Showing papers on "Solution set published in 2000"
TL;DR: In this article, a proximal-point algorithm for finding zeros of maximal monotone operators in an infinite-dimensional Hilbert space is proposed, which converges strongly to a solution provided the problem has a solution.
Abstract: This paper concerns with convergence properties of the classical proximal point algorithm for finding zeroes of maximal monotone operators in an infinite-dimensional Hilbert space. It is well known that the proximal point algorithm converges weakly to a solution under very mild assumptions. However, it was shown by Guler [11] that the iterates may fail to converge strongly in the infinite-dimensional case. We propose a new proximal-type algorithm which does converge strongly, provided the problem has a solution. Moreover, our algorithm solves proximal point subproblems inexactly, with a constructive stopping criterion introduced in [31]. Strong convergence is forced by combining proximal point iterations with simple projection steps onto intersection of two halfspaces containing the solution set. Additional cost of this extra projection step is essentially negligible since it amounts, at most, to solving a linear system of two equations in two unknowns.
343 citations
10 Sep 2000
TL;DR: Several new set quality metrics are introduced that can be used to assess the ‘‘goodness’’ of an observed Pareto solution set, which should enable a designer to either monitor the quality of an observation P solution set as obtained by a multiobjective optimization method, or compare the sets as reported by different multiObjective optimization methods.
Abstract: In this paper, several new set quality metrics are introduced that can be used to eva the ‘‘goodness’’ of an observed Pareto solution set. These metrics, which are formu in closed-form and geometrically illustrated, include hyperarea difference, Pareto spr accuracy of an observed Pareto frontier, number of distinct choices and cluster. metrics should enable a designer to either monitor the quality of an observed P solution set as obtained by a multiobjective optimization method, or compare the qu of observed Pareto solution sets as reported by different multiobjective optimization m ods. A vibrating platform example is used to demonstrate the calculation of these m for an observed Pareto solution set. @DOI: 10.1115/1.1329875 #
282 citations
01 Jan 2000
TL;DR: In this chapter, fuzzy relational equations are approached from an order-theoretical point of view and it is shown how all inverse problems can be reduced to systems of polynomial lattice equations.
Abstract: Fuzzy relational equations are without doubt the most important inverse problems arising from fuzzy set theory, and in particular from fuzzy relational calculus. Indeed, the calculus of fuzzy relations is a powerful one, with applications in fuzzy control and fuzzy systems modelling in general, approximate reasoning, relational databases, clustering, etc. In this chapter, fuzzy relational equations are approached from an order-theoretical point of view. It is shown how all inverse problems can be reduced to systems of polynomial lattice equations. The exposition is limited to the description of exact solutions, and analytical ways are presented for obtaining the complete solution set when working in a broad and interesting class of distributive lattices. Ample literature pointers to approximate solution methods and application areas are provided.
215 citations
TL;DR: This article presents algorithms, based on homotopy continuation, that compute much of the geometric information contained in the primary decomposition of the solution set of polynomial systems, by finding, at each dimension, generic points on each component.
Abstract: In engineering and applied mathematics, polynomial systems arise whose solution sets contain components of different dimensions and multiplicities In this article we present algorithms, based on homotopy continuation, that compute much of the geometric information contained in the primary decomposition of the solution set In particular, ignoring multiplicities, our algorithms lay out the decomposition of the set of solutions into irreducible components, by finding, at each dimension, generic points on each component As by-products, the computation also determines the degree of each component and an upper bound on its multiplicity The bound is sharp (ie, equal to one) for reduced components The algorithms make essential use of generic projection and interpolation, and can, if desired, describe each irreducible component precisely as the common zeroes of a finite number of polynomials
154 citations
TL;DR: It is proved that the proposed neural network can converge globally to the solution set of the problem when the matrix involved in the problem is positive semidefinite and can converge exponentially to a unique solution when the Matrix is positive definite.
147 citations
TL;DR: This paper studies Lyapunov stability of fuzzy DEs and the periodicity of the fuzzy solution set for both the time-dependent and autonomous cases.
Abstract: Formulations of fuzzy differential equations (DEs) in terms of the Hukuhara derivative do not reproduce the rich and varied behavior of crisp DEs. Another interpretation in terms of differential inclusions, expressed level setwise, overcomes much of this deficiency and opens up for profitable investigation such properties as stability, attraction, periodicity, and the like. This is especially important for investigating continuous systems, which are uncertain or incompletely specified. This paper studies studies Lyapunov stability of fuzzy DEs and the periodicity of the fuzzy solution set for both the time-dependent and autonomous cases.
