Showing papers on "Strongly monotone published in 1999"

A smoothing method for mathematical programs with equilibrium constraints

Abstract: The mathematical program with equilibrium constraints (MPEC) is an optimization problem with variational inequality constraints. MPEC problems include bilevel programming problems as a particular case and have a wide range of applications. MPEC problems with strongly monotone variational inequalities are considered in this paper. They are transformed into an equivalent one-level nonsmooth optimization problem. Then, a sequence of smooth, regular problems that progressively approximate the nonsmooth problem and that can be solved by standard available software for constrained optimization is introduced. It is shown that the solutions (stationary points) of the approximate problems converge to a solution (stationary point) of the original MPEC problem. Numerical results showing viability of the approach are reported.

ε-Enlargements of Maximal Monotone Operators in Banach Spaces

Abstract: Given a maximal monotone operator T in a Banach space, we consider an enlargement Te, in which monotonicity is lost up to e, in a very similar way to the e-subdifferential of a convex function. We establish in this general framework some theoretical properties of Te, like a transportation formula, local Lipschitz continuity, local boundedness, and a Brondsted–Rockafellar property.

A hybrid Newton method for solving the variational inequality problem via the D-gap function

Abstract: The variational inequality problem (VIP) can be reformulated as an unconstrained minimization problem through the D-gap function It is proved that the D-gap function has bounded level sets for the strongly monotone VIP A hybrid Newton-type method is proposed for minimizing the D-gap function Under some conditions, it is shown that the algorithm is globally convergent and locally quadratically convergent

Maximal Monotone Multifunctions of Brøndsted–Rockafellar Type

Abstract: We consider whether the “inequality-splitting” property established in the Brondsted–Rockafellar theorem for the subdifferential of a proper convex lower semicontinuous function on a Banach space has an analog for arbitrary maximal monotone multifunctions. We introduce the maximal monotone multifunctions of type (ED), for which an “inequality-splitting” property does hold. These multifunctions form a subclass of Gossez"s maximal monotone multifunctions of type (D); however, in every case where it has been proved that a multifunction is maximal monotone of type (D) then it is also of type (ED). Specifically, the following maximal monotone multifunctions are of type (ED): • ultramaximal monotone multifunctions, which occur in the study of certain nonlinear elliptic functional equations; • single-valued linear operators that are maximal monotone of type (D); • subdifferentials of proper convex lower semicontinuous functions; • “subdifferentials” of certain saddle-functions. We discuss the negative alignment set of a maximal monotone multifunction of type (ED) with respect to a point not in its graph – a mysterious continuous curve without end-points lying in the interior of the first quadrant of the plane. We deduce new inequality-splitting properties of subdifferentials, almost giving a substantial generalization of the original Brondsted–Rockafellar theorem. We develop some mathematical infrastructure, some specific to multifunctions, some with possible applications to other areas of nonlinear analysis: • the formula for the biconjugate of the pointwise maximum of a finite set of convex functions – in a situation where the “obvious” formula for the conjugate fails; • a new topology on the bidual of a Banach space – in some respects, quite well behaved, but in other respects, quite pathological; • an existence theorem for bounded linear functionals – unusual in that it does not assume the existence of any a priori bound; • the 'big convexification" of a multifunction.

Degenerate Convection-Diffusion Equations and Implicit Monotone Difference Schemes

Abstract: We analyse implicit monotone finite difference schemes for nonlinear, possibly strongly degenerate, convection-diffusion equations in one spatial dimension. Since we allow strong degeneracy, solutions can be discontinuous and are in general not uniquely determined by their data. We thus choose to work with weak solutions that belong to the BV (in space and time) class and, in addition, satisfy an entropy condition. The difference schemes are shown to converge to the unique BV entropy weak solution of the problem. This paper complements our previous work [8] on explicit monotone schemes.

On smoothness of carrying simplices

Abstract: We consider dissipative strongly competitive systems &i = Xi fi (x) of ordinary differential equations. It is known that for a wide class of such systems there exists an invariant attracting hypersurface Z, called the carrying simplex. In this note we give an amenable condition for Z to be a Cl submanifold-with-corners. We also provide conditions, based on a recent work of M. Benaim (On invariant hypersurfaces of strongly monotone maps, J. Differential Equations 136 (1997), 302-319), guaranteeing that Z is of class 0k+1

A Linearly Convergent Derivative-Free Descent Method for Strongly Monotone Complementarity Problems

Abstract: We establish the first rate of convergence result for the class of derivative-free descent methods for solving complementarity problems. The algorithm considered here is based on the implicit Lagrangian reformulation [26, 35] of the nonlinear complementarity problem, and makes use of the descent direction proposed in [42], but employs a different Armijo-type linesearch rule. We show that in the strongly monotone case, the iterates generated by the method converge globally at a linear rate to the solution of the problem.

