Browsing by Author "Das, Indraneel"

Now showing 1 - 7 of 7
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    A Closer Look at Drawbacks of Minimizing Weighted Sums of Objectives for Pareto Set Generation in Multicriteria Optimization Problems
    (1996-12) Das, Indraneel; Dennis, J.E. Jr.
    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. Secondly, it is a frequent observation that even for convex Pareto curves, an evenly distributed set of weights fails to produce an even distribution of points from all parts of the Pareto set. This article aims to identify the mechanism behind this occurrence. Roughly, the weight is related to the slope of the Pareto curve in the objective space in a way such that an even spread of Pareto points actually corresponds to often very uneven distributions of weights. Several examples are provided showing assumed shapes of Pareto curves and the distribution of weights corresponding to an even spread of points on those Pareto curves.
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    A Piecewise Linear Approach to Component Reliability Analysis
    (1996-09) Das, Indraneel
    A standard intermediate step in component reliability analysis involves constructing a first-order or second-order approximation to the limit state surface at one design point. This paper looks into the possibilities for building a several-point, piecewise linear approximation to the limit state surface with the aim of obtaining a better estimate of the probability of failure of the component. The piecewise linear approximation is built at the points of intersection between the limit state surface and the coordinate axes and the one closest to the origin in the standard normal space. Details of finding these points are mentioned and computational results on several problems reported. The experiments suggest the method to be a promising one if care is exercised in applying it, and situations where one needs to be careful are described.
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    An Interior Point Algorithm for the General Nonlinear Programming Problem with Trust Region Globalization
    (1996-07) Das, Indraneel
    This paper attempts to develop an SQP-based interior point technique for solving the general nonlinear programming problem using trust region globalization and the Coleman-Li scaling. The SQP subproblem is decomposed into a normal and a reduced tangential subproblem in the tradition of numerous works on equality constrained optimization, and strict feasibility is maintained with respect to the bounds. This is intended to be an extension of previous work by Coleman & Li and Vicente. Even though no theoretical proofs of convergence are provided, some computational results are presented which indicate that this algorithm holds promise. The computational experiments have been geared towards improving the semi-local convergence of the algorithm; in particular high sensitivity of the speed of convergence with respect to the fraction of the trust region radius allowed for the normal step and with respect to the initial trust region radius are observed. The chief advantages of this algorithm over primal-dual interior point algorithms are better handling of the `sticking problem' and a reduction in the number of variables by elimination of the multipliers of bound constraints.
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    Nonlinear Multicriteria Optimization and Robust Optimality
    (1997-04) Das, Indraneel
    This dissertation attempts to address two important problems in systems engineering, namely, multicriteria optimization and robustness optimization. In fields ranging from engineering to the social sciences, designers are very often required to make decisions that attempt to optimize several criteria or objectives at once. Mathematically this amounts to finding the Pareto optimal set of points for these constrained multiple criteria optimization problems which happen to be nonlinear in many realistic situations, particularly in engineering design. Traditional techniques for nonlinear multicriteria optimization suffer from various drawbacks. The popular method of minimizing weighted sums of the multiple objectives suffers from the deficiency that choosing an even spread of `weights' does not yield an even spread of points on the Pareto surface and further this spread is often quite sensitive to the relative scales of the functions. A continuation/homotopy based strategy for tracing out the Pareto curve tries to make up for this deficiency, but unfortunately requires exact second derivative information and further cannot be applied to problems with more than two objectives in general. Another technique, goal programming, requires prior knowledge of feasible goals which may not be easily available for more than two objectives. Normal-Boundary Intersection (NBI), a new technique introduced in this dissertation, overcomes all of the difficulties inherent in the existing techniques by introducing a better parametrization of the Pareto set. It is rigorously proved that NBI is completely independent of the relative scales of the functions and is quite successful in producing an evenly distributed set of points on the Pareto set given an evenly distributed set of `NBI parameters' (comparable to the `weights' in minimizing weighted sums of objectives). Further, this method can easily handle more than two objectives while retaining the computational efficiency of continuation-type algorithms, which is an improvement over homotopy techniques for tracing the trade-off curve. Various aspects of NBI including computational issues and its relationships with minimizing convex combinations and goal programming are discussed in this dissertation. Finally some case studies from engineering disciplines are performed using NBI. The other facet of this dissertation deals with robustness optimization, a concept useful in quantifying the stability of an optimum in the face of random fluctuations in the design variables. This robustness optimization problem is presented as an application of multicriteria optimization since it essentially involves the simultaneous minimization of two criteria, the objective function value at a point and the dispersion in the function values in a neighborhood of the point. Moreover, a formulation of the robustness optimization problem is presented so that it fits the framework of constrained, nonlinear optimization problems, which is an improvement on existing formulations that deal with either unconstrained nonlinear formulations or constrained linear formulations.
