Dhish Kumar Saxena

Objective Reduction in Many-Objective Optimization: Linear and Nonlinear Algorithms

The difficulties faced by existing multiobjective evolutionary algorithms (MOEAs) in handling many-objective problems relate to the inefficiency of selection operators, high computational cost, and difficulty in visualization of objective space. While many approaches aim to counter these difficulties by increasing the fidelity of the standard selection operators, the objective reduction approach attempts to eliminate objectives that are not essential to describe the Pareto-optimal front (POF). If the number of essential objectives is found to be two or three, the problem could be solved by the existing MOEAs. It implies that objective reduction could make an otherwise unsolvable (many-objective) problem solvable. Even when the essential objectives are four or more, the reduced representation of the problem will have favorable impact on the search efficiency, computational cost, and decision-making. Hence, development of generic and robust objective reduction approaches becomes important. This paper presents a principal component analysis and maximum variance unfolding based framework for linear and nonlinear objective reduction algorithms, respectively. The major contribution of this paper includes: 1) the enhancements in the core components of the framework for higher robustness in terms of applicability to a range of problems with disparate degree of redundancy; mechanisms to handle input data that poorly approximates the true POF; and dependence on fewer parameters to minimize the variability in performance; 2) proposition of an error measure to assess the quality of results; 3) sensitivity analysis of the proposed algorithms for the critical parameter involved, and the characteristics of the input data; and 4) study of the performance of the proposed algorithms vis-à-vis dominance relation preservation based algorithms, on a wide range of test problems (scaled up to 50 objectives) and two real-world problems.

On Handling a Large Number of Objectives A Posteriori and During Optimization

Dimensionality reduction methods are used routinely in statistics, pattern recognition, data mining, and machine learning to cope with high-dimensional spaces. Also in the case of high-dimensional multiobjective optimization problems, a reduction of the objective space can be beneficial both for search and decision making. New questions arise in this context, e.g., how to select a subset of objectives while preserving most of the problem structure. In this chapter, two different approaches to the task of objective reduction are developed, one based on assessing explicit conflicts, the other based on principal component analysis (PCA). Although both methods use different principles and preserve different properties of the underlying optimization problems, they can be effectively utilized either in an a posteriori scenario or during search. Here, we demonstrate the usability of the conflict-based approach in a decision-making scenario after the search and show how the principal-component-based approach can be integrated into an evolutionary multicriterion optimization (EMO) procedure.

Dimensionality reduction of objectives and constraints in multi-objective optimization problems: A system design perspective

The notion of optimal system design holds that in order to dasiatrulypsila maximize/minimize an objective function, the feasible set needs to be optimized. Inspired by it, the attempt in our recent work was to incorporate constraint-reduction in our earlier proposed procedures on dimensionality reduction of objectives. In that, while targetting constrained single-objective optimization problems (SOPs), we could arrive at a critical set of constraints and also their importance based rank-ordering. This information was used to study the shift from the constrained to the unconstrained optima. The methodology above was based on treating the a priori stated constraints as objectives besides the original-objective, and on applying (K. Deb et al., 2006), (D.K. Saxena et al., 2007) to this combined objective set-but-without constraints. In this work, the endeavor is to extend the above notion to the realm of multi-objective optimization problems (MOPs). Towards it, while we hire much from the above methodology, we make a fundamental shift, in that, we retain the a priori stated constraints, while evaluating the combined objective set. The motivation for this shift lies, in that, it allows more effective realization of the notion of system design than the approach in (D.K. Saxen et al., 2007). Reasonable effort has been spent on establishing this argument. Incorporating this change, a procedure for simultaneous reduction in objectives and constraints (for both SOPs, MOPs) is proposed, which also defines a realizable path towards optimal system design. Finally, the procedure is demonstrated on two test problems and one real world problem.

