Analytic Hierarchy Process is one of the most known multicriteria decision aid methods. Nevertheless, as it relies on decision makers (DM) pairwise comparisons, a problem may occur if some comparisons are not well don...
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Analytic Hierarchy Process is one of the most known multicriteria decision aid methods. Nevertheless, as it relies on decision makers (DM) pairwise comparisons, a problem may occur if some comparisons are not well done. This issue, known as inconsistency, appears when an inconsistency threshold is violated. One way to deal with inconsistency is to redo all judgments, as many times as needed, in order to reach acceptable levels. This work proposes a nonlinear programming model that reduces inconsistency to zero or near zero, without needing to redo all judgments. The reduction is achieved by adjusting the original judgments in a minimum way, keeping the DM's decisions within a tolerable range. Only discrete values are generated, so the solution respects the limits of the Saaty scale (1-9). To illustrate the efficiency of the nonlinear model, a comparison between the proposed model and other models taken from recent literature was made. The results show that the proposed model performed better, since the original judgments were changed in a minimum way, also the inconsistency was completely removed. Alternatively, if some inconsistency is allowed more original judgments can be preserved.
The problem of determining an optimal set of values for the parameters of a manufactured product (and/or a manufacturing process), from the point of views of its robustness against various sources of noise, is commonl...
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The problem of determining an optimal set of values for the parameters of a manufactured product (and/or a manufacturing process), from the point of views of its robustness against various sources of noise, is commonly referred to as the the parameter design problem. We propose an algorithmic strategy for solving the parameter design problem, and present a specific implementation of this strategy which we refer to as the Successive Quadratic Variance Approximation Method (SQVAM). SQVAM is based on conventional optimization techniques, and assumes that the functional relationship between the input parameters and the performance characteristic of interest is either known or could be well approximated. An illustrative numerical example, discussing the design of a Wheatstone bridge, is then presented. This is the same example used by Taguchi to illustrate the orthogonal array method.
The paper proposes network analysis methods that generalize the work begun in [1]. The application of specific nonlinear objective functions in network optimization problems has led to the development of a general the...
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The paper proposes network analysis methods that generalize the work begun in [1]. The application of specific nonlinear objective functions in network optimization problems has led to the development of a general theory that describes networks of various types. The proposed procedure also can be used to construct efficient numerical algorithms for practical network calculations. Some previously published results are given here with their proofs in order to ensure closure of presentation and to extend the authors' unified terminology to the entire subject.
A minimax approach to nonlinear programming is presented. The original nonlinear programming problem is formulated as an unconstrained minimax problem. Under reasonable restrictions, it is shown that a point satisfyin...
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A minimax approach to nonlinear programming is presented. The original nonlinear programming problem is formulated as an unconstrained minimax problem. Under reasonable restrictions, it is shown that a point satisfying the necessary conditions for a minimax optimum also satisfies the Kuhn-Tucker necessary conditions for the original problem. A least pth type of objective function for minimization with extremely large values of p is proposed to solve the problem. Several numerical examples compare the present approach with the well-known SUMT method of Fiacco and McCormick. In both cases, a recent minimization algorithm by Fletcher is used.
Starting from one extension of the Hahn-Banach theorem, the Mazur-Orlicz theorem, and a not very restrictive concept of convexity, that arises naturally in minimax theory, infsup-convexity, we derive an equivalent ver...
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Starting from one extension of the Hahn-Banach theorem, the Mazur-Orlicz theorem, and a not very restrictive concept of convexity, that arises naturally in minimax theory, infsup-convexity, we derive an equivalent version of that fundamental result for finite dimensional spaces, which is a sharp generalization of Konig's Maximum theorem. It implies several optimal statements of the Lagrange multipliers, Karush/Kuhn-Tucker, and Fritz John type for nonlinear programs with an objective function subject to both equality and inequality constraints.
Methods are considered for solving nonlinear programming problems using an exactl 1 penalty function. LP-like subproblems incorporating a trust region constraint are solved successively both to estimate the active set...
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Methods are considered for solving nonlinear programming problems using an exactl 1 penalty function. LP-like subproblems incorporating a trust region constraint are solved successively both to estimate the active set and to provide a foundation for proving global convergence. In one particular method, second order information is represented by approximating the reduced Hessian matrix, and Coleman-Conn steps are taken. A criterion for accepting these steps is given which enables the superlinear convergence properties of the Coleman-Conn method to be retained whilst preserving global convergence and avoiding the Maratos effect. The methods generalize to solve a wide range of composite nonsmooth optimization problems and the theory is presented in this general setting. A range of numerical experiments on small test problems is described.
A model algorithm based on the successive quadratic programming method for solving the general nonlinear programming problem is presented. The objective function and the constraints of the problem are only required to...
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A model algorithm based on the successive quadratic programming method for solving the general nonlinear programming problem is presented. The objective function and the constraints of the problem are only required to be differentiable and their gradients to satisfy a Lipschitz condition. The strategy for obtaining global convergence is based on the trust region approach. The merit function is a type of augmented Lagrangian. A new updating scheme is introduced for the penalty parameter, by means of which monotone increase is not necessary. Global convergence results are proved and numerical experiments are presented.
We explore properties of nonlinear programming problems (NLPs) that arise in the formulation of NMPC subproblems and show their influence on stability and robustness of NMPC. NLPs that satisfy linear independence cons...
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We explore properties of nonlinear programming problems (NLPs) that arise in the formulation of NMPC subproblems and show their influence on stability and robustness of NMPC. NLPs that satisfy linear independence constraint qualification (LICQ), second order sufficient conditions (SOSC) and strict complementarity (SC), have solutions that are continuous and differentiable with perturbations of the problem data. As a result, they are important prerequisites for nominal and ISS stability of NMPC controllers. Moreover, we show that ensuring these properties is possible through reformulation of the NLP subproblem for NMPC, through the addition of (1 penalty and barrier terms. We show how these properties also establish ISS of related sensitivity-based NMPC controllers, such as asNMPC and amsNMPC. Finally, we demonstrate the impact of our reformulated NLPs on several examples that have shown nonrobust performance on earlier NMPC strategies. (C) 2015, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved.
In some constrained nonlinear programming problems possessing several local optima, a local optimum can be recognized as the global optimum by looking closely at the Lagrangian, the augmented function. Similarly, clas...
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