Showing papers on "Strongly monotone published in 1996"
TL;DR: There is a correspondence between monotone metrics and operator means in the sense of Kubo and Ando, and it follows that all proposals of Morozova and Chentsov are indeed monot one metrics.
540 citations
TL;DR: A class of parametric smooth functions that approximate the fundamental plus function, (x)+=max{0, x}, by twice integrating a probability density function leads to classes of smooth parametric nonlinear equation approximations of nonlinear and mixed complementarity problems (NCPs and MCPs).
Abstract: We propose a class of parametric smooth functions that approximate the fundamental plus function, (x)+=max{0, x}, by twice integrating a probability density function. This leads to classes of smooth parametric nonlinear equation approximations of nonlinear and mixed complementarity problems (NCPs and MCPs). For any solvable NCP or MCP, existence of an arbitrarily accurate solution to the smooth nonlinear equations as well as the NCP or MCP, is established for sufficiently large value of a smoothing parameter α. Newton-based algorithms are proposed for the smooth problem. For strongly monotone NCPs, global convergence and local quadratic convergence are established. For solvable monotone NCPs, each accumulation point of the proposed algorithms solves the smooth problem. Exact solutions of our smooth nonlinear equation for various values of the parameter α, generate an interior path, which is different from the central path for interior point method. Computational results for 52 test problems compare favorably with these for another Newton-based method. The smooth technique is capable of solving efficiently the test problems solved by Dirkse and Ferris [6], Harker and Xiao [11] and Pang & Gabriel [28].
465 citations
TL;DR: In this article, the authors proposed projection-type methods for variational inequality with monotone underlying functions, in which the projection direction is modified by a strongly-monotone mapping of the form $I - \alpha F or, if the underlying function F is affine with underlying matrix M, the mapping can be computed in the form of $I+ \alpha M^T, with $\alpha \in (0,\infty)
Abstract: We propose new methods for solving the variational inequality problem where the underlying function $F$ is monotone. These methods may be viewed as projection-type methods in which the projection direction is modified by a strongly monotone mapping of the form $I - \alpha F$ or, if $F$ is affine with underlying matrix $M$, of the form $I+ \alpha M^T$, with $\alpha \in (0,\infty)$. We show that these methods are globally convergent, and if in addition a certain error bound based on the natural residual holds locally, the convergence is linear. Computational experience with the new methods is also reported.
293 citations
TL;DR: It is shown that any stationary point of the unconstrained objective function is a solution of NCP if the mapping F involved in NCP is continuously differentiable and monotone, and that the level sets are bounded if F is continuous and strongly monotones.
Abstract: A reformulation of the nonlinear complementarity problem (NCP) as an unconstrained minimization problem is considered. It is shown that any stationary point of the unconstrained objective function is a solution of NCP if the mapping F involved in NCP is continuously differentiable and monotone, and that the level sets are bounded if F is continuous and strongly monotone. A descent algorithm is described which uses only function values of F. Some numerical results are given.
185 citations
TL;DR: In this paper, the global discretisation error for strong time discretisations of finite dimensional Ito stochastic differential equations (SDEs) with a strongly monotone linear operator with eigenvalues λ 1 ≤ λ 2 ≤ … in its drift term is estimated.
Abstract: The global discretisation error is estimated for strong time discretisations of finite dimensional Ito stochastic differential equations (SDEs) which are Galerkin approximations of a class of parabolic stochastic partial differential equation (SPDE) with a strongly monotone linear operator with eigenvalues λ1 ≤ λ2 ≤ … in its drift term. If an order γ strong Taylor scheme with time-step δ is applied to the N dimensional Ito-Galerkin SDE, the discretisation error is bounded above bywhere [x] is the integer part of the real number x and the constant K depends on the initial value, bounds on the other coefficients in the SPDE and the length of the time interval under consideration.
128 citations
TL;DR: The accuracy of this two-level method for the discretization and solution of nonlinear boundary value problems and the scaling between the fine and coarse mesh widths required to ensure optimal accuracy of the fine mesh solution is derived as a byproduct of the error estimates herein.
