Showing papers on "Solution set published in 1987"
TL;DR: The uncovered set is a generalization of the core known as the uncovered set as mentioned in this paper, and it has been shown that it is a solution set for majority voting games in a variety of institutional settings.
Abstract: Recent work in social choice theory has focused on an important generalization of the core known as the uncovered set. Miller (1977, 1980), working with finite alternative spaces, and McKelvey (1986), working with infinite alternative spaces, argue that the uncovered set serves as a general solution set for majority voting games. They and others have shown that, under a variety of institutional settings, game theoretic behavior by participants leads to outcomes in the uncovered set. Perhaps the simplest example of this is two-candidate competition under the plurality rule. In this context, Miller (1980) observed that an electoral strategy is undominated (in the usual game theoretic sense) if and only if it is an element of the uncovered set, and McKelvey (1986), building on a previous paper by McKelvey and Ordeshook (1976), demonstrated that the uncovered set contains the support set of any mixed strategy equilibrium. In the context of multicandidate competition, Cox (1985) showed that, under certain Condorcet voting procedures, undominated strategies will be in the uncovered set. The relationship of the urfcovered set to sophisticated voting and agendas has also been explored. Shepsle and Weingast (1984) have applied and extended Miller's results to derive bounds on agenda reachable outcomes in multidimensional choice spaces, while McKelvey has shown in related work that the uncovered set contains any outcome reachable by sophisticated voting when agendas are endogenously generated. The importance of the uncovered set as a solution concept naturally motivates interest in its size and properties. Miller has demonstrated, in the case of tournaments, that the uncovered set coincides with the core, when a core exists. He has conjectured that the uncovered set is generally
105 citations
TL;DR: It is shown that it is possible to construct linear, convex, and quasiconvex representations for linear, convolutional, and convex vector problems, respectively, which are satisfactory for finding all the optimal solutions of a vector problem.
Abstract: In this paper, we investigate the scalar representation of vector optimization problems in close connection with monotonic functions. We show that it is possible to construct linear, convex, and quasiconvex representations for linear, convex, and quasiconvex vector problems, respectively. Moreover, for finding all the optimal solutions of a vector problem, it suffices to solve certain scalar representations only. The question of the continuous dependence of the solution set upon the initial vector problems and monotonic functions is also discussed.
59 citations
TL;DR: In this paper, the multivalued Cauchy problem is solved in Cχ in a Hausdorff upper semicontinuous way. And the solution set of the multiview Cauche problem set is shown to be in C Ω(Cχ).
Abstract: Let X be a separable Banach space and F(•,•) an orientor field from [0,b]xX into the nonempty, compact, convex subsets of X, which is completely Hausdorff upper semicontinuous and We show that solution set of the multivalued Cauchy problem set in Cχ
46 citations
TL;DR: A theorem of Gay on the quadratic approximation property of the preconditioned fixpoint inverse is strengthened and bounds are given for the overestimation of the solution set of a system of linear interval equations by various inclusion intervals for this set.
Abstract: Bounds are given for the overestimation of the solution set of a system of linear interval equations by various inclusion intervals for this set. In particular, a theorem of Gay [this Journal, 19 (1982), pp. 858–870] on the quadratic approximation property of the preconditioned fixpoint inverse is strengthened.
42 citations
TL;DR: In this article, the convergence and optimality properties of the modified two-step algorithm for on-line determination of the optimum steady-state operating point of an industrial process were investigated.
Abstract: This paper investigates convergence and optimality properties of the modified two-step algorithm for on-line determination of the optimum steady-state operating point of an industrial process. Mild sufficient conditions are derived for the convergence and feasibility of the algorithm. It is shown that every point within the solution set of the algorithm satisfies first-order necessary conditions for optimality, and that every optimal solution belongs to this set. It is also shown that there are advantages to be gained by using a linear mathematical model of the process within the implementation of the algorithm.
36 citations
TL;DR: In this article, the authors considered matrix equations of the form AX = B and YA = B in a class of complete and completely distributive lattices, where the multiplication operation of matrices is defined either by the Max-Min or by the Min-Max way.
33 citations
TL;DR: A set of coefficient matrices which are invariant to a solution set of a simple fuzzy relation equation is considered and the equipollency of the cardinal numbers of solution sets ofsimple fuzzy relation equations for two constant vectors is shown.
26 citations
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TL;DR: In this paper, the authors present a practical three-dimensional gravity ideal body computer code, IDB, that can optimize a mesh with over 105 cells, based on the theory of Parker's ideal body, which characterizes the solution with the smallest possible maximum density.
Abstract: The interpretation of gravity anomaly data suffers from a fundamental nonuniqueness, even when the solution set is bounded by physical or geologic constraints. Therefore, constructing a single solution that fits or approximately fits the data is of limited value. Consequently, much effort has been applied in recent years to developing inverse techniques for rigorous deduction of properties common to all possible solutions. To this end, Parker developed the theory of an ideal body, which characterizes the extremal solution with the smallest possible maximum density. Gravity ideal‐body analysis is an excellent reconaissance exploration tool because it is especially well suited for handling sparse data contaminated with noise, for finding useful, rigorous bounds on the infinite solution set, and for predicting accurately what data need to be collected in order to tighten those bounds. We present a practical three‐ dimensional gravity ideal‐body computer code, IDB, that can optimize a mesh with over 105 cells...
20 citations
TL;DR: In this paper, the interrelation between interval Gauss elimination and interval iteration is investigated, and a new existence theorem for interval Gaussian elimination (in the guise of a perturbation theorem), a convergence and comparison theorem for a general family of interval iteration schemes, and an alternative method for the calculation of the hull of the solution set of linear interval equations with inverse positive coefficient matrix is presented.
