Showing papers on "Solution set published in 1990"
TL;DR: A projection operator which is not identically zero, but which is guaranteed to vanish on all the proper components of the system f"i=0, which fills the role of a general affine projection operator or variable elimination ''black box'' which can be used for arbitrary polynomial systems.
116 citations
TL;DR: Algorithms have been proposed which enable us to modify the original equation in a way leading to its genuine solution or at least an approximate one and distortion of the fuzzy set Y forms a cornerstone of the proposed procedures.
84 citations
TL;DR: An iterative, adaptive, space-variant restoration incorporating both regularization and artifact-suppression constraints is developed, which bound the norm of the variation of the image from its local mean using regionally adaptive bounds.
Abstract: The authors develop an iterative, adaptive, space-variant restoration incorporating both regularization and artifact-suppression constraints. The algorithm is formulated using the method of projections onto convex sets (POCS). They introduce a closed-convex regularization constraint set called the partial Wiener solution set. Projection onto this set forces the solution to be equal to the Wiener solution over a predetermined set of frequencies that lie outside the neighborhoods of the zeros of the degradation transfer function. In the neighborhoods of the zeros, the Wiener solution contains significant errors that contribute to artifacts. Hence, in these neighborhoods the Wiener solution is discarded, and the missing frequency components are determined so that the entire solution is consistent with the artifact-suppression constraints and other a priori information. The proposed artifact-suppression constraints bound the norm of the variation of the image from its local mean using regionally adaptive bounds. The high quality of the resulting restorations is noteworthy. >
79 citations
TL;DR: An interactive method to generate an approximation of the noninferior solution set for two objective shortest path problems is proposed to assist the decision maker in selecting the preferred or best compromise solution from among the non inferior solutions.
61 citations
TL;DR: In this paper, a sensitivity analysis of numerical problems with independent, non-correlated data for general systems of linear and nonlinear equations is presented, with the additional property that, due to an automatic error control mechanism, every computed result is guaranteed to be correct.
Abstract: Methods are presented for performing a rigorous sensitivity analysis of numerical problems with independent, noncorrelated data for general systems of linear and nonlinear equations. The methods may serve for the following two purposes. First, to bound the dependency of the solution on changes in the input data. In contrast to condition numbers a componentwise sensitivity analysis of the solution vector is performed. Second, to estimate the true solution set for problems the input data of which are afflicted with tolerances. The methods presented are very effective with the additional property that, due to an automatic error control mechanism, every computed result is guaranteed to be correct. Examples are given for linear systems demonstrating that the computed bounds are in general very sharp. Interesting comparisons to traditional condition numbers are given.
52 citations
TL;DR: Error bounds and upper Lipschitz continuity results are given for monotone linear complementarity problems with a nondegenerate solution because of the polyhedral characterization of the solution set as the intersection of the feasible region with a single linear affine inequality constraint.
Abstract: Error bounds and upper Lipschitz continuity results are given for monotone linear complementarity problems with a nondegenerate solution. The existence of a nondegenerate solution considerably simplifies the error bounds compared with problems for which all solutions are degenerate. Thus when a point satisfies the linear inequalities of a nondegenerate complementarity problem, the residual that bounds the distance from a solution point consists of the complementarity condition alone, whereas for degenerate problems this residual cannot bound the distance to a solution without adding the square root of the complementarity condition to it. This and other simplified results are a consequence of the polyhedral characterization of the solution set as the intersection of the feasible region {z∣Mz + q ≥ 0, z ≥ 0} with a single linear affine inequality constraint.
47 citations
TL;DR: In this article, the authors introduce the notion of likelihood for a solution of a differential inclusion and formulates a corresponding class of estimation, prediction, and filtering problems, which is independent of probability theory, relying solely on the metric structure.
26 citations
03 Jan 1990
TL;DR: An interactive sequential goal programming is proposed which introduces the method of displaced ideal with fuzzy set theory for interactively solving MODM problems and extends ISGP to attain the best compromise solution from nondominated solution set by evaluating quantitatively.
