Showing papers on "Solution set published in 1986"
01 Jan 1986
TL;DR: In this paper, the problem of enumerating solutions to layout problems has been addressed by drawing a clean distinction between the quantitative and continuous properties of a solution (such as the dimensions of the spaces allocated) and some of its qualitative or discrete properties (particularly the geometric or spatial relations between the allocated spaces).
Abstract: Starting with [Grason 70] and [Steadman 70] and continued through [Mitchell et al. 76] and [Flemming 78], work on the exhaustive enumeration of solutions to layout problems has produced a particular approach which, by now, has established itself as a fully developed paradigm. This paradigm achieves great conceptual clarity by drawing a clean distinction between, on the one hand, the quantitative and continuous properties of a solution (such as the dimensions of the spaces allocated) and, on the other hand, some of its qualitative or discrete properties (particularly the geometric or spatial — not necessarily topological — relations between the allocated spaces). The paradigm stresses the importance of using a formalized representation for properties of the second type and calls for an explicit specification of the necessary and sufficient conditions under which such representations are to be considered well-formed or syntactically correct; that is, every representation of a solution satisfies these conditions and every object that satisfies these conditions represents a solution. In the enumeration of solution sets, these representations play a crucial role in two ways:
(1)
Each representation is an abstraction since it supresses certain properties of the solution it describes. Different solutions can therefore have the same representation, and each representation consequently describes not a single solution, but an entire class or subset of solutions. Under a suitably selected representation, the (possibly infinite) set of solutions is divided into a finite set of subsets which can be enumerated by generating all well-formed representations as objects.
91 citations
TL;DR: A bound on the distance between an arbitrary point and the solution set of a monotone linear complementarity problem is given in terms of a condition constant that depends on the problem data only and a residual function of the violations of the complementary problem conditions by the point considered.
Abstract: We give a bound on the distance between an arbitrary point and the solution set of a monotone linear complementarity problem in terms of a condition constant that depends on the problem data only and a residual function of the violations of the complementary problem conditions by the point considered. When the point satisfies the linear inequalities of the complementarity problem, the residual consists of the complementarity condition plus its square root. This latter term is essential and without it the error bound cannot hold. We also show that another natural residual that has been employed to bound errors for strictly monotone linear complementarity problems fails to bound errors for the monotone case considered here.
69 citations
68 citations
TL;DR: In this paper, a-Pareto optimality is extended based on the α-level sets of the fuzzy numbers and a new interactive decision-making method for obtaining the satisficing solution of the decision maker (DM) on the basis of the linear programming method.
Abstract: In this paper, we focus on multiobjective linear programming problems with fuzzy parameters and present a new interactive decision making method for obtaining the satisficing solution of the decision maker (DM) on the basis of the linear programming method. The fuzzy parameters in the objective functions and the constraints are characterized by fuzzy numbers. The concept of a-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 decision making method, in order to generate a candidate for the satisficing solution which is also α-Pareto optimal, if the DM specifies the degree a of the α-level sets and the reference objective values, the minimax problem is solved by making use of the linear programming method, and the DM is supplied with the corresponding α-Pareto optimal solution together with the trade-off rates among the values of the objective functions and the degree α. Then by considering the current values of the objective functions and a as well as the trade-off rates, the DM acts on this solution by updating his/her reference objective values and/or degree α. In this way the satisficing solution for the DM can be derived efficiently from among an α-Pareto optimal solution set. A numerical example illustrates various aspects of the results developed in this paper.
34 citations
TL;DR: In this article, the authors considered the task of identifying a causal, linear, dynamic, multivariable system excited by stationary, zero-mean noise of unknown spectrum, and given measurements of the system inputs and outputs contaminated by independent, additive noise also of unknown spectra.
Abstract: This note considers the task of identifying a causal, linear, dynamic, multivariable system excited by stationary, zero-mean noise of unknown spectrum, and given measurements of the system inputs and outputs contaminated by independent, additive noise also of unknown spectra. Although the solution is in general not unique, finite-dimensional parameterizations of the solution set are given, even though the various spectra may not be rational.
29 citations
TL;DR: In this paper, the authors analyse the solution set of first-order initial value differential problems of the form dy dx = ƒ(x, y), y(0) = 0 in the context of combinatorial species in the sense of A. Joyal (Adv. in Math. 42 (1981), 1−82).
27 citations
TL;DR: In this article, the possibility of non-radial solutions of bifurcating from the radial solutions family was investigated and the global structure of the nonradial solution set was investigated.
Abstract: : This document considers a nonlinear elliptic problem. Suppose this problem has a family of positive radial solutions parameterized by R, i.e., UR (X). This paper studies the possibility of the existence of nonradial solutions of bifurcating from the radial solutions family. Answering a question posed by Smoller and Wasserman, it is shown this happens if f satisfies suitable assumptions. Therefore, the global structure of the nonradial solution set is investigated.
26 citations
01 Jan 1986
TL;DR: In this paper, a geometrical characterization of the solution set of a symmetric positive definite n×n matrix is presented, where orthant and null invariance are introduced.