Using objective reduction and interactive procedure to handle many-objective optimization problems

A number of practical optimization problems are posed as many-objective (more than three objectives) problems. Most of the existing evolutionary multi-objective optimization algorithms, which target the entire Pareto-front are not equipped to handle many-objective problems. Though there have been copious efforts to overcome the challenges posed by such problems, there does not exist a generic procedure to effectively handle them. This paper presents a simplify and solve framework for handling many-objective optimization problems. In that, a given problem is simplified by identification and elimination of the redundant objectives, before interactively engaging the decision maker to converge to the most preferred solution on the Pareto-optimal front. The merit of performing objective reduction before interacting with the decision maker is two fold. Firstly, the revelation that certain objectives are redundant, significantly reduces the complexity of the optimization problem, implying lower computational cost and higher search efficiency. Secondly, it is well known that human beings are not efficient in handling several factors (objectives in the current context) at a time. Hence, simplifying the problem a priori addresses the fundamental issue of cognitive overload for the decision maker, which may help avoid inconsistent preferences during the different stages of interactive engagement. The implementation of the proposed framework is first demonstrated on a three-objective problem, followed by its application on two real-world engineering problems.

Framework for many-objective test problems with both simple and complicated pareto-set shapes

Test problems have played a fundamental role in understanding the strengths and weaknesses of the existing Evolutionary Multiobjective Optimization (EMO) algorithms. A range of test problems exist which have enabled the research community to understand how the performance of EMO algorithms is affected by the geometrical shape of the Pareto front (PF), i.e., PF being convex, concave or mixed. However, the shapes of the Pareto Set (PS) of most of these test problems are rather simple (linear or quadratic), even though the real-world engineering problems are expected to have complicated PS shapes. The state-of-the-art in many-objective optimization problems (those involving four or more objectives) is rather worse. There is a dearth of test problems (even those with simple PS shapes) and the algorithms that can handle such problems. This paper proposes a framework for continuous many-objective test problems with arbitrarily prescribed PS shapes. The behavior of two popular EMO algorithms namely NSGAII and MOEA/D has also been studied for a sample of the proposed test problems. It is hoped that this paper will promote an integrated investigation of EMO algorithms for their scalability with objectives and their ability to handle complicated PS shapes with varying nature of the PF.

An evolutionary multi-objective framework for business process optimisation

This paper aims to investigate the application of evolutionary multi-objective optimisation to the new domain of business process optimisation. Business process optimisation is considered as the problem of constructing feasible business process designs with optimum attribute values such as duration and cost. The feasibility of a process design is based on: (i) the process requirements such as the required input and the expected output resources and (ii) the connectivity of the participating tasks in the process design through their input and output resources. Due to the multi-objective and discrete nature of the problem and the resulting fragmented search space, discovering feasible business process designs is one of the main challenges. The proposed approach involves the application of a series of evolutionary multi-objective optimisation algorithms (EMOAs) in an attempt to generate a series of diverse optimised business process designs for given process requirements. The proposed optimisation framework introduces a quantitative representation of business processes involving two matrices one for capturing the process design and one for calculating and evaluating the process attributes. It also introduces an algorithm that checks the feasibility of each candidate solution (i.e. process design). The results for two real-life scenarios demonstrate how the proposed framework produces a number of optimised design alternatives. NSGA-II proves unfit for the specific problem whilst PESA-II shows the best results due to its sophisticated region-based selection technique.

Constrained many-objective optimization: A way forward

Many objective optimization is a natural extension to multi-objective optimization where the number of objectives are significantly more than five. The performance of current state of the art algorithms (e.g. NSGA-II, SPEA2) is known to deteriorate significantly with increasing number of objectives due to the lack of adequate convergence pressure. It is of no surprise that the performance of NSGA-II on some constrained many-objective optimization problems [7] (e.g., DTLZ5-(5,M), M = 10, 20) in an earlier study [18] was far from satisfactory. Till date, research in many-objective optimization has focussed on two major areas (a) dimensionality reduction in the objective space and (b) preference ordering based approaches. This paper introduces a novel evolutionary algorithm powered by epsilon dominance (implemented within the framework of NSGA-II) and controlled infeasibility for improved convergence while the critical set of objectives is identified through a nonlinear dimensionality reduction scheme. Since approaching the Pareto-optimal front from within the feasible search space will need to overcome the problems associated with low selection pressure, the mechanism to approach the front from within the infeasible search space is promising as illustrated in this paper. The performance of the proposed algorithm is compared with NSGA-II (original, with crowding distance measure) and NSGA-II (epsilon dominance) on the above set of constrained multiobjective problems to highlight the benefits.