Abstract: We consider a two-level method for the discretization and solution of nonlinear boundary value problems. The method basically involves (i) solving the nonlinear problem on a {\it very} coarse mesh, (ii) linearizing about the coarse mesh solution, and solving the linearized problem on the fine mesh, one time! We analyze the accuracy of this procedure for strongly monotone nonlinear operators (\S2) and general semilinear elliptic boundary value problems (without monotonicity assumptions). In particular, the scaling between the fine and coarse mesh widths required to ensure optimal accuracy of the fine mesh solution is derived as a byproduct of the error estimates herein.
83 citations
TL;DR: The Choquet integral determines a new nonnegative monotone set function that is absolutely continuous with respect to the original one and is a useful method to define sound fuzzy measures in various applications.
71 citations
TL;DR: The existence of globally attractive order intervals for strongly monotone discrete dynamical systems in ordered Banach spaces is first proved under some appropriate conditions in this article, and threshold results on global asymptotic stability are then obtained.
Abstract: The existence of globally attractive order intervals for some strongly monotone discrete dynamical systems in ordered Banach spaces is first proved under some appropriate conditions. With the strict sublinearity assumption, threshold results on global asymptotic stability are then obtained. As applications, the global asymptotic behaviours of nonnegative solutions for time-periodic parabolic equations and cooperative systems of ordinary differential equations are discussed and some biological interpretations and concrete application examples are also given.
51 citations
30 citations
30 citations
TL;DR: In this article, a subtler weakening of one of the characterizations of monotone normality leads to a new class of spaces called monotonically T2, which are called mi-spaces.
01 Jan 1996
TL;DR: In this article, the authors generalize the Brezis-Haraux theorem on the range of the sum of monotone operators from a Hilbert space to general Banach spaces and show that the range is topologically almost equal to the sum R(A) + R(B) where r is a compatible topology in X** x X* as proposed by Gossez.
Abstract: The purpose of this paper is to generalize the Brezis-Haraux theorem on the range of the sum of monotone operators from a Hilbert space to general Banach spaces. The result obtained provides that the range R(A + BT) is topologically almost equal to the sum R(A) + R(B) where r is a compatible topology in X** x X* as proposed by Gossez. To illustrate the main result we consider some basic properties of densely maximal monotone operators.
TL;DR: In this paper, the existence and uniqueness of the solution for the corresponding initial value problem were proved. And the authors provided an approximation via finite differences in time and convergence results along with error estimates.
Abstract: In this paper we deal with the equation L (d2u/dt2) +B (du/dt)+Au f, where L and A are linear positive selfadjoint operators in a Hilbert space H and from a Hilbert space V C H to its dual space V, respectively, and B is a maximal monotone operator from V to V. By assuming some coerciveness on L + B and A, we state the existence and uniqueness of the solution for the corresponding initial value problem. An approximation via finite differences in time is provided and convergence results along with error estimates are presented.
TL;DR: In this paper, the continuity of the parallel sum of two monotone operators with respect to set convergence was studied from a quantitative point of view, and its stability was examined.
TL;DR: In this paper, the authors show a strong relation between Φ-approximation by a class of real A-measurable functions and best-step monotone approximations by constants on subsets of a measure space.
TL;DR: In this article, it was shown that every monotone function can be interpolated by a polynomial on any set of cardinal vectors of size κ on any latticeL 0 and for every cardinal κ there is a lattice\(L \supseteq L_0 \) on which every monotonous function on any cardinal vector can be computed.
Abstract: We show that for every latticeL0 and for every cardinal κ there is a lattice\(L \supseteq L_0 \) on which every monotone function can be interpolated by a polynomial on any set of size κ.
TL;DR: In this article, the authors give conditions under which a monotone operator T + ∆ ∆ f is maximal monotonous in the Banach space, where ∆ is the subdifferential of a proper lower semicontinuous convex function f from Xinto R ∪{+∞}.
01 Jan 1996
TL;DR: In this paper, it was shown that every monotone operator on a subspace of a Banach space containing densely a continuous image of an Asplund space is singlevalued on the whole space except a σ-cone supported set.
Abstract: We extend Zaj́ıček’s theorem from [Za] about points of singlevaluedness of monotone operators on Asplund spaces. Namely we prove that every monotone operator on a subspace of a Banach space containing densely a continuous image of an Asplund space (these spaces are called GSG spaces) is singlevalued on the whole space except a σ-cone supported set.
