Indraneel Das

Bio: Indraneel Das is an academic researcher from United Technologies. The author has contributed to research in topics: Quadratic programming & Pareto principle. The author has an hindex of 14, co-authored 23 publications receiving 3785 citations. Previous affiliations of Indraneel Das include University of Houston & Mobil.

Normal-Boundary Intersection: A New Method for Generating the Pareto Surface in Nonlinear Multicriteria Optimization Problems

Abstract: This paper proposes an alternate method for finding several Pareto optimal points for a general nonlinear multicriteria optimization problem. Such points collectively capture the trade-off among the various conflicting objectives. It is proved that this method is independent of the relative scales of the functions and is successful in producing an evenly distributed set of points in the Pareto set given an evenly distributed set of parameters, a property which the popular method of minimizing weighted combinations of objective functions lacks. Further, this method can handle more than two objectives while retaining the computational efficiency of continuation-type algorithms. This is an improvement over continuation techniques for tracing the trade-off curve since continuation strategies cannot easily be extended to handle more than two objectives.

A closer look at drawbacks of minimizing weighted sums of objectives for Pareto set generation in multicriteria optimization problems

Abstract: A standard technique for generating the Pareto set in multicriteria optimization problems is to minimize (convex) weighted sums of the different objectives for various different settings of the weights. However, it is well-known that this method succeeds in getting points from all parts of the Pareto set only when the Pareto curve is convex. This article provides a geometrical argument as to why this is the case.

On characterizing the “knee” of the Pareto curve based on Normal-Boundary Intersection

Abstract: This paper deals with the issue of generating one Pareto optimal point that is guaranteed to be in a “desirable” part of the Pareto set in a given multicriteria optimization problem. A parameterization of the Pareto set based on the recently developed normal-boundary intersection technique is used to formulate a subproblem, the solution of which yields the point of “maximum bulge”, often referred to as the “knee of the Pareto curve”. This enables the identification of the “good region” of the Pareto set by solving one nonlinear programming problem, thereby bypassing the need to generate many Pareto points. Further, this representation extends the concept of the “knee” for problems with more than two objectives. It is further proved that this knee is invariant with respect to the scales of the multiple objective functions. The generation of this knee however requires the value of each objective function at the minimizer of every objective function (the pay-off matrix). The paper characterizes situations when approximations to the function values comprising the pay-off matrix would suffice in generating a good approximation to the knee. Numerical results are provided to illustrate this point. Further, a weighted sum minimization problem is developed based on the information in the pay-off matrix, by solving which the knee can be obtained.

Normal-boundary intersection: an alternate method for generating pareto optimal points in multicriteria optimization problems

Abstract: This paper proposes an alternate method for finding several Pareto optimal points for a general nonlinear multicriteria optimization problem, aimed at capturing the tradeoff among the various conflicting objectives. It can be rigorously proved that this method is completely independent of the relative scales of the functions and is quite successful in producing an evenly distributed set of points in the Pareto set given an evenly distributed set of `weights'', a property which the popular method of linear combinations lacks. Further, this method can be easily extended in case of more than two objectives while retaining the computational efficiency of continuation-type algorithms, which is an improvement over homotopy techniques for tracing the tradeoff curve.

A preference ordering among various Pareto optimal alternatives

Abstract: It is often necessary to choose a Pareto optimal point from a set of many. This paper introduces the concept of order of efficiency, which provides a notion that is stronger than Pareto optimality and allows us to set up a preference ordering amongst various alternatives that are Pareto optimal. This approach does not resort to setting up a ranking on the basis of an arbitrary “criterion of merit” obtained by combining the multiple decision criteria into one scalar index. Examples are cited and it is argued that using the procedure described in this paper, it is possible to rule out Pareto alternatives with “extreme components” and retain alternatives “in the middle” of the Pareto set without the help of plots or other visualization aids. This makes the approach applicable for cases where the number of criteria is very high and visualization is intractable.

