Bounds on efficient outcomes in interactive multiple criteria decision making problems are derived. Bounds are dynamic, i.e., they become stronger with the growing number of explicitly identified outcomes. They are al...
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Bounds on efficient outcomes in interactive multiple criteria decision making problems are derived. Bounds are dynamic, i.e., they become stronger with the growing number of explicitly identified outcomes. They are also parametric with respect to weighting coefficients. Computational cost to calculate bounds is negligible. Bounds of the sort offer a breakthrough for prohibitive size and/or solution time bottlenecks by allowing a decision maker to interact with an approximation of the underlying mathematical model rather the model itself. Possible applications of bounds to existing interactive decision making algorithms are discussed. Illustrative numerical examples are given. (C) 2002 Published by Elsevier Science B.V.
The main goal of this note is to give a counterexample to the Triality Theorem in Gao and Ruan (Math Methods Oper Res 67:479-491, 2008). This is done first by considering a more general optimization problem with the a...
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The main goal of this note is to give a counterexample to the Triality Theorem in Gao and Ruan (Math Methods Oper Res 67:479-491, 2008). This is done first by considering a more general optimization problem with the aim to encompass several examples from Gao and Ruan (Math Methods Oper Res 67:479-491, 2008) and other papers by Gao and his collaborators (see f.i. Gao Duality principles in nonconvex systems. Theory, methods and applications. Kluwer, Dordrecht, 2000;Gao and Sherali Advances in applied mathematics and global optimization. Springer, Berlin, 2009). We perform a thorough analysis of the general optimization problem in terms of local extrema while presenting several counterexamples.
Traditional plane-based clustering methods measure the within-cluster or between-cluster scatter by linear, quadratic or some other unbounded functions, which are sensitive to the samples far from the cluster center. ...
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Traditional plane-based clustering methods measure the within-cluster or between-cluster scatter by linear, quadratic or some other unbounded functions, which are sensitive to the samples far from the cluster center. This paper introduces the ramp functions into plane-based clustering and proposes a ramp-based twin support vector clustering (RampTWSVC). RampTWSVC is very robust to the samples far from the cluster center, because its within-cluster and between-cluster scatters are measured by the bounded ramp functions. Thus, it is easier to find the intrinsic clusters than other plane-based clustering methods. The nonconvex programming problem in RampTWSVC is solved efficiently through an alternating iteration algorithm, and its local solution can be obtained in a finite number of iterations theoretically. In addition, its nonlinear manifold clustering formation is also proposed via a kernel trick. Experimental results on several benchmark datasets show the better performance of our RampTWSVC compared with other plane-based clustering methods.
Using the concept of Φ-conjugate functions, a wide class of nonconvex optimization problems can be investigated. By generalized Lagrangians, the problems indicated and a generalized notion of stability can be treated...
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Using the concept of Φ-conjugate functions, a wide class of nonconvex optimization problems can be investigated. By generalized Lagrangians, the problems indicated and a generalized notion of stability can be treated. It is possible to give duality theorems for these problems such that the approach given by Rockafellar (1967) for convex problems can be extended to the nonconvex case.
We analyze the anomalous wave structure appearing in flow dynamics under the influence of magnetic field in materials described by non-ideal equations of state. We consider the system of magnetohydrodynamics equations...
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We analyze the anomalous wave structure appearing in flow dynamics under the influence of magnetic field in materials described by non-ideal equations of state. We consider the system of magnetohydrodynamics equations closed by a general equation of state (EOS) and propose a complete spectral decomposition of the fluxes that allows us to derive an expression of the nonlinearity factor as the mathematical tool to determine the nature of the wave phenomena. We prove that the possible formation of non-classical wave structure is determined by both the thermodynamic properties of the material and the magnetic field as well as its possible rotation. We demonstrate that phase transitions induced by material properties do not necessarily imply the loss of genuine nonlinearity of the wavefields as is the case in classical hydrodynamics. The analytical expression of the nonlinearity factor allows us to determine the specific amount of magnetic field necessary to prevent formation of complex structure induced by phase transition in the material. We illustrate our analytical approach by considering two non-convex EOS that exhibit phase transitions and anomalous behavior in the evolution. We present numerical experiments validating the analysis performed through a set of one-dimensional Riemann problems. In the examples we show how to determine the appropriate amount of magnetic field in the initial conditions of the Riemann problem to transform a thermodynamic composite wave into a simple nonlinear wave. (C) 2014 AIP Publishing LLC.