136 citations
TL;DR: An algorithm for finding an explicit description of solution sets of systems of strict polynomial inequalities, correct up to lower dimensional algebraic sets is presented, based on the cylindrical algebraic decomposition algorithm.
107 citations
TL;DR: In this article, a characterization of the nonemptiness of the solution set for generalized non-coercive equilibrium problems (an extension of generalized quasi-variational inequalities) defined in reflexive Banach spaces in the quasi-convex case is provided.
Abstract: We provide a characterization of the nonemptiness of the solution set for generalized noncoercive equilibrium problems (an extension of generalized quasi-variational inequalities) defined in reflexive Banach spaces in the quasi-convex case. In addition, several necessary and sufficient conditions for the set of solutions to these problems to be nonempty and bounded are also given. Our approach is based on recession notions which proved to be very useful in the study of noncoercive minimization problems. In fact, we find some particular cones as estimates for the recession cone of the solution set. These cones (for the ones containing the latter set) are proved to be sharp enough to encompass several special situations found in the literature.
102 citations
09 Jul 2000
TL;DR: It is shown that the Set Covering problem with intersection 1 cannot be approximated within a o(log n) factor in random polynomial time unless N P ⊆ ZTIME(nO(log log n)), and it is observed that this problem is Max-SNP-Hard.
Abstract: We consider a restricted version of the general Set Covering problem in which each set in the given set system intersects with any other set in at most 1 element. We show that the Set Covering problem with intersection 1 cannot be approximated within a o(log n) factor in random polynomial time unless N P ⊆ ZTIME(nO(log log n)). We also observe that the main challenge in derandomizing this reduction lies in finding a hitting set for large volume combinatorial rectangles satisfying certain intersection properties. These properties are not satisfied by current methods of hitting set construction.
An example of a Set Covering problem with the intersection 1 property is the problem of covering a given set of points in two or higher dimensions using straight lines; any two straight lines intersect in at most one point. The best approximation algorithm currently known for this problem has an approximation factor of θ(log n), and beating this bound seems hard. We observe that this problem is Max-SNP-Hard.
91 citations
TL;DR: This work introduces and studies the Abadie constraint qualification, the weak Pshenichnyi--Levin--Valadier property, and related constraint qualifications for semi-infinite systems of convex inequalities and linear inequalities, and gives some applications of constraint qualifications to linear representations of conveX inequality systems.
Abstract: We introduce and study the Abadie constraint qualification, the weak Pshenichnyi--Levin--Valadier property, and related constraint qualifications for semi-infinite systems of convex inequalities and linear inequalities. Our main results are new characterizations of various constraint qualifications in terms of upper semicontinuity of certain multifunctions. Also, we give some applications of constraint qualifications to linear representations of convex inequality systems, to convex Farkas--Minkowski systems, and to formulas for the distance to the solution set. Some of our concepts and results are new even in the particular case of finite inequality systems.
88 citations
TL;DR: In this article, the Lie point symmetry transformations acting simultaneously on difference equations and lattices, while leaving the solution set of the corresponding difference scheme invariant, are used to obtain particular solutions of differential-difference equations.
Abstract: A method is presented for finding the Lie point symmetry transformations acting simultaneously on difference equations and lattices, while leaving the solution set of the corresponding difference scheme invariant. The method is applied to several examples. The found symmetry groups are used to obtain particular solutions of differential-difference equations.
TL;DR: First, the existence of solutions to a parabolic hemivariational inequality which contains nonlinear evolution operator is established and the upper semicontinuity property of the solution set of the inequality is proved.
Abstract: In this paper we study the optimal control of systems driven by parabolic hemivariational inequalities. First, we establish the existence of solutions to a parabolic hemivariational inequality which contains nonlinear evolution operator. Introducing a control variable in the second member and in the multivalued term, we prove the upper semicontinuity property of the solution set of the inequality. Then we use this result and the direct method of the calculus of variations to show the existence of optimal admissible state–control pairs.
TL;DR: If the velocity of a differential inclusion is regularly quasi-concave in a parameter, the solution set and attainability set are also dependent upon the parameter in like manner and the result is applied to give a vastly improved notion of fuzzy differential equations.
01 Jan 2000
TL;DR: For some Newton-type methods such as primal-dual interior-point following methods and Chen-Mangasarian smoothing methods, local superlinear convergence can be shown without assuming the solutions are isolated.