TL;DR: In this article, it was shown that the results for reflexive spaces can be recovered for general Banach spaces by using monotone operator of type "(NI)", a class of multifunctions from E into E* which includes the subdifferentials of all proper, convex, lower semicontinuous functions on E, all surjective operators and, if E is reflexive, all maximal monotones operators.
Abstract: Let E be a real Banach space with dual E*. We associate with any nonempty subset H of E×E* a certain compact convex subset of the first quadrant in ℝ2, which we call the picture of H, Π(H). In general, Π(H) may be empty, but Π(M) is nonempty if M is a nonempty monotone subset of E×E*. If E is reflexive and M is maximal monotone then Π(M) is a single point on the diagonal of the first quadrant of ℝ2. On the other hand, we give an example (for E the nonreflexive space L1[0,1]) of a maximal monotone subset M of E×E* such that (0,1)∈Π(M) and (1,1)∈Π(M) but (1,0)∉Π(M). We show that the results for reflexive spaces can be recovered for general Banach spaces by using monotone operator of type ‘(NI)’ — a class of multifunctions from E into E* which includes the subdifferentials of all proper, convex, lower semicontinuous functions on E, all surjective operators and, if E is reflexive, all maximal monotone operators. Our results lead to a simple proof of Rockafellar's result that if E is reflexive and S is maximal monotone on E then S+J is surjective. Our main tool is a classical minimax theorem.
Journal Article•
TL;DR: This paper characterise the class of pairs of functions satisfying the relationship and shows that it extends and encapsulates previous results concerning translations from combi- national to monotone networks.
Abstract: This paper considers a particular relationship defined over pairs of n-argument monotone Boolean functions. The relationship is of interest since we can show that if ( g, h ) satisfy it then for any n-argument monotone Boolean function f there is a close rela- tionship between the combinational and monotone network complexities of the function ( f/ \ g) \/ h. We characterise the class of pairs of functions satisfying the relationship and show that it extends and encapsulates previous results concerning translations from combi- national to monotone networks.
TL;DR: In this article, the existence theorems of variational inequalities associated with perturbed maximal monotone mappings are discussed. But they do not discuss the surjectivity results for perturbed MMMM mappings.
01 Jan 1996
TL;DR: In this article, the authors considered the asymptotic behavior of solutions of a class of ordinary dmerentialequations x = F(x) in the nonnegative orthant R7 and showed that every bounded solution to such a system converges to a single equilibrium.
Abstract: The author considers the asymptotic behavior of solutions of a class of ordinary dmerentialequations x = F(x) in the nonnegative orthant R7. Suppose that F(o) = o, Fi(x1,..., xn) isnondecreasing in xk for all k≠ i and that F possesses an order-increasing invarian function.Then it is shown that every bounded solution to such a system converges to a single equilibrium.
Journal Article•
Journal Article•
TL;DR: In this paper, it was shown that there are many spaces of arbitrarily large cardinality satisfying all above listed axioms except linear ordering, and that the latter condition is not redundant.
Abstract: The real line R may be characterized as the unique nonatomic directed partially ordered abelian group which is monotone sigma-complete (countable increasing bounded sequences have suprema), satisfies the countable refinement property (countable sums $\sum_ma_m=\sum_nb_n$ of positive elements $a_m$, $b_n$ have common refinements) and that is linearly ordered. We prove here that the latter condition is not redundant, thus solving an old problem by A. Tarski, by proving that there are many spaces (in particular, of arbitrarily large cardinality) satisfying all above listed axioms except linear ordering.
TL;DR: In this article, it was shown that the picture of a maximal monotone subset can be quite substantial in the case of non-reflexive S+J, where S: E→2E* is a singleton.
Abstract: Let E be a nontrivial Banach space. The concept of ‘picture’ has been used to provide a new proof of the surjectivity of S+J, for E reflexive and S: E→2E* maximal monotone. It is known that if E is reflexive, then the picture of a maximal monotone subset of E×E* is a singleton. We calculate an example showing that in the nonreflexive case, the picture of a maximal monotone subset can be quite substantial.