Showing papers on "Solution set published in 1988"
TL;DR: In this paper, it was shown that the gradient of the objective function of a convex quadratic program is constant on the solution set of the problem, and that the optimal solution set lies in an affine subspace orthogonal to this constant gradient.
146 citations
01 Mar 1988
TL;DR: It is shown that 2 is the largest value ofk such that all valid inequalities for the set covering polytope with integer coefficients no greater thank are contained in the elementary closure of the constraint set.
Abstract: : While the set packing polytope, through its connection with vertex packing, has lent itself to fruitful investigations, little is known about the set covering polytope. We characterize the class of valid inequalities for the set covering polytope with coefficients equal to 0, 1, or 2, and give necessary and sufficient conditions for such an inequality to be minimal and to be facet defining. We show that all inequalities in the above class are contained in the elementary closure of the constraint set, and that 2 is the largest value of k such that all valid inequalities for the set covering polytope with integer coefficients no greater than k are contained in the elementary closure. We point out a connection between minimal inequalities in the class investigated and certain circulant submatrices of the coefficient matrix. Finally, we discuss a procedure for generating all the inequalities in the above class, as well as inequalities that cut off a fractional solution to the linear programming relaxation of the set covering problem, and inequalities whose addition to the constraint set improves the lower bound given by a feasible solution to the dual of the linear programming relaxation. (Author)
92 citations
TL;DR: A new interactive fuzzy decision making method for solving multiobjective linear fractional programming problems by assuming that the decision maker (DM) has fuzzy goals for each of the objective functions.
82 citations
TL;DR: In this article, an existence theorem of Browder-Hartman-Stampacchia variational inequalities is extended to multi-valued monotone operators and properties of solution sets are discussed.
62 citations
TL;DR: In this article, a homotopy method for solving polynomial systems of equations is presented, which is linear with respect to the homogonality of the problem and only one auxiliary parameter is needed to regularize the problem.
Abstract: A new homotopy method for solving systems of polynomial equations is presented. The homotopy equation is extremely simple: It is linear with respect to the homotopy parameter and only one auxiliary parameter is needed to regularize the problem. Within some limits, an arbitrary starting problem can be chosen, as long as its solution set is known. No restrictions on the polynomial systems are made. A few numerical tests are reported which show the influence of the auxiliary parameter, resp. the starting problem, upon the computa- tional cost of the method.
47 citations
01 Jun 1988
TL;DR: This chapter discusses low-dimensional polynomial systems, which are solved easily and reliably by the covering method, and how to save searching for large empty regions simply by choosing the next box near a box containing a solution point.
Abstract: Publisher Summary This chapter discusses covering methods for the enclosure of the solution set of a finite-dimensional system of nonlinear equations. The chapter discusses low-dimensional polynomial systems, which are solved easily and reliably by the covering method. There is no problem in solving polynomial systems piecewise when the slopes are calculated as in Neumaier. This makes the method suitable for problems in a computer-aided geometrical design. The situation may be different for high-dimensional systems, particularly if these involve functions not defined for all values of the variables; because of the overestimation, the method may generate an exponential number of boxes. In this case, a natural approach may be to combine the covering method with continuation techniques to save searching for large empty regions simply by choosing the next box near a box containing a solution point, taking into account the direction in which the curve leaves the current box.
46 citations
TL;DR: In this article, the authors deal with a statistical approach to stability analysis in nonlinear stochastic programming, where the distribution function of the underlying random variable is estimated by the empirical distribution function, and the problem of estimated parameters is considered.
Abstract: The paper deals with a statistical approach to stability analysis in nonlinear stochastic programming. Firstly the distribution function of the underlying random variable is estimated by the empirical distribution function, and secondly the problem of estimated parameters is considered. In both the cases the probability that the solution set of the approximate problem, is not contained in an l-neighbourhood of the solution set to the original problem is estimated, and under differentiability properties an asymptotic expansion for the density of the (unique) solution to the approximate problem is derived.
34 citations
01 Jan 1988
TL;DR: The Fenchel-Moreau theorem for set functions is proved in this paper, and some properties of subdifferential and conjugate functional of set functions are investigated. But none of these properties can be found in this paper.
Abstract: The Fenchel-Moreau theorem for set functions is proved, and some properties of subdifferential and conjugate functional of set functions are investigated.
29 citations
11 Apr 1988
TL;DR: Develops an iterative, space-variant, adaptive restoration algorithm incorporating both regularization and ringing suppression constraints, formulated using the method of projections onto convex sets (POCS).