Abstract: In this paper, a novel approach is provided to an important but unsolved mathematical problem that occurs in a wide variety of applications: Given a symmetric positive definite n×n matrix ?, determine all diagonal nonnegative matrices ?~ so that the difference matrix ?= ?- ?~ is nonnegative definite and its rank is minimal. In this paper, we explore the geometrical properties of the solution vectors x satisfying ?.x=0. New concepts such as orthant and null invariance are introduced. The results in this paper are of key importance in the analysis of noisy linear equations and factor analysis. They Hill lead to a complete geometrical characterization of the solution set, which will be described in a forthcoming paper.
21 citations
TL;DR: In this paper, the authors address the issue of solution and estimation of linear rational expectations models and provide evidence in the form of a Monte Carlo experiment that one may recover accurate estimates of the parameters that characterize the solution set with which the data were generated.
18 citations
TL;DR: In this article, the authors give an approach to the study of both differential inclusions and ordinary differential equations in a Banach space X. The central point concerns the question of the existence and properties of the solution set of a differential inclusion whose right-hand side has the weak Scorza Dragoni property.
Abstract: This article gives an approach to the study of both differential inclusions and ordinary differential equations in a Banach space X. The central point concerns the question of the existence and properties of the solution set of a differential inclusion whose right-hand side has the weak Scorza Dragoni property.Bibliography: 37 titles.
16 citations
TL;DR: It is shown that under certain conditions this set of first-order solutions of a family of nonlinear programs in which the inequality constraints are fixed but the right hand side of the equality constraints varies varies is a topological manifold.
Abstract: This paper studies the set of first-order solutions of a family of nonlinear programs in which the inequality constraints are fixed but the right hand side of the equality constraints varies. It is shown that under certain conditions this set is a topological manifold. The results are applied to the problem of describing the set of first-order Pareto optima in a pure exchange economy with monotone utility functions.
TL;DR: In this paper, conditions for the convergence of perturbed solutions to a point of the reduced problem solution set, if the small parameter tends to zero, are given for a singular situation where the dual solution set is unbounded.
Abstract: Mathematical programming (MP) problems depending on a small parameter are investigated. Attention is paid to the cases where the solutions to the reduced program and/or the solutions to the dual reduced program are not unique. Conditions are given for the convergence of perturbed solutions to a point of the reduced problem solution set, if the small parameter tends to zero. It is shown how to find this point and how to construct an approximate solution to the perturbed program. A singular situation may appear if the dual solution set is unbounded. In this case, a gap between perturbed and reduced solutions may arise. However, it is shown that the perturbed solutions are close to the solutions of some modified reduced problem. The practical usefulness of perturbation theory is demonstrated by considering the two LP problems. Decomposition and aggregation procedures are constructed on the base of general results to find suboptimal solutions of these problems.
TL;DR: In this paper, an extremization problem in the general form max x ϵ X ǫ(x) is treated as the description of a parametric family of problems where the extremized function is the same but the set X is a variable parameter.
01 Jan 1986
TL;DR: An interactive fuzzy decisionmaking method is presented, which assumes that the decisionmaker (DM) has fuzzy goals for each of the objective functions in multiobjective nonlinear programming problems and can be derived efficiently from among a Pareto optimal solution set.
Abstract: An interactive fuzzy decisionmaking method is presented, which assumes that the decisionmaker (DM) has fuzzy goals for each of the objective functions in multiobjective nonlinear programming problems. Having determined the membership functions for each of the objective functions, if the DM selects an appropriate standing membership function and specifies his/her aspiration levels of achievement of the other membership functions, called constraint membership values, the corresponding constraint problem is solved, and the DM is supplied with the Pareto optimal solution together with the trade-off rates between the membership functions. Then by considering the current values of the membership functions as well as the trade-off rates, the DM acts on this solution by updating his/her constraint membership values. In this way, the satisficing or compromise solution for the DM can be derived efficiently from among 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 output.
TL;DR: In this paper, the authors consider a class of semilinear elliptic boundary value problems depending on a parameter, which arise in the theory of combustion and show a connection between the solution set of the boundary value problem and a simple scalar equation (the Semenov approximation).
Abstract: We consider a class of semilinear elliptic boundary value problems depending on a parameter, which arise in the theory of combustion. Based on the results in another paper by the same author, a rigorous quantitative connection is shown between the solution set of the boundary value problem and that of a simple scalar equation (the Semenov approximation).
TL;DR: In this article, an interval operator of a function strip with inverse-nonnegative Lipschitz matrix is discussed, which can construct a sequence of intervals which converges to a limit interval if the set of solutions of a given function strip is not empty.
Abstract: An interval operator of a function strip with inverse-nonnegative Lipschitz matrix is discussed. Using this operator we can construct a sequence of intervals which converges to a limit interval, if the set of solutions of a given function strip is not empty. The limit interval optimally encloses the set of solutions.
01 Jan 1986
TL;DR: An interactive method for solving multiobjective nonlinear programming problems with fuzzy parameters characterized by fuzzy numbers and the concept of α-Pareto optimality is introduced in which the ordinary Pareto Optimality is extended based on the a-level sets of the fuzzy numbers.