Abstract: Recently, multiple objective decision making(MODM) has been well established as a practical approach to seek a satisfactory solution to a decision making problem in a limited resources, information, and cognitive ability of the Decision Maker(DM). In this paper, we propose an interactive sequential goal programming(ISGP) which introduce the method of displaced ideal(MDI) with fuzzy set theory for interactively solving MODM problems. That is, we extend ISGP to attain the best compromise solution from nondominated solution set by evaluating quantitatively, which have been evaluated by the DM.
20 citations
TL;DR: In this paper, a general interpolation problem is defined which contains as special cases the M-Pade approximation problem as well as classical interpolation problems, e.g., Newton-PADE approximation.
19 citations
TL;DR: In this paper, a new solution set for majority voting tournaments was proposed, and the internal structure of the solution set was examined, showing that, in the absence of a Condorcet winner, there is always a cycle including precisely the points in the Banks set.
Abstract: We consider a new solution set for majority voting tournaments recently proposed by Banks (1985), and we examine its internal structure. In particular, we demonstrate that, in the absence of a Condorcet winner, there is always a cycle including precisely the points in the Banks set. We introduce the concept of "external stability" in order to facilitate analysis.
19 citations
TL;DR: Some insights into the extent of constraint checking involved in and feasibility of more checking for three or more enzymes are discussed and the trade-off between computation time and the soundness of the solution set is examined.
Abstract: Restriction site mapping requires a generator to put forward possible maps and a constraint checker to reject false maps. Ideally these combine to give an algorithm which calculates a sound and complete solution set. Three algorithms for generation are presented and compared. Two decompose a multi-enzyme problem (greater than or equal to 3) into subproblems. The constraint checker is based on separation theory. Some insights into the extent of constraint checking involved in and feasibility of more checking for three or more enzymes are discussed. The trade-off between computation time and the soundness of the solution set is examined.
01 Jan 1990
TL;DR: In this paper, a continuous-time version of the subgradient projection algorithm is applied to find equlibria of non-cooperative games, and convergence in norm is established under a strict montonicity assumption.
Abstract: This paper applies a continuous-time version of the subgradient projection algorithm to find equlibria of non-cooperative games. Under monotonicity assumptions this algorithm is known to generate trajectories which Cesaro converge weakly to the solution set. Convergence in norm is established under a strict montonicity assumption. Strong monotonicity is shown to entail norm convergence at an exponential rate. A sharpness condition yields norm convergence in finite time, and the necessary lapse is estimated.
TL;DR: In this article, the authors consider the notion of cycle avoiding trajectories in majority voting tournaments and show that they underlie and guide several apparently disparate voting processes, including sincere and sophisticated voting.
Abstract: This paper considers the notion of cycle avoiding trajectories in majority voting tournaments and shows that they underlie and guide several apparently disparate voting processes. The set of alternatives that are maximal with respect to such trajectories constitutes a new solution set of considerable significance. It may be dubbed the Banks set, in recognition of the important paper by Banks (1985) that first made use of this set. The purpose of this paper is to informally demonstrate that the Banks set is a solution set of broad relevance for understanding group decision making in both cooperative and non-cooperative settings and under both sincere and sophisticated voting. In addition, we show how sincere and sophisticated voting processes can be viewed as mirror images of one another — embodying respectively, “dmemory” and “foresight.” We also show how to develop the idea of a “sophisticated agenda,” one in which the choice of what alternatives to propose is itself a matter of strategic calculation.
TL;DR: It is shown that the cardinal numbers of solution sets of fuzzy relation equations for two constant vectors with the same partition on the index set are equal to each other.
TL;DR: In this paper, the authors give necessary and sufficient conditions for the solution set of a system of linear interval equations to be nonconvex, and derive some consequences for the resulting solution.
Abstract: We give necessary and sufficient conditions for the solution set of a system of linear interval equations to be nonconvex and derive some consequences.