Abstract: We show that, for some Newton-type methods such as primal-dual interior-point path following methods and Chen-Mangasarian smoothing methods, local superlinear convergence can be shown without assuming the solutions are isolated The analysis is based on local error bounds on the distance from the iterates to the solution set
01 Jan 2000
TL;DR: It is shown, among other things, that a local error bound of order 1/2 holds for a system defined by linear inequalities and a single (nonconvex) quadratic equality, and a sharpening of Lojasiewicz’ error bound for piecewise Quadratic functions is given.
Abstract: In this paper we consider the problem of estimating the distance from a given point to the solution set of a quadratic inequality system. We show, among other things, that a local error bound of order 1/2 holds for a system defined by linear inequalities and a single (nonconvex) quadratic equality. We also give a sharpening of Lojasiewicz’ error bound for piecewise quadratic functions. In contrast, the early results for this problem further require either a convexity or a nonnegativity assumption.
TL;DR: A new algo-rithm is proposed and it is proved that it generates a sequence convergent to a solution point of the problem also when the operator is point-to-set.
Abstract: In this paper we study extragradient-type methods to solve variational inequality problems involving maximal monotone point-to-set operators. We will show, using an example, that, in this case, to achieve convergence to the solution set of sequences generated by extragradient methods it is necessary, in the search of the auxiliary point and its suitable image by the involved operator, to enlarge the set of possible search directions. Such an enlargement, a generalization of the e-subgradient, is known and we use it to overcome this failure of the extragradient-methods. We propose a new algo-rithm and prove that it generates a sequence convergent to a solution point of the problem also when the operator is point-to-set
TL;DR: This paper adopts the generalized Newton method proposed by De Luca, Facchinei, and Kanzow to solve subproblems and adopt some NCP functions to check the criteria, and shows that the PPA converges globally provided that the solution set of the problem is nonempty.
Abstract: In this paper, we consider a proximal point algorithm (PPA) for solving monotone nonlinear complementarity problems (NCP). PPA generates a sequence by solving subproblems that are regularizations of the original problem. It is known that PPA has global and superlinear convergence properties under appropriate criteria for approximate solutions of subproblems. However, it is not always easy to solve subproblems or to check those criteria. In this paper, we adopt the generalized Newton method proposed by De Luca, Facchinei, and Kanzow to solve subproblems and adopt some NCP functions to check the criteria. Then we show that the PPA converges globally provided that the solution set of the problem is nonempty. Moreover, without assuming the local uniqueness of the solution, we show that the rate of convergence is superlinear in a genuine sense, provided that the limit point satisfies the strict complementarity condition.
TL;DR: The trajectory defined by the augmented smoothing equations and global convergence of the method under an assumption that the original P0 matrix linear complementarity problem has a solution are studied.
Abstract: In this paper, we propose a Big-$\Gamma$ smoothing method for solving the P0 matrix linear complementarity problem We study the trajectory defined by the augmented smoothing equations and global convergence of the method under an assumption that the original P0 matrix linear complementarity problem has a solution The method has been tested on the P0 matrix linear complementarity problem with unbounded solution set Preliminary numerical results indicate the robustness of the method
TL;DR: The authors developed a two-stage algorithm to generate and evaluate DMUs, which derives a solution set instead of an optimal solution in single objective ones due to the multiple objectives of problems.
TL;DR: Acydicity of solution sets to asymptotic problems, when the value is prescribed either at the origin or at infinity, is proved for differential inclusions and discontinuous autonomous differentialinclusions as mentioned in this paper.
Abstract: Acydicity of solution sets to asymptotic problems, when the value is prescribed either at the origin or at infinity, is proved for differential inclusions and discontinuous autonomous differential inclusions. Existence criteria showing that such sets are non-empty are obtained as well.
TL;DR: In this paper, the Riccati transformation is used in the reduction of order of second and third order ordinary differential equations of maximal symmetry, and the sl(2, R) subalgebra is preserved under this transformation.
TL;DR: Optimal values and solution sets are regarded as mappings of the expected recourse functions and their perturbations, respectively and conditions are identified implying that these mappings are directionally differentiable and semidifferentiable on appropriate functional spaces.
Abstract: Two-stage stochastic programs with random right-hand side are considered. Optimal values and solution sets are regarded as mappings of the expected recourse functions and their perturbations, respectively. Conditions are identified implying that these mappings are directionally differentiable and semidifferentiable on appropriate functional spaces. Explicit formulas for the derivatives are derived. Special attention is paid to the role of a Lipschitz condition for solution sets as well as of a quadratic growth condition of the objective function.
01 Jan 2000
TL;DR: In this paper, it was shown that the set of all the efficient points (Pareto solutions) of a linear fractional vector optimization problem with a bounded constraint set is connected.