The Analytic Center Quadratic Cut Method for Strongly Monotone Variational Inequality Problems

Abstract: Convergence of an algorithm for strongly monotone variational inequality problems (VIPs) is investigated At each iteration, the algorithm adds a quadratic cut through the analytic center of the consequently shrinking convex set It is shown that the sequence of analytic centers converges to the unique solution in ${\cal O}(1/\sqrt{k})$, where k is the number of iterations

Totally Monotone Core and Products of Monotone Measures.

Abstract: Several approaches to the product of non-additive monotone measures (or capacities) are discussed and a new approach is proposed. It starts with the Mobius product [E. Hendon, H.J. Jacobsen, B. Sloth, T. Tranaes, The product of capacities and belief functions, Mathematical Social Sciences 32 (1996) 95–108] of totally monotone measures and extends it by means of a supremum to general monotone measures. The sup runs over sets of totally monotone measures. These sets are defined like the core of monotone measures (or cooperative games). The new product is compatible with the partial order ⩽ for arbitrary monotone measures.

Explicit Mathematics with the Monotone Fixed Point Principle. II: Models

Abstract: This paper continues investigations of the monotone fixed point principle in the context of Feferman's explicit mathematics begun in [14]. Explicit mathematics is a versatile formal framework for representing Bishop-style constructive mathematics and generalized recursion theory. The object of investigation here is the theory of explicit mathematics augmented by the monotone fixed point principle, which asserts that any monotone operation on classifications (Feferman's notion of set) possesses a least fixed point. To be more precise, the new axiom not merely postulates the existence of a least solution, but, by adjoining a new constant to the language, it is ensured that a fixed point is uniformly presentable as a function of the monotone operation. Let T0 + UMID denote this extension of explicit mathematics. [14] gave lower bounds for the strength of two subtheories of To + UMID in relating them to fragments of second order arithmetic based on comprehension. [14] showed that To ↾ + UMID and To ↾ + INDk + UMID have at least the strength of ( – CA), and ↾ and – CA), respectively. Here we are concerned with the exact reversals. Let UMIDN be the monotone fixed-point principle for subclassifications of the natural numbers. Among other results, it is shown that To ↾ + UMIDN and To ↾ + INDN + UMIDN have the same strength as ( – CA) ↾ and ( – CA), respectively. The results are achieved by constructing set-theoretic models for the aforementioned systems of explicit mathematics in certain extensions of Kripke-Platek set theory and subsequently relating these set theories to subsystems of second arithmetic.

Global stability and permanence for a class of type K monotone systems

Abstract: In this paper, the convergence of solutions of type K monotone systems is studied. The basic assumption is that the Jacobian matrix is stable for every point in $R_+^n$. The main results are the following. If the system has a positive steady state, then it is globally asymptotically stable in $\mbox{Int} R_+^n$. A sufficient and necessary condition for a nonnegative steady state of the system to be globally asymptotically stable is presented. Moreover, we provide sufficient conditions for type K monotone systems to be permanent and for existence and uniqueness of a positive steady state.

Nowhere monotone functions and functions of nonmonotonic type

Abstract: We investigate the relationships between the notions of a continuous function being monotone on no interval, monotone at no point, of monotonic type on no interval, and of monotonic type at no point. In particular, we characterize the set of all points at which a function that has one of the weaker properties fails to have one of the stronger properties. A theorem of Garg about level sets of continuous, nowhere monotone functions is strengthened by placing control on the location in the domain where the level sets are large. It is shown that every continuous function that is of monotonic type on no interval has large intersection with every function in some second category set in each of the spaces pn, Cn, and Lip1. 1. NONMONOTONICITY PROPERTIES In a series of interesting papers [4], [5], [6], [7], [8], Garg investigated level set structures and derivate structures of continuous functions. This investigation was continued in a paper by Bruckner and Garg [2]. These articles considered several notions that measure degrees of pathology in the class of continuous, nowhere monotone functions. In this section we further study the relationships among these properties. We use C and BV to denote the collections of functions from [0, 1] into R, the reals, that are continuous and of bounded variation, respectively. Df (x) and Df (x) denote the lower and upper (two-sided) Dini derivates, respectively, of a function f at a number x (see [1]). We use standard terms such as perfect sets, first category sets, sets with the Baire property, etc., whose definitions may be found in [9]. Following Bruckner and Garg [2], we say that a function f is nondecreasing at x if f(t)-f(x) > 0 for all t $& x in some neighborhood of x. That f is nonincreasing t-xat x is defined with the obvious modification. If f is either nonincreasing at x or nondecreasing at x, then we say that f is monotone at x. That f is nonmonotone at x means that f is not monotone at x. If f is a function and m E R then, following Garg [8], denote by f+m and f-r the functions defined by f+r(x) = f (x) + mx and f-m(x) = f(x) mx. Inclusion of the "+" and "-" avoids confusion with Received by the editors August 20, 1996 and, in revised form, May 7, 1997. 1991 Mathematics Subject Classification. Primary 26A48; Secondary 26A24.