01 Jan 1990
TL;DR: In this paper, an interactive decision-making method for multiobjective linear, linear fractional and nonlinear programming problems with fuzzy parameters is presented, based on the α-level sets of the fuzzy numbers, the concept of α-multiobjective programming and local M-α-Pareto optimality is introduced.
Abstract: This paper presents interactive decision making methods for multiobjective linear, linear fractional and nonlinear programming problems with fuzzy parameters. On the basis of the α-level sets of the fuzzy numbers, the concept of α-multiobjective programming and (local) M-α-Pareto optimality is introduced. Through the interaction with the decision maker (DM), the fuzzy goals of the DM for each of the objective functions in α-multiobjective programming are quantified by eliciting the corresponding membership functions. After determining the membership functions, in order to generate a candidate for the (local) satisficing solution which is also (local) M-α-Pareto optimal, if the DM specifies the degree α of the α-level sets and the reference membership values, the (augmented) minimax problem is solved and the DM is supplied with the corresponding (local) M-α-Pareto optimal solution together with the trade-off rates among the values of the membership functions and the degree α. Then by considering the current values of the membership functions and α as well as the trade-off rates, the DM responds by updating his/her reference membership values and/or the degree α. In this way the (local) satisficing solution for the DM can be derived efficiently from among an M-α-Pareto optimal solution set. Based on the proposed methods for multiobjective linear, linear fractional and nonlinear programming problems with fuzzy parameters, interactive computer programs are developed and an illustrative numerical example for nonlinear case is demonstrated.
05 Apr 1990
TL;DR: A formula is derived, within a Bayesian framework, and it provides an estimate of the number of solutions which can be constructed from a given dictionary for any given crossword puzzle without direct recourse to computation of all solutions.
Abstract: There exist implemented algorithms which can provide all possible solutions to any crossword puzzle grid for any given set of words. A formula is derived, within a Bayesian framework, and it provides an estimate of the number of solutions which can be constructed from a given dictionary for any given crossword puzzle without direct recourse to computation of all solutions. The formula is constructed to account for any puzzle geometry and any given dictionary, natural language or otherwise. The number of solutions estimated by the formula, for a variety of puzzles and dictionaries, is compared with the actual number in each solution set. The formula is shown to be only partially effective but to be capable of further development. The formula is useful as an indicator of the time required for computing all solutions, if any exist. >
TL;DR: The main result is based on the theory of infinite systems of linear inequalities and the second-order necessary optimality conditions are established under a regularity assumption for a problem of minimizing a function over the solution set of an inclusion system.
Abstract: Second-order necessary optimality conditions are established under a regularity assumption for a problem of minimizing a functiong over the solution set of an inclusion system 0 ∈F(x), x ∈ M, whereF is a set-valued map between finite-dimensional spaces andM is a given subset. The proof of the main result of the paper is based on the theory of infinite systems of linear inequalities.
01 Jan 1990
TL;DR: The objective is to explain how numerical continuation and bifurcation techniques can be used to investigate the parametric nonlinear programming problem in a global sense and a class of efficient predictor-corrector continuation procedures for tracing solution paths of the system F(z,alpha) = O which are tailored specifically to theParametric programming problem.
Abstract: The parametric nonlinear programming problem is that of determining the behavior of solution(s) as a parameter or vector of parameters alpha belonging to R(sup r) varies over a region of interest for the problem: Minimize over x the set f(x, alpha):h(x, alpha) = 0, g(x, alpha) is greater than or equal to 0, where f:R(sup (n+r)) approaches R, h:R(sup (n+r)) approaches R(sup q) and g:R(sup (n+r)) approaches R(sup p) are assumed to be at least twice continuously differentiable. Some of these parameters may be fixed but not known precisely and others may be varied to enhance the performance of the system. In both cases a fundamentally important problem in the investigation of global sensitivity of the system is to determine the stability boundaries of the regions in parameter space which define regions of qualitatively similar solutions. The objective is to explain how numerical continuation and bifurcation techniques can be used to investigate the parametric nonlinear programming problem in a global sense. Thus, first the problem is converted to a closed system of parameterized nonlinear equations whose solution set contains all local minimizers of the original problem. This system, which will be represented as F(z,alpha) = O, will include all Karush-Kuhn-Tucker and Fritz John points, both feasible and infeasible solutions, and relative minima, maxima, and saddle points of the problem. The local existence and uniqueness of a solution path (z(alpha), alpha) of this system as well as the solution type persist as long as a singularity in the Jacobian D(sub z)F(z,alpha) is not encountered. Thus the nonsingularity of this Jacobian is characterized in terms of conditions on the problem itself. Then, a class of efficient predictor-corrector continuation procedures for tracing solution paths of the system F(z,alpha) = O which are tailored specifically to the parametric programming problem are described. Finally, these procedures and the obtained information are illustrated within the context of design optimization.
