulations, according to different subclasses of OWA functions. Furthermore, we provide various preprocessing procedures, the validity scopes of which depend again on the considered subclass of OWA functions. For designing such procedures, we propose generalized optimality conditions and efficiently computable bounds. These procedures enable to reduce the size of the instances before launching an off-the-shelf software for solving the mixed integer program. Their impact on the resolution time is evaluated on the basis of numerical experiments.     

A hybrid system for multiobjective problems – A case study in NP-hard problems

In attempt to solve multiobjective problems, various mathematical and stochastic methods have been developed. The methods operate based on mathematical models while in most cases these models are drastically simplified imagine of real world problems.In this study, a hybrid intelligent system is used instead of mathematical models. The main core of the system is fuzzy rule base which maps decision space (Z) to solution space (X). The system is designed on noninferior region and gives a big picture of this region in the pattern of fuzzy rules. Since some solutions may be infeasible; then specified feedforward neural network is used to obtain noninferior solutions in an exterior movement.In addition, numerical examples of well-known NP-hard problems (i.e. multiobjective traveling salesman problem and multiobjective knapsack problem) are provided to clarify the accuracy of developed system.