This paper is concerned with the global optimization of polynomial programming problems of the type that arise in various location allocation, chemical process, and engineering design contexts. For such problems, we h...
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This paper is concerned with the global optimization of polynomial programming problems of the type that arise in various location allocation, chemical process, and engineering design contexts. For such problems, we have previously presented a generic reformulation-linearization technique (RLT) to obtain global optimal solutions. We now introduce several new classes of constraints for both univariate and multivariate versions of this problem, including certain simple convex variable bounding types of restrictions that can be used to augment the basic (linear) RLT relaxation in order to yield tighter lower bounds. A constraint selection strategy is also developed to predict and generate only a useful subset of these constraints, so that the size of the resulting relaxation remains manageable. These relaxations are embedded within a globally convergent branch-and-bound algorithm that additionally incorporates certain range-reduction strategies and a new branching variable selection procedure. Computational results using various chemical process, pooling, and engineering design problems from the literature are also presented to demonstrate the viability of the proposed approach. (C) 1997 Elsevier Science B.V.
We study the weight minimization problem in a dual setting. We propose new dual formulations for non-linear multipoint approximations with diagonal approximate Hessian matrices, which derive from separable series expa...
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We study the weight minimization problem in a dual setting. We propose new dual formulations for non-linear multipoint approximations with diagonal approximate Hessian matrices, which derive from separable series expansions in terms of exponential intervening variables. These, generally, nonconvex approximations are formulated in terms of intervening variables with negative exponents, and are therefore applicable to the solution of the weight minimization problem in a sequential approximate optimization (SAO) framework. Problems in structural optimization are traditionally solved using SAO algorithms, like the method of moving asymptotes, which require the approximate subproblems to be strictly convex. Hence, during solution, the nonconvex problems are approximated using convex functions, and this process may in general be inefficient. We argue, based on Falk's definition of the dual, that it is possible to base the dual formulation on nonconvex approximations. To this end we reintroduce a nonconvex approach to the weight minimization problem originally due to Fleury, and we explore certain convex and nonconvex forms for subproblems derived from the exponential approximations by the application of various methods of mixed variables. We show in each case that the dual is well defined for the form concerned, which may consequently be of use to the future code developers. Copyright (C) 2009 John Wiley & Sons, Ltd.
In this paper, a finite branch-and-bound algorithm is developed for the minimization of a concave power law over a polytope. Linear terms are also included in the objective function. Using the first order necessary co...
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In this paper, a finite branch-and-bound algorithm is developed for the minimization of a concave power law over a polytope. Linear terms are also included in the objective function. Using the first order necessary conditions of optimality, the optimization problem is transformed into an equivalent problem consisting of a linear objective function, a set of linear constraints, a set of convex constraints, and a set of bilinear complementary constraints. The transformed problem is then solved using a finite branch-and-bound algorithm that solves two convex problems at each of its nodes. The method is illustrated by means of an example from the literature.
The paper investigates the long time average of the solutions of Hamilton-Jacobi equations with a noncoercive. nonconvex Hamiltonian in the torus R-2/Z(2). We give nonresonance conditions under which the long-time ave...
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The paper investigates the long time average of the solutions of Hamilton-Jacobi equations with a noncoercive. nonconvex Hamiltonian in the torus R-2/Z(2). We give nonresonance conditions under which the long-time average converges to a constant. In the resonant case, we show that the limit still exists, although it is nonconstant in general. We compute the limit at points where it is not locally constant. (C) 2009 Elsevier Masson SAS. All rights reserved.
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