Abstract: As it was shown by C. Malivert (1996) and other Authors, in a Linear Fractional Vector Optimization Problem (for short, LFVOP) any point satisfying the first-order necessary optimality condition (a stationary point) is a solution. Therefore, solving such a problem is equivalent to solve a monotone affine vector variational inequality of a special type. This observation allows us to apply the existing results on monotone affine variational inequality to establish some facts about connectedness and stability of the solution sets in LFVOP. In particular, we are able to solve a question raised by E. U. Choo and D. R. Atkins (1983) by proving that the set of all the efficient points (Pareto solutions) of a LFVOP with a bounded constraint set is connected.
01 Jan 2000
TL;DR: In this article, it was shown that the solution sets of a strongly monotone vector variational inequality and of its relaxed inequality can be different from each other, by constructing an example.
Abstract: By constructing an example we show that the solution sets of a strongly monotone vector variational inequality and of its relaxed inequality can be different from each other. A sufficient condition for the coincidence of these solution sets is given for general vector variational inequalities; connectedness and path-connectedness of the solution sets for some kinds of monotone problems in Hilbert spaces are studied in detail.
TL;DR: In this article, the authors examined nonlinear evolution hemivariational inequality defined on a Gelf and fivefold of spaces and showed that the problem with multivalued and $L$-pseudomonotone operator and zero initial data has a solution and established the existence result when the operator is single valued of Leray-Lions type and the initial condition is nonzero.
Abstract: In the paper we examine nonlinear evolution hemivariational
inequality defined on a Gelfand fivefold of spaces. First
we show that the problem with multivalued and $L$-pseudomonotone
operator and zero initial data has a solution. Then
the existence result is established in the case when the operator is
single valued of Leray-Lions type and the initial condition is nonzero.
Finally, the asymptotic behavior of solutions of hemivariational
inequality with operators of divergence form is considered and
the result on upper semicontinuity of the solution set is given.
TL;DR: In this paper, the authors analyzed the set of scalar algebraic Riccati equations that play an important role in finding feedback Nash equilibria of the scalar N-player linear-quadratic differential game and showed that in general there exist at most 2/sup N/-1 solutions of the ARE that give rise to a Nash equilibrium.
Abstract: We analyze the set of scalar algebraic Riccati equations (ARE) that play an important role in finding feedback Nash equilibria of the scalar N-player linear-quadratic differential game. We show that in general there exist at most 2/sup N/-1 solutions of the ARE that give rise to a Nash equilibrium. In particular we analyze the number of equilibria as a function of the autonomous growth parameter and present both necessary and sufficient conditions for the existence of a unique solution of the ARE.
01 Jan 2000
TL;DR: In this article, a generalization of Adamjan-Arov-Krein theorem is formulated for general case (the classical theorem concerns the completely indeterminate matrix case only).
Abstract: This paper deals with operator Nehari problem. The main objective is to adopt the feedback coupling method developed earlier in analytic (orthogonal scattering) case to nonanalytic (non-orthogonal scattering) case. The exposition goes in terms slightly different from the ones of Adamjan and Arov and leads to a parametrization of the solution set in general case. The parametrization formula is analogue of Arov—Grossman formula for nonanalytic case. A generalization of Adamjan—Arov—Krein theorem is formulated in the last section for general case (the classical theorem concerns the completely indeterminate matrix case only). The proof will be published elsewhere. It depends upon studying the residual part of the feedback coupling of scattering systems.
TL;DR: A new fusion algorithm is derived for estimating the state of a linear dynamic system by fusing uncertain observations, which suffer from two types of uncertainties simultaneously, and combines set theoretic and stochastic estimation in a rigorous manner.
TL;DR: The suggested bounds on weight parameters guarantee that the discrete Hopfield network will locate a solution after each relaxation for a class of optimization problems of any size, although the solutions will be average quality rather than optimum.
28 Jun 2000
TL;DR: In this article, an analytical version of this criterion is extended to the case of not necessarily Hurwitz polynomials, and a generalization is shown to linearize some control synthesis problems in the sense that the set of stabilizing controllers is obtained as the solution set of a number of linear equations.
Abstract: Mikhailov's criterion states that the hodograph /spl delta/(j/spl omega/) of a real, n/sup th/ degree, Hurwitz stable polynomial /spl delta/(s) turns strictly counterclockwise and goes through n quadrants as /spl omega/ runs from 0 to +/spl infin/. In this paper we first give an analytical version of this criterion and then extend this analytical condition to the case of not necessarily Hurwitz polynomials. This generalization is shown to "linearize" some control synthesis problems in the sense that the set of stabilizing controllers is obtained as the solution set of a number of linear equations.