16 citations
01 Jan 1987
TL;DR: A new interactive satisficing method for multiobjective nonlinear programming problems with fuzzy parameters in which the ordinary Pareto optimality is extended based on the α-level sets of the fuzzy numbers.
Abstract: This paper presents a new interactive satisficing method for multiobjective nonlinear programming problems with fuzzy parameters. The fuzzy parameters in the objective functions and the constraints are characterized by fuzzy numbers. The concept of α-Pareto optimality is introduced in which the ordinary Pareto optimality is extended based on the α-level sets of the fuzzy numbers. In our interactive satisficing method if the decision maker (DM) specifies the degree a of the a-level sets and the reference objective values, the augmented minimax problem is solved and the DM is supplied with the corresponding a-Pareto optimal solution together with the trade-off rates among the values of the objective functions and the degree a. Then by considering the current values of the objective functions and a as well as the trade-off rates, the DM responds by updating his reference objective values and/or the degree a. In this way the satisficing solution for the DM can be derived efficiently from among an a-Pareto optimal solution set. On the basis of the proposed method, a time-sharing computer program is written and an illustrative numerical example is demonstrated along with the computer outputs.
16 citations
01 Jan 1987
TL;DR: This paper shows how it is possible to impose a certain order over nondominated elements of a set, based only on the information contained in the formulation of the corresponding vector optimization problem, and presents a numerically attractive method to determine nested subsets of properly efficient elements.
Abstract: Given a vector optimization problem, the solution set is composed of nondominated elements of a set. All nondominated elements are equivalent in the sense that each of them is a solution as good as any other one. However, if the vector optimization problem is a formalization of a practical multiple objective decision problem, one may expect that some solutions are more preferred than others. To assist a decision maker, most of methods for multiple objective decision making rely upon additional information, to be obtained from the decision maker, to that contained in the definition of the vector optimization problem. In contrast to that, in this paper we show how it is possible to impose a certain order over nondominated elements of a set, based only on the information contained in the formulation of the corresponding vector optimization problem. We also present a numerically attractive method to determine nested subsets of properly efficient elements.
TL;DR: It is proved boundedness of the feasible domain when the quadratic problem is concave, and easily computable bounds for the solution norm for the convex case are given.
TL;DR: In this paper, a nonlinear boundary value problem exhibiting a nonlocal bifurcation in its solution set is considered, and it is shown that for some values of the problem parameters there is only the zero solution, while for other values of these parameters two nontrivial solutions also appear.
01 Jan 1987
TL;DR: Investigations in the field of vector optimization and its practical application on a structural mechanics system showed that the efficiency of the single preference function is problem-dependent and furthermore dependent on the adapting on special optimization algorithms (Methods of Mathematical Programming).
Abstract: This paper reports on investigations carried out in the field of vector optimization and its practical application on a structural mechanics system. The main reason is that in addition to the minimization of costs for developing and manufacturing machines and plants, some other objectives such as shape accuracy, reliability and others are playing an important role as well. These problems can be formulated as “Optimization Problems with Multiple Objectives” (Vector-Optimization, PARETO-Optimization). Most of the objectives are non-linear and besides that competing. Thus, they do not lead to one or several solutions for the optimum but rather to a “functional-efficient” solution set, i.e. the decision maker is able to select out of this set the most efficient compromise solution. Moreover, the vector optimization problem is transformed into a scalar substitute problem by means of a preference function. This so-called optimization strategy is an important part of the modelling. For carrying out the transformation, several preference functions, e.g. objective weighting, distance functions, constraint oriented transformation (Trade-off Method), and Min-Max-Formulation have been analysed and tested. The investigations showed that the efficiency of the single preference function is problem-dependent and furthermore dependent on the adapting on special optimization algorithms (Methods of Mathematical Programming). Within the scope of this contribution the application of multicriteria optimization techniques in structural mechanics systems is shown, e.g. the shape optimization of a special shell structure (conveyer belt drum).
01 Jan 1987
TL;DR: The existence of nonlinear Neumann problems with inhomogeneous boundary conditions is established and the solution set is described in this paper, where the asymptotic behaviour of the time-dependent parabolic equation is studied.
Abstract: The existence of nonlinear Neumann problems with inhomogeneous boundary conditions is established and the solution set is described. Then the asymptotic behaviour of the time-dependent parabolic equation is studied On etablit l'existence de solutions de problemes de Neumann non lineaires inhomogenes et on discute l'ensemble des solutions. On etudie le comportement asymptotique des solutions du probleme parabolique associe
TL;DR: In this paper, the problem of constructing a finite set which approximates a set of Pareto-optimal solutions is examined, and the problem is shown to be NP-hard.
Abstract: The problem of constructing a finite set which approximates a set of Pareto-optimal solutions is examined.
TL;DR: In this article, sufficient conditions are given to guarantee the closedness and uniform boundedness of the solution sets corresponding to a pair of dual parametrized minimax problems, where the parameter of the minimax problem belongs to a metrizable space and affects not only the objective function, but also the feasible sets.
Abstract: Sufficient conditions are given to guarantee the closedness and uniform boundedness of the solution sets corresponding to a pair of dual parametrized minimax problems. The parameter of the minimax problems belongs to a metrizable space and affects not only the objective function, but also the feasible sets.
01 Mar 1987
TL;DR: The correspondences between the primal and dual solution sets are state: under certain conditions the primal solution sets correspond andDual solution sets coincide or are equivalent.
Abstract: A wide variety of applied problems involve programs trading off one objective against another, such as time versus cost. The present article does this by interchanging the objective function and one constraint. The approach is a practical formulation of the bi-objective program. We state the correspondences between the primal and dual solution sets: under certain conditions the primal solution sets correspond and dual solution sets coincide or are equivalent. The article concludes with numerical illustrations and a large-scale application to portfolio analysis.