Showing papers on "Solution set published in 1980"

A New Solution Set for Tournaments and Majority Voting: Further Graph- Theoretical Approaches to the Theory of Voting

Abstract: The Condorcet, or minimal undominated, set of proposals has been identified as a solution set for majority voting. Unfortunately, the Condorcet set may be very large and may include Pareto-inefficient proposals. An alternative solution set is proposed here: the "uncovered set," from which every other proposal in the tournament representing majority preference is reachable in no more than two steps. It is shown that this set has a number of desirable properties, and that several important voting processes lead to decisions within it. Ihis article extends some earlier work by further applying concepts and theorems from the theory of directed graphs to characterize and analyze the structure of majority preference.

Stability of nondominated solutions in multicriteria decision-making

Abstract: Decision-making problems with multiple noncommensurable objectives are specified by two factors, i.e., the set of all feasible solutions and the domination structure. The solutions are characterized as nondominated points. Hence, in these problems, there may exist two parameter vectors, according to which the above two factors change. The stability of the solution set for perturbations of these parameters is investigated in this paper. The analysis is guided by using the concept of continuity of the solution map defined on the two parameter spaces.

Concerning the solution set ofAx=b where P≦A≦Q and p≦b≦q

Abstract: LetP?Q ben×n real matrices so that ifP?A?Q for some matrixA, thenA is nonsingular. Letp andq ben-dimensional real column vectors. This paper determines the set of all solutionsx to the equationAx=b for allA andb so thatP?A?Q andp?b?q.

On the structure of the solution set for periodic boundary value problems

Abstract: and we give some applications of the obtained results to periodic boundary value problems. The simple solvability of (1) has been intensively studied in recent years, particularly in the case where L is a Fredholm operator of index 0 and in connection with boundary value problems. See for example the extensive bibliographies in J. Mawhin [lo], L. Cesari, R. Kannan and J. Schuur [3] and M. Furi, M. Martelli and A. Vignoli [S]. In the case where E = F, L is the identity operator and h is continuous and compact the set of solutions of (1) (i.e. the set of fixed points of h) is a continuum provided that h can be uniformly approximated by a sequence {h,} of completely continuous maps such that the set of fixed points of h, is a continuum for each n. This result is due to Krasnosel’skii and Perov and is an abstract version of the classical Hukuhara-Kneser property for ordinary differential equations (see P. Rabinowitz 1111). With essentially the same assumptions M. Aronszajin [l] proved that the set of fixed points of h is nonempty, compact and acyclic. (see Lasry and Robert [S]). In this paper, by means of a theorem proved in [5] (see also R. Gaines and J. Mawhin [6]), we extend the result of Krasnosel’skii and Perov and the result of Aronszajn to the more general setting of linear Fredholm operators of index 0. As a consequence we show that the set of solutions for particular periodic boundary value problems for ordinary differential equations is nonempty, compact, and acyclic.

Representation of Convex Sets

Abstract: It is well known that a closed convex set in ℝd can be represented as the solution set of a semi-infinite system of linear inequalities. The topic of this paper is to investigate the connections between the point set in ℝd and the defining system of inequalities. Typical problems of this type are redundancy and minimality of the system and the dimension of the solution set. Furthermore it is investigated what relations can be stated between two polyhedral sets being dual to each other in the linear programming sense. Applications to linear complementarity problems are indicated.

Exact bounds for the solution branches of nonlinear eigenvalue problems

Abstract: A method is constructed which yields a strip containing the full solution sets of nonlinear eigenvalue problems of the formu=?Tu. The strip can be narrowed iteratively, and the method applies for both stable and unstable branches. Its high degree of accuracy is demonstrated by numerical examples. In particular, a lower bound is given for the critical value at which criticality is lost in the thermal ignition problem for the unit ball.

Continuity of the Optimal Value Function Under the Mangasarian-Fromovitz Constraint Qualification.

Abstract: : We demonstrate the continuity of the optimal value of general inequality-constrained parametric nonlinear programs having a solution point that satisfies the Mangasarian-Fromovitz Constraint Qualification (MFCQ). This result is applied to prove the continuity of the optimal value function for general inequality-equality parametric programs under MFCQ, by using a variable reduction approach. An abstract set (not defined by functional inequalities or equalities) is allowed to enter (nontrivially into the definition of the feasible region and results regarding the behavior of the parameter dependent feasible region and solution set are also obtained. (Author)

On the structure of minimal spectral factors

Abstract: Two different well known approches to the spectral factorization problem ?(s)=W(s)W' (-s) are connected together by relating the geometric properties of the solution set of the underlying Algebraic Riccati Equation to the structure of the "all pass" factor of each minimal solution W(s).

