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1.
This paper presents some new results in the theory of Newton-type methods for variational inequalities, and their application to nonlinear programming. A condition of semistability is shown to ensure the quadratic convergence of Newton's method and the superlinear convergence of some quasi-Newton algorithms, provided the sequence defined by the algorithm exists and converges. A partial extension of these results to nonsmooth functions is given. The second part of the paper considers some particular variational inequalities with unknowns (x, ), generalizing optimality systems. Here only the question of superlinear convergence of {x k } is considered. Some necessary or sufficient conditions are given. Applied to some quasi-Newton algorithms they allow us to obtain the superlinear convergence of {x k }. Application of the previous results to nonlinear programming allows us to strengthen the known results, the main point being a characterization of the superlinear convergence of {x k } assuming a weak second-order condition without strict complementarity.  相似文献   

2.
Summary In this paper, we study the convergence of formal power series solutions of functional equations of the formP k(x)([k](x))=(x), where [k] (x) denotes thek-th iterate of the function.We obtain results similar to the results of Malgrange and Ramis for formal solutions of differential equations: if(0) = 0, and(0) =q is a nonzero complex number with absolute value less than one then, if(x)=a(n)x n is a divergent solution, there exists a positive real numbers such that the power seriesa(n)q sn(n+1)2 x n has a finite and nonzero radius of convergence.
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3.
If (O) is a quadratic cone in PG(3,q), with vertex x, then a flock of (O) is a partition of (O)-{x} into q disjoint conics. With such a flock there correspond a translation plane of order q 2 and a generalized quadrangle of order (q 2, q). Here we determine all flocks of (O) for q 8.  相似文献   

4.
This paper studies the convergence properties of algorithms belonging to the class of self-scaling (SS) quasi-Newton methods for unconstrained optimization. This class depends on two parameters, say k and k , for which the choice k =1 gives the Broyden family of unscaled methods, where k =1 corresponds to the well known DFP method. We propose simple conditions on these parameters that give rise to global convergence with inexact line searches, for convex objective functions. The q-superlinear convergence is achieved if further restrictions on the scaling parameter are introduced. These convergence results are an extension of the known results for the unscaled methods. Because the scaling parameter is heavily restricted, we consider a subclass of SS methods which satisfies the required conditions. Although convergence for the unscaled methods with k 1 is still an open question, we show that the global and superlinear convergence for SS methods is possible and present, in particular, a new SS-DFP method.  相似文献   

5.
For an odd prime powerq the infinite field GF(q 2 )= n0 GF (q 2n ) is explicitly presented by a sequence (f n)1 ofN-polynomials. This means that, for a suitably chosen initial polynomialf 1, the defining polynomialsf nGF(q)[x] of degrees2 n are constructed by iteration of the transformation of variablexx+1/x and have linearly independent roots over GF(q). In addition, the sequences are trace-compatible in the sense that the relative traces map the corresponding roots onto each other. In this first paper the caseq1 (mod 4) is considered and the caseq3 (mod 4) will be dealt with in a second paper. This specific construction solves a problem raised by A. Scheerhorn in [11].  相似文献   

6.
This paper concerns solving an overdetermined linear systemA T x=b in the leastl 1-norm orl -norm sense, whereA m×n ,m<n. We show that the primal-dual interior point approach for linear programming can be applied, in an effective manner, to linear programming versions of thel 1 andl -problems. The resulting algorithms are simple to implement and can attain quadratic or superlinear convergence rate. At each iteration, the algorithms must solve a linear system with anm×m positive-definite coefficient matrix of the formADA T , whereD is a positive diagonal matrix. The preliminary numerical results indicate that the proposed algorithms offer considerable promise.This research was supported in part by Grants NSF DMS-91-02761 and DOE DE-FG05-91-ER25100.  相似文献   

7.
A family of conics in PG(2,q) is called saturated if any line LPG(2,q) is incident with at least one conic of the family. Then, if <(q+1)/2, the support of is a (k,n)-blocking set. It is shown that in this way one can get blocking sets whose character n is small compared to q; it is also shown that cannot be taken independent of q, but must necessarily increase as q does.  相似文献   

8.
Summary A one-dimensional chain of nearest neighbor linearly interacting oscillators {q x } x is studied. The set of all its extremal DLR measures is characterized in terms of a parameter 2. For each there is a Gaussian DLR measure with support on the set of configurations determined by the rate of growth of¦q x¦. It is then finally proved that there is only one translationally invariant DLR measure. This proves the following conjecture: invariant DLR measures give uniformly finite first moment to ¦q x¦.  相似文献   

9.
The Newton's method for finding the root of the equation (t)=0 can be easily generalized to the case where is monotone, convex, but not differentiable. Then, the convergence is superlinear. The purpose of this note is to show that the convergence is only superlinear. Indeed, for all (1, 2), we exhibit an example where the convergence of the iterates is exactly .  相似文献   

10.
Moser-type estimates for functions whose gradient is in the Lorentz space L(n, q), 1q, are given. Similar results are obtained for solutions uH inf0 sup1 of Au=(f i ) x i , where A is a linear elliptic second order differential operator and |f|L(n, q), 2q.Work partially supported by MURST (40%).  相似文献   

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