An integration of efficiency projections into the Geoffrion approach for multiobjective linear programming

Winkels, Heinz-Michael; Meika, Manfred

doi:10.1016/0377-2217(84)90319-9

Cited by 20 publications

(3 citation statements)

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“…Once the rank is determined, in view of Equations (18) and (19), the different possible solutions of the linear system are uniformly given by:…”

Section: Parametric Solving Of Systems Of Linear Equationsmentioning

confidence: 99%

“…Solving parametric linear systems of equations is an interesting computational issue encountered, for example, when computing the characteristic solutions for differential equations (see [5]), when dealing with coloured Petri nets (see [13]) or in linear prediction problems and in different applications of operation research and engineering (see, for example, [6], [8], [12], [15] or [19] where the elements of the matrix have a linear affine dependence on a set of parameters). More general situations where the elements are polynomials in a given set of parameters are encountered in computer algebra problems (see [1] and [18]).…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Generalizing Cramer's Rule: Solving Uniformly Linear Systems of Equations

Diaz–Toca¹,

González-Vega²,

Lombardi³

2005

SIAM J. Matrix Anal. & Appl.

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“…Once the rank is determined, in view of Equations (18) and (19), the different possible solutions of the linear system are uniformly given by:…”

Section: Parametric Solving Of Systems Of Linear Equationsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Generalizing Cramer's Rule: Solving Uniformly Linear Systems of Equations

Diaz–Toca¹,

González-Vega²,

Lombardi³

2005

SIAM J. Matrix Anal. & Appl.

View full text Add to dashboard Cite

show abstract

“…Applications and modifications of the GDF method are described in [4,40,42,51,53,73,79,84,143,144,164,186,197,203,219,264].…”

Section: Geoffrion-dyer-feinberg Methodsmentioning

confidence: 99%

Interactive Nonlinear Multiobjective Optimization Methods

Miettinen

Hakanen

Podkopaev

2016

International Series in Operations Research &Amp; Management Science

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Interactive Nonlinear Multiobjective Procedures

Miettinen

International Series in Operations Research &Amp; Management Science

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Keywords:An overview of the interactive methods for solving nonlinear multiple criteria decision making problems is given. In interactive methods, the decision maker progressively provides preference information so that the most satisfactory compromise can be found. The basic features of several methods are introduced and some theoretical results are provided. In addition, references to modifications and applications as well as to other methods are indicated.Multiple criteria decision making, Multiobjective optimization, Nonlinear optimization, Interactive methods. IntroductionNonlinear multiobjective optimization means multiple criteria decision making involving nonlinear functions of continuous decision variables. In these problems, the best possible compromise is to be found from an infinite number of alternatives represented by decision variables restricted by constraint functions.Solving multiobjective optimization problems usually requires the participation of a human decision maker who is supposed to have better insight into the problem and to express preference relations between alternative solutions. The methods can be divided into four classes according to the role of the decision maker in the solution process. If the decision maker is not involved, we use methods where no articulation of preference information is used, in other words, no-preference methods. If the decision maker expresses preference information after the solution process, we speak about a posteriori methods whereas a priori methods require articulation of preference information before the solution process. The most extensive method class is interactive methods where the decision maker specifies preference information progressively during the solution process. Here we concentrate on this last-mentioned class and introduce several examples of interactive methods.Many real-world phenomena behave in a nonlinear way. Besides, linear problems can always be solved using methods created for nonlinear problems but not vice versa. For these reasons, we here devote ourselves to nonlinear problems. We assume that all the information involved is deterministic and that we have a single decision maker. Further information about the topics treated here can be found in [105]. 228 MULTIPLE CRITERIA OPTIMIZATION ConceptsLet us begin by introducing several concepts and definitions. We study multiobjective optimization problems of the form involving objective functions that we want to minimize simultaneously. The decision (variable) vectors belong to the (nonempty) feasible regionThe feasible region is formed by constraint functions but we do not fix them here.We denote the image of the feasible region by and call it a feasible objective region. Objective (function) values form objective vectorsNote that if is to be maximized, it is equivalent to minimizeWe call a multiobjective optimization problem convex if all the objective functions and the feasible region are convex. On the other hand, the problem is nondifferentiable if at least one o...

show abstract

An integration of efficiency projections into the Geoffrion approach for multiobjective linear programming

Cited by 20 publications

References 15 publications

Generalizing Cramer's Rule: Solving Uniformly Linear Systems of Equations

Generalizing Cramer's Rule: Solving Uniformly Linear Systems of Equations

Interactive Nonlinear Multiobjective Optimization Methods

Interactive Nonlinear Multiobjective Procedures

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