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    Nonlinear multicriteria optimization and robust optimality
    (1997) Das, Indraneel; Dennis, John E., Jr.
    This dissertation attempts to address two important problems in systems engineering, namely, multicriteria optimization and robustness optimization. In fields ranging from engineering to the social sciences designers are very often required to make decisions that attempt to optimize several criteria or objectives at once. Mathematically this amounts to finding the Pareto optimal set of points for these constrained multiple criteria optimization problems which happen to be nonlinear in many realistic situations, particularly in engineering design. Traditional techniques for nonlinear multicriteria optimization suffer from various drawbacks. The popular method of minimizing weighted sums of the multiple objectives suffers from the deficiency that choosing an even spread of 'weights' does not yield an even spread of points on the Pareto surface and further this spread is often quite sensitive to the relative scales of the functions. A continuation/homotopy based strategy for tracing out the Pareto curve tries to make up for this deficiency, but unfortunately requires exact second derivative information and further cannot be applied to problems with more than two objectives in general. Another technique, goal programming, requires prior knowledge of feasible goals which may not be easily available for more than two objectives. Normal-Boundary Intersection (NBI), a new technique introduced in this dissertation, overcomes all of the difficulties inherent in the existing techniques by introducing a better parametrization of the Pareto set. It is rigorously proved that NBI is completely independent of the relative scales of the functions and is quite successful in producing an evenly distributed set of points on the Pareto set given an evenly distributed set of 'NBI parameters' (comparable to the 'weights' in minimizing weighted sums of objectives). Further, this method can easily handle more than two objectives while retaining the computational efficiency of continuation-type algorithms, which is an improvement over homotopy techniques for tracing the trade-off curve. Various aspects of NBI including computational issues and its relationships with minimizing convex combinations and goal programming are discussed in this dissertation. Finally some case studies from engineering disciplines are performed using NBI. The other facet of this dissertation deals with robustness optimization, a concept useful in quantifying the stability of an optimum in the face of random fluctuations in the design variables. This robustness optimization problem is presented as an application of multicriteria optimization since it essentially involves the simultaneous minimization of two criteria, the objective function value at a point and the dispersion in the function values in a neighborhood of the point. Moreover, a formulation of the robustness optimization problem is presented so that it fits the framework of constrained, nonlinear optimization problems, which is an improvement on existing formulations that deal with either unconstrained nonlinear formulations or constrained linear formulations.
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    Normal-Boundary Intersection: A New Method for Generating the Pareto Surface in Nonlinear  Multicriteria Optimization Problems
    (1996-07) Das, Indraneel; Dennis, J.E. Jr.
    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.
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    Robustness Optimization for Constrained, Nonlinear Programming Problems
    (1997-03) Das, Indraneel
    In realistic situations, engineering designs should take into consideration random aberrations from the stipulated design variables arising from manufacturing variability. Moreover, many environmental parameters are often stochastic in nature. Traditional nonlinear optimization attempts to find a deterministic optimum of a cost function and does not take into account the effect of these random variations on the objective. This paper attempts to devise a technique for finding optima of constrained nonlinear functions that are robust with respect to such variations. The expectation of the function over a domain of aberrations in the parameters is taken as a measure of `robustness' of the function value at a point. It is pointed out that robustness optimization is ideally an attempt to trade off between `optimality' and `robustness'. A newly-developed multicriteria optimization technique, known as Normal-Boundary Intersection, is used here to find evenly-spaced points on the Pareto curve for the `optimality' and `robustness' criteria. This Pareto curve enables the user to make the trade-off decision explicitly free of arbitrary `weighting' parameters. This paper also formulates a derivative-based approximation for evaluating the expected value of the objective function on the nonlinear manifold defined by the state equations for the system. Existing procedures for evaluating the expectation usually involve numerical integration techniques requiring many solutions of the state equations for one evaluation of the expectation. The procedure presented here bypasses the need for multiple solutions of the state equations and hence provides a cheaper and more easily optimizable approximation to the expectation. Finally, this paper discusses how nonlinear inequality constraints should be treated in the presence of random parameters in the design. Computational results are presented for finding a robust optimum of a structural optimization problem.