Abstract: An interactive method for solving multiobjective nonlinear programming problems with fuzzy parameters characterized by fuzzy numbers is presented and examined. The concept of α-Pareto optimality is introduced in which the ordinary Pareto optimality is extended based on the a-level sets of the fuzzy numbers. In our interactive method, if the decision maker (DM) specifies the degree a of the α-level sets and the reference objective values, the minimax problem is solved and the DM is supplied with the corresponding α-Pareto optimal solution together with the trade-off rates among the objective functions and the degree a. Then by considering the current values of the objective functions and as well as the trade-off rates, the DM responds by updating his reference objective values and/or the degree α. In this way the satisficing solution for the DM can be derived from among an α-Pareto optimal solution set. A numerical example illustrates various aspects of the results developed in this paper.
TL;DR: A methodology is described for allocating reliability to various nuclear reactor systems, subsystems, components, operations, and structures consistent with a set of global safety criteria that are not rigid.
Abstract: A methodology is described for allocating reliability to various nuclear reactor systems, subsystems, components, operations, and structures consistent with a set of global safety criteria that are not rigid. The problem is formulated as a multiattribute decision analysis paradigm; the multiobjective optimization, which is performed on a probabilistic risk assessment model and reliability cost functions, serves as the guiding principle for reliability and risk allocation. The concept of ''noninferiority'' is used in the multiobjective optimization problem. Finding the noninferior solution set is the main theme of the current approach. The assessment of the decision-maker's preferences could then be performed more easily on the noninferior solution set.
TL;DR: In this article, sufficient conditions for semicontinuity of the solution multifunction without closedness assumptions are given. But they do not consider the generalization of the marginal function.
Abstract: Studies on generalizations of the marginal function and the solution multifunction in vector optimization is given. We present sufficient conditions for semicontinuity of the solution multifunction without closedness assumptions.
TL;DR: This paper presents a heuristic method for reducing the computational burden in multiple objective dynamic programming (MODP), using techniques originally suggested for multiple objective linear programming, to give a representative subset of the set of efficient ways of attaining that state (stage).
01 Jan 1986
TL;DR: In this article, the basic steps that are used to solve linear equations and inequalities in one variable are discussed, and a common mistake in solving inequalities is to forget to reverse the direction of the inequality symbol when multiplying both sides by a negative number.
Abstract: This chapter discusses the basic steps that are used to solve linear equations and inequalities in one variable. The solution set for an equation is the set of all numbers that, when used in place of the variable, make the equation a true statement. Two or more equations with the same solution set are called “equivalent equations.” Addition property of equality states that the same quantity can be added to both sides of an equation without changing the solution set. Multiplication property of equality states that multiplying both sides of an equation by the same non-zero quantity never changes the solution set. The addition property for inequalities states that adding the same quantity to both sides of an inequality never changes the solution set. Multiplication property for inequalities states that both sides of an inequality can be multiplied by the same non-zero number without changing the solution set as long as each time it is multiplied by a negative number, the direction of the inequality symbol is also reversed. A common mistake in solving inequalities is to forget to reverse the direction of the inequality symbol when multiplying both sides by a negative number. When this mistake occurs, the graph of the solution set is always to the wrong side of the end point.
TL;DR: In this article, the continuous affine functionals on families of analytic functions that are defined by integrating a kernel with respect to probability measures on the unit circle are characterized. And the solution set is characterized for the maximum of a continuously affine functional over these families, including closed convex hulls of convex and starlike mappings.
01 Feb 1986
TL;DR: In this paper, the authors consider the problem of finding the tube of all solutions to a nonlinear multistage inclusion that arise from a given set and also satisfy an additional phase constraint.
Abstract: One of the means of modelling a system with an uncertainty in the parameters or in the inputs is to consider a multistage inclusion or a differential inclusion. These types of models may serve to describe an uncertainty for which the only available data is a set-membership description of the admissible constraints on the unknown parameters.
A problem under discussion here deals with the specification of the "tube" of all solutions to a nonlinear multistage inclusion that arise from a given set and also satisfy an additional phase constraint. The description of this "solution tube" is important for solving problems of guaranteed estimation of the dynamics of uncertain systems as well as for the solution of other "viability" problems for systems described by equations involving multivalued maps.
TL;DR: In this paper, the Schrodinger equation has been transformed into a set of coupled partial differential equations having hyper-variables as arguments and a procedure for embedding the boundary conditions into the N -body scattering solution by using homogeneous linear algebraic equations is proposed.
TL;DR: The method of interchanging networks resulting from structural investigations is described to solve problems, where the solution set is given by the set of all permutations of melements.
Abstract: In this paper the method of interchanging networks resulting from structural investigations is described to solve such problems, where the solution set is given by the set of all permutations of melements. Some applications of this method are discussed. With regard to the application for solving the traveling salesman problem a coupling of the method of interchanging networks with the extension principle algorithm is described and computational results are presented.