The Weighted Sperner's Set Problem

Abstract: A polynomial time bounded algorithm is presented for solving the Weighted Sperner's Set Problem, that is, the problem of computing an independent Set of maximal weight on a weighted partially ordered set.

A computer-aided design of robust regulators

Abstract: The objective of this paper is to present a design methodology for linear and nonlinear regulators by applying mathematical programming techniques. Starting with the linear vector equations an algorithmic procedure is outlined for selecting the regulator parameter vector r within the framework of classical time and frequency domain specifications, as well as the quadratic optimality criterion. For nonlinear regulators described by nonlinear vector equations attention is focused on the design of absolutely stable regulators subject to prescribed exponential stability and sector maximization requirements. Reformulating the performance specifications in terms of a set of inequalities, a feasible region is delineated in the regulator parameter space. Then, maximization of the volume of an imbedded hypercube (or hypersphere) inside the region via mathematical programming methods, results in an easily visualized solution set, this feature being particularly attractive in building robust regulators to meet real-world implementation tolerances and system parameter uncertainities.

Synthesis of optimization algorithms by concatenation

Abstract: Based on the idea that classes of algorithms can be modeled with abstract schemas, a systematic approach to algorithm analysis has been developed over the past two decades [6, 7, 9, 12, 13, 14, 18, 19, 22]. The approach consists of identifying a given algorithm with the appropriate schema, and in using the convergence proof-pattern associated with that schema to structure and simplify the analysis of the properties of the given algorithm. It stands to reason that a similar approach may prove useful in simplifying and systematizing the synthesis of algorithms. Assuming that a collection of problem models, and the corresponding algorithm schemas is available, the synthesis approach will consist of identifying the given problem with the appropriate problem model, and then synthesizing the desired algorithm by incorporating the specific properties of the problem into the schema or schemas which correspond to the problem model. Some problem models and associated algorithm schemas already exist. The best known problem model is the fixed point formulation. In this model it is assumed that the solution set of the problem is described as the fixed point set of a point-to-point or point-to-set map. The associated schemas are the Picard iteration [17], and its variations through application of averaging processes [3, 4, 8, 10]. In this paper we examine a different problem model, namely the intersection model. This model assumes that the solution set of the problem is described as the intersection of a finite family of component sets, and the corresponding algorithm schema consists of the concatenation of a finite family of autonomous algorithm components, each of which is capable of finding points in one of the component sets. The intersection model is particularly well suited for the analysis and synthesis of relaxation methods. These methods, proposed originally by Jacobi and Seidel for solving linear systems of equations, were extended so that nonlinear systems could be solved [16], and then generalized to chaotic procedures [2, 15, 21, 21] . Relaxation methods have also been used on structured optimization problems. For example, in [1] and [11] the problem considered is that of minimizing a functional on a product space, in [9] various coordinate descent algorithms are analyzed using Zangwill's idea of composition of point-to-set maps, and in [5], Fiorot and Huard present a theory for the analysis of algorithms for optimization which are obtained by cumposition of component algorithms.

Lipschitz-Stabilität von Optimierungs- und Approximationsaufgaben / Lipschitz-Stability for Programming and Approximation Problems

Abstract: This paper deals with the Lipschitz stability of the feasible region, the solution set and the minimum value of the objective function for convex programming problems when the data are subjected to small perturbations We show that a certain regularity condition is necessary and sufficient for the Lipschitz continuity of the feasible region We get the Lipschitz continuity of the minimum value and the set of e-solutions Several examples show that in general the Lipschitz upper semicontinuity doesn’t hold for the exact solution set However we prove for weak Chebyshev systems in C[a,b] with unique alternation element gf for each f ? C[a,b], that the selection s: f → gf is Lipschitz continuous Consequences resulting from rounaing errors are discussed for numerical methods

A Functional Inequality Arising in Combinatorics

Abstract: In this paper, we discuss the functional inequality \( p(n + m){\kern 1pt} \le {\kern 1pt} (_{n}^{{n + m}}){\kern 1pt} p(n){\kern 1pt} p(m) \), which arises in tournament theory and other parts of combinatorics. A simple transformation removes the binomial coefficient, and then the solution set divides naturally into three classes of functions. One class consists of all the nonpositive functions since this inequality puts no restriction on such functions. The counting-function solutions, i.e., the nonnegative solutions, all lie in the other two classes and satisfy easily obtainable exponential growth bounds. This set of solutions also possesses a structure in the sense that various combinations of these solutions, e.g., sums and products, are again in the set. Various solution functions and properties of solutions are obtained by introducing a slack function to convert the functional inequality to a functional equation. The general solution to this functional equation is obtained by transforming it to another functional equation whose general solution is known. Solution functions found in this manner occur in pairs and are sometimes even from different solution classes. This slack-function concept has modifications, so it can be applied in other ways to the functional inequality and to other inequalities.