01 Jan 1990
TL;DR: A procedure for computing the solution set of parametric equations describing a sensor-object imaging relationship and an application of these techniques which demonstrates the use of task-related information and set-based decision-making methods are described.
Abstract: We view the problem of sensor-based decision-making in terms of two components: a sensor fusion component that isolates a set of models consistent with observed data, and an evaluation component that uses this information and task-related information to make model-based decisions. This paper describes a procedure for computing the solution set of parametric equations describing a sensor-object imaging relationship. Given a parametric form with s parameters, we show that this procedure can be implemented using a parallel array of 6s2 processors. We then describe an application of these techniques which demonstrates the use of task-related information and set-based decision-making methods. Comments University of Pennsylvania Department of Computer and Information Science Technical Report No. MSCIS-90-80. This technical report is available at ScholarlyCommons: http://repository.upenn.edu/cis_reports/749 Interval-Based Techniques for Sensor Data Fusion MS-CIS-90-80 GRASP LAB 241
01 Apr 1990
TL;DR: In this article, the authors examine infinite-dimensional control systems governed by semilinear evolution equations and having both state and control constraint, and show that the original trajectories are dense in an appropriate function space in the relaxed ones.
Abstract: In this paper we examine infinite-dimensional control systems governed by semilinear evolution equations and having both state and control constraint. We introduce the relaxed system and show that the original trajectories are dense in an appropriate function space in the relaxed ones. We also determine the dependence of the solution set on the initial conditions. Then using those results we establish necessary and sufficient conditions for optimality for some optimization problems. Finally we prove some controllability results.
01 May 1990
TL;DR: A new algorithm for finding all DC solution points of nonlinear circuit equations is presented that guarantees that all components of solution set M contained in an initial bounded rectangular region will be found within a prescribed accuracy.
Abstract: A new algorithm for finding all DC solution points of nonlinear circuit equations is presented. The algorithm makes use of interval analysis techniques. Unlike other known algorithms, the present algorithm guarantees that all components of solution set M contained in an initial bounded rectangular region will be found within a prescribed accuracy. Moreover, the new algorithm requires far less computer time. >
TL;DR: It is shown how to efficiently combine the ellipsoid method for computing bounds for the solution set with the path-following algorithm proposed by the authors for the LCP.
Abstract: This paper deals with the LCP (linear complementarity problem) with a positive semi-definite matrix. Assuming that a strictly positive feasible solution of the LCP is available, we propose ellipsoids each of which contains all the solutions of the LCP. We use such an ellipsoid for computing a lower bound and an upper bound for each coordinate of the solutions of the LCP. We can apply the lower bound to test whether a given variable is positive over the solution set of the LCP. That is, if the lower bound is positive, we know that the variable is positive over the solution set of the LCP; hence, by the complementarity condition, its complement is zero. In this case we can eliminate the variable and its complement from the LCP. We also show how we efficiently combine the ellipsoid method for computing bounds for the solution set with the path-following algorithm proposed by the authors for the LCP. If the LCP has a unique non-degenerate solution, the lower bound and the upper bound for the solution, computed at each iteration of the path-following algorithm, both converge to the solution of the LCP.
TL;DR: In this article, a variational principle for continuum states is given that permits numerical solution by the Ritz method, which allows one to maximize the accuracy of the solution in preselected regions of space, and also allows the selection of that solution from the perhaps infinitely degenerate solution set that is needed in the particular application.