MOEA/D: A Multiobjective Evolutionary Algorithm Based on Decomposition

Abstract: Decomposition is a basic strategy in traditional multiobjective optimization. However, it has not yet been widely used in multiobjective evolutionary optimization. This paper proposes a multiobjective evolutionary algorithm based on decomposition (MOEA/D). It decomposes a multiobjective optimization problem into a number of scalar optimization subproblems and optimizes them simultaneously. Each subproblem is optimized by only using information from its several neighboring subproblems, which makes MOEA/D have lower computational complexity at each generation than MOGLS and nondominated sorting genetic algorithm II (NSGA-II). Experimental results have demonstrated that MOEA/D with simple decomposition methods outperforms or performs similarly to MOGLS and NSGA-II on multiobjective 0-1 knapsack problems and continuous multiobjective optimization problems. It has been shown that MOEA/D using objective normalization can deal with disparately-scaled objectives, and MOEA/D with an advanced decomposition method can generate a set of very evenly distributed solutions for 3-objective test instances. The ability of MOEA/D with small population, the scalability and sensitivity of MOEA/D have also been experimentally investigated in this paper.

Survey of multi-objective optimization methods for engineering

Abstract: A survey of current continuous nonlinear multi-objective optimization (MOO) concepts and methods is presented. It consolidates and relates seemingly different terminology and methods. The methods are divided into three major categories: methods with a priori articulation of preferences, methods with a posteriori articulation of preferences, and methods with no articulation of preferences. Genetic algorithms are surveyed as well. Commentary is provided on three fronts, concerning the advantages and pitfalls of individual methods, the different classes of methods, and the field of MOO as a whole. The Characteristics of the most significant methods are summarized. Conclusions are drawn that reflect often-neglected ideas and applicability to engineering problems. It is found that no single approach is superior. Rather, the selection of a specific method depends on the type of information that is provided in the problem, the user’s preferences, the solution requirements, and the availability of software.

An Evolutionary Many-Objective Optimization Algorithm Using Reference-Point-Based Nondominated Sorting Approach, Part I: Solving Problems With Box Constraints

Abstract: Having developed multiobjective optimization algorithms using evolutionary optimization methods and demonstrated their niche on various practical problems involving mostly two and three objectives, there is now a growing need for developing evolutionary multiobjective optimization (EMO) algorithms for handling many-objective (having four or more objectives) optimization problems. In this paper, we recognize a few recent efforts and discuss a number of viable directions for developing a potential EMO algorithm for solving many-objective optimization problems. Thereafter, we suggest a reference-point-based many-objective evolutionary algorithm following NSGA-II framework (we call it NSGA-III) that emphasizes population members that are nondominated, yet close to a set of supplied reference points. The proposed NSGA-III is applied to a number of many-objective test problems with three to 15 objectives and compared with two versions of a recently suggested EMO algorithm (MOEA/D). While each of the two MOEA/D methods works well on different classes of problems, the proposed NSGA-III is found to produce satisfactory results on all problems considered in this paper. This paper presents results on unconstrained problems, and the sequel paper considers constrained and other specialties in handling many-objective optimization problems.

Normal-Boundary Intersection: A New Method for Generating the Pareto Surface in Nonlinear Multicriteria Optimization Problems

Abstract: This paper proposes an alternate method for finding several Pareto optimal points for a general nonlinear multicriteria optimization problem. Such points collectively capture the trade-off among the various conflicting objectives. It is proved that this method is independent of the relative scales of the functions and is successful in producing an evenly distributed set of points in the Pareto set given an evenly distributed set of parameters, a property which the popular method of minimizing weighted combinations of objective functions lacks. Further, this method can handle more than two objectives while retaining the computational efficiency of continuation-type algorithms. This is an improvement over continuation techniques for tracing the trade-off curve since continuation strategies cannot easily be extended to handle more than two objectives.