Abstract: Develops an iterative, space-variant, adaptive restoration algorithm incorporating both regularization and ringing suppression constraints. The algorithm is formulated using the method of projections onto convex sets (POCS). In the proposed method, regularized restoration is implemented by projecting at each iteration onto the extended Wiener solution set. Projection onto this set forces the solution to be equal to the Wiener solution at a set of frequencies over which the magnitude of the degradation transfer function is above a predetermined threshold. Ringing suppression is achieved by adaptively bounding the regional image variance in an attempt to reduce the energy of the ringing artifacts. Three corresponding convex constraint sets are defined to impose tight, moderate, and loose bounds on image variances over the uniform, texture and edge regions, respectively. >
22 citations
TL;DR: In this article, the authors examined the solution set of differential inclusions in separable Banach spaces with a Caratheodory orientor field and showed that the set is an Rδ set in C(T,X).
Abstract: We examine the solution set of differential inclusions in separable Banach spaces with a Caratheodory orientor field i.e. teF(t ,x) is measurable and xeF(t,x) is completely u.s.c. We show that the solution set is an Rδ set in C(T,X), extending this way earlier results by Lasry-Robert, Himmelberg-Van Vleck, De Blasi - Myjak and Papageorgiou
6 citations
TL;DR: In this paper, a survey is presented of some recently obtained results in the problem of identifying linear relations from noisy data, starting from a geometrically inspired definition of noise and linear relations, the mathematical problem is formulated.
Journal Article•
TL;DR: In this paper, Wu et al. proposed the Most Probable Point Locus (MPPL) method for constructing a CDF efficiently by minimizing the number of solution points required.
Abstract: The statistical distribution of cycle life, N, could be easily constructed if N were an explicit function of the design factors. A simple strategy for constructing the distribution of N is to obtain a set of solutions for perturbed values of the design factors, fitting an approximating polynomial for N, thereby obtaining an explicit form on which the reliability methods can be applied. But the method is considered to be inefficient because there is no strong rational basis for selecting the values of the design factors for the solution set. The most probable point locus (MPPL) method has been proposed by Y.T. Wu as a method for constructing a CDF efficiently by minimizing the number of solution points required. The method is summarized.
TL;DR: In this paper, the minimum number of equations needed to define a given set-theoretically defined subset of an algebraically closed field has been studied, and it has been shown that if B is n-dimensional, then n-f 1 equations would suffice for every subset.
Abstract: Let B be a commutative Noetherian ring, X = Spec B the associated affine scheme, I C B an ideal and V = V{I) C X the closed subset defined by I. DEFINITION. Elements / i , . . . , / a € I define V set-theoretically (equivalent^, V is defined set-theoretically by s equations f\\ = 0, f2 = 0 , . . . , fs = 0) (M/i.---./.) = V7. Hubert's Nullstellensatz implies that in the case when B is a finitely generated algebra over an algebraically closed field fc, this definition agrees with the usual one, i.e., all / i , . . . , fs vanish at a fc-rational point if and only if it belongs to V. In the sequel \"defined\" always means \"defined set-theoretically\". The question we are dealing with here concerns the minimum number of equations needed to define a given V C X. A classical result that goes back to L. Kronecker [Kr] says that if B is n-dimensional, then n-f 1 equations would suffice for every V C X. Our first theorem describes those V C X which can be defined by n equations. THEOREM A. Let k be an algebraically closed field, X a smooth affine n-dimensional variety over k with coordinate ring B, and V = V' U P\\ U P2 U • • • U Pr an algebraic subset of X = SpecB, where V' is the union of irreducible components of positive dimensions and Pi)P2^...1Pr some isolated closed points (which do not belong to V). Then V can be defined by n equations if and only if one of the following conditions holds. (i) r = 0, i.e., V consists only of irreducible components of positive dimension. (ii) V' is empty y i.e., V consists only of closed points and there exist positive integers ni, ri2, . . . ,n r such that niP\\ -fri2P2 H H nTPr = 0 in Ao(X). (iii) V' is nonempty, r > 1 and there exist positive integers ni,ri2,.. . ,nr such that rtiPi + n2^2 H h nTPr belongs to the image of the natural map AQ(V') —* Ao(X) induced by the inclusion V' —• X.
TL;DR: A perturbation method for the solution of the Kramers Eq. for a particle moving in a cosine potential where the amplitude of the potential is a cissoidal function of time is described in this article.
Abstract: A perturbation method for the solution of the Kramers Eq. for a particle moving in a cosine potential where the amplitude of the potential is a cissoidal function of time is described. The solution is effected by expanding the distribution function in the usual Fourier-Hermite series yielding a set of ordinary differential-difference Eqs. giving the exact time dependence of the ensemble averages. These Eqs. are a double matrix set inn the order of the Hermite functions andp the order of the circular functions. The perturbation is applied by expanding the solution set in powers of the small parameter, amplitude of the potential/kT. This allows one to systematically uncouple then and thep dependencies in the original set, thus that set may be solved to any order inn by limiting the size of then matrix.