Regularization of Fixed-Point Problems and Staircase Iteration

Abstract: Let C be a nonempty closed convex subset of the Hilbert space H and P be a nonexpansive mapping from C into C. The Tikhonov regularization method is extended to the fixed point problem for P. This method generates a family of strongly contractive mappings P r from H into H by composition of P with the projector onto C and with the resolvent of a given maximal and strongly monotone operator R on H with positive parameter r. If the fixed point set 5 of P is nonempty (for instance C bounded), then, as r tends to zero, u r converges to u ⋆ in S the unique solution to the variational inequality defined by R and the closed convex subset S. Moreover, the iteration method suitably combined, by a staircase technique, with approximation of P by a sequence of nonexpansive mappings P n and with regularization generates a sequence that converges strongly to u ⋆. Applications to some variational problems are considered

On a central limit theorem for monotone noise

Abstract: We study a special central limit theorem for monotone noise. If we represent all operators on suitable symmetric Fock spaces we obtain a statement of strong convergence.

Existence and proximal point algorithms for nonlinear monotone complementarity problems

Abstract: The purpose of the present paper is a statement of the nonlinear complementarity problem associated to monotone operators in Hilbert spaces. Existence results are proved, and proximal point algorithms are given

Degenerate Sweeping Processes

Abstract: The aim of this paper is to summarize some recent results concerning the evolutionary differential inclusions {fy(1)|31-1} where A is a maximal monotone and strongly monotone operator in a real Hilbert space H, and t →C(t) is a set-valued mapping, cf. the precise assumptions below. Moreover, as usually, denotes the cone of normals to the closed convex set C(t) at the point v G C(t).

Stable Solution of Variational Inequalities with Composed Monotone Operators

Abstract: Convergence of proximal-based methods is analysed for variational inequalities with operators of the type T 0 +∂F, where T 0 is a single-valued, hemicontinuous and monotone operator and ∂F is the subdifferential of a proper convex lower semicontinuous functional. The analysis is oriented to methods coupling successive approximation of the variational inequality with the proximal point algorithm as well as to related methods using regularization on a subspace and weak regularization. Conditions ensuring linear convergence are established. Finally, we observe briefly some classes of problems which can be solved by means of the methods considered.

A Combined Feasible-Infeasible Point Continuation Method for Strongly Monotone Variational Inequality Problems

Abstract: In this paper, we discuss the variational inequality problems VIP(X, F), where F is a strongly monotone function and the convex feasible set X is described by some inequaliy constraints. We present a continuation method for VIP(X, F), which solves a sequence of perturbed variational inequality problems PVIP(X, F, e, μ) depending on two parameters e ≥ 0 and μ>0. It is worthy to point out that the method will be a feasible point type when ee0 and an infeasible point type when e>0, i.e., it is a combined feasible–infeasible point (CFIFP for short) method. We analyse the existence, uniqueness and continuity of the solution to PVIP(X, F, e, μ), and prove that any sequence generated by this method converges to the unique solution of VIP(X, F). Moreover, some numerical results of the algorithm are reported which show the algorithm is effective.

Existence and iterative algorithms of solutions for generalized mixed quasi-variational inequalities

Abstract: In this paper,. by using the auxiliary technique of variational inequalities, the existence and iterative algorithms of solutions for a class of generalized mixed quasi-variational inequalities are studied. Our results answer the open problems mentioned by Noor, improve and generalize some recent known results.

The iterative solution for a class of monotone operator equations

Abstract: The iterative solution for a class of multivalued monotone operator equations just likeA(u)∈−B(u) is discussed, whereA is a positive definite linear single-valued operator,B is a bounded and monotone multivalued operator. The existence and convergence of approximate solutions are proved. The method of numerical realization is demonstrated in some examples.

The perturbed generalized tikhonov's algorithm

Abstract: We work on the research of a zero of a maximal monotone operator on a real Hilbert space. Following the recent progress made in the context of the proximal point algorithm devoted to this problem, we introduce simultaneously a variable metric and a kind of relaxation in the perturbed Tikhonov's algorithm studied by P. Tossings. So, we are led to work in the context of the variational convergence theory.