Abstract: A variational principle for continuum states is given that permits numerical solution by the Ritz method. It allows one to maximize the accuracy of the solution in preselected regions of space, and also allows the selection of that solution from the perhaps infinitely degenerate solution set that is needed in the particular application.
03 Dec 1990
TL;DR: A solution procedure from the defined quasi-characteristic matrix is proposed that not only determines the solution set but also defines the types of solution set before the solutionSet is determined.
Abstract: Concerns a fuzzy relation equation xoA=b where x and b are input and output vectors and A is a state matrix with max-min operator 'o'. The authors extend their previous work by relaxing the condition that the elements in vector b are strictly distinct and allowing the same values of membership to exist in b. They propose a solution procedure from the defined quasi-characteristic matrix that not only determines the solution set but also defines the types of solution set before the solution set is determined. Therefore, it provides an insight into the structure of fuzzy relation equations. Also by means of the quasi-characteristic matrix, the authors perform a sensitivity analysis on A to determine the set of state-matrices which has the same solution set with respect to the given output-vector. >
Dissertation•
01 Jan 1990
TL;DR: In this article, the authors proposed an optimal solution to the Fredholm integral equation (FIE) and analyzed the size of the solution set subject to linear inequality constraints, and showed the effect of dependent errors on the information provided by the data.
Abstract: The determination of an aerosol size distribution is presently difficult because current aerosol instruments cannot perfectly discriminate aerosols based on size and because a only limited number of data can be obtained. As a result, for a given set of data the relationship between the unknown distribution and the data is a finite Fredholm integral equation. If the size distribution is desired, then one should answer the following
• What measurements should be taken?
• How should the measurements be used to determine a size distribution? In this thesis, we shed some light on the answers to these questions by finding optimal solutions to the Fredholm integral equation, and by characterizing the size of the solution set. The questions of existence and uniqueness of solutions subject to linear inequality constraints are examined. Optimal solutions based on regularization are developed, and numerical methods for finding these solutions are described. Numerical experiments are presented that demonstrate the importance of
• describing dependent error sources.
• considering the magnitude of the errors in the data when there are few data.
• using generalized cross validation when there are many data and the magnitude of the errors is unknown. An analysis that uses some simple information concepts is presented for examining the size of the solution set. An example is presented that demonstrates the effect of dependent errors on the information provided by the data, and some illustrative experiment design studies are presented.
TL;DR: In this article, a method for constructing linear optimal (in the minimax sense) solution procedures for linear operator equations of the first kind with random errors in the initial data is proposed, and the analytical form of an optimal solution procedure for an arbitrary absolutely convex set of admissible solutions (a well-posed set) is described.
Abstract: A method is proposed for constructing linear optimal (in the minimax sense) solution procedures for linear operator equations of the first kind with random errors in the initial data. The analytical form of an optimal solution procedure for an arbitrary absolutely convex set of admissible solutions (a well-posed set) is described, as well as a method of constructing this set.
01 Sep 1990
TL;DR: In this article, necessary and sufficient conditions for nonlinear programming problem stability (in the nonconvex case), with Pareto set stability, with weak efficiency, efficiency and proper efficiency criteria are considered.
Abstract: Some geometric properties of the solution set for nonlinear and multicriteria programming problems and the related numeric algorithms are considered. The author deals with necessary and sufficient conditions for nonlinear programming problem stability (in the nonconvex case), with Pareto set stability, Pareto set connectedness conditions, with weak efficiency, efficiency and proper efficiency criteria. A study of numerical algorithms based on geometric properties of the so-called convolutions function is also considered. Necessary and sufficient convergence conditions for large classes of algorithms are presented and easy to check sufficient conditions are given. Further results deal with problems of using local unconstrained minimization algorithms to solve quasi-convex problems and the problem of using some convolution functions for constructing decision making procedures. New classes of inverse nonlinear programming problems are discussed and software implementations of DISO/PC-MCNLP are presented.