Entropy-Based Termination Criterion for Multiobjective Evolutionary Algorithms

Multiobjective evolutionary algorithms evolve a population of solutions through successive generations toward the Pareto-optimal front (POF). One of the most critical questions faced by the researchers and practitioners in this domain relates to the number of generations that may be sufficient for an algorithm to offer a good approximation of the POF for a given problem. Ironically, to date, this question largely remains unanswered and the number of generations are arbitrarily fixed a priori, with potentially punitive implications. If the a priori fixed generations are insufficient, then the algorithm reports suboptimal solutions. In contrast, if the a priori fixed generations are far too many, it implies waste of computational resources. This paper proposes a novel entropy-based dissimilarity measure that helps identify on the fly the number of generations beyond which an algorithm stabilizes, implying that either a good approximation has been obtained or that it cannot be obtained due to the stagnation of the algorithm in the search space. Given that in either case no further improvement in the approximation can be obtained, despite additional computational expense, the proposed dissimilarity measure provides a termination criterion and facilitates a termination detection algorithm. The generality, on-the-fly implementation, low-computational complexity, and the demonstrated efficacy of the proposed termination detection algorithm, on a wide range of multiobjective and many-objective test problems, define the novel contribution of this paper.

Machine learning based decision support for many-objective optimization problems

Multiple Criteria Decision-Making (MCDM) based Multi-objective Evolutionary Algorithms (MOEAs) are increasingly becoming popular for dealing with optimization problems with more than three objectives, commonly termed as many-objective optimization problems (MaOPs). These algorithms elicit preferences from a single or multiple Decision Makers (DMs), a priori or interactively, to guide the search towards the solutions most preferred by the DM(s), as against the whole Pareto-optimal Front (POF). Despite its promise for dealing with MaOPs, the utility of this approach is impaired by the lack of-objectivity; repeatability; consistency; and coherence in DM?s preferences. This paper proposes a machine learning based framework to counter the above limitations. Towards it, the preference-structure of the different objectives embedded in the problem model is learnt in terms of: a smallest set of conflicting objectives which can generate the same POF as the original problem; the smallest objective sets corresponding to pre-specified errors; and the objective sets of pre-specified sizes that correspond to minimum error. While the focus is on demonstrating how the proposed framework could serve as a decision support for the DM, its performance is also studied vis-i?-vis an alternative approach (based on dominance relation preservation), for a wide range of test problems and a real-world problem. The results mark a new direction for MCDM based MOEAs for MaOPs.

Identifying the redundant, and ranking the critical, constraints in practical optimization problems

This article presents a procedure for identification of the redundant constraints and ranking of the critical constraints by order of their importance, in single- and multi-objective optimization problems. The revelation of the redundant constraints throws light on the physics of the problem which may otherwise not be obvious to the engineers. Furthermore, the ranking of critical constraints allows for an exploration of the potential gain in objective value(s) through a logical elimination of certain constraints. Given a constrained optimization problem, the proposed procedure transforms the constraints into additional objectives (constraint objectives) and obtains a set of non-dominated solutions for the transformed problem by using a multi-objective evolutionary algorithm. Then, operating on the objective vectors of the obtained solutions, the procedure identifies the redundant, and ranks the critical constraints, based on the range of the constraint objectives and their correlations. The utility of the proposed procedure is demonstrated on four practical optimization problems.