Global Convergence Results Research Articles

Global convergence toward traveling fronts in nonlinear parabolic systems with a gradient structure

We consider nonlinear parabolic systems of the form ut=−∇V(u)+uxx, where u∈Rn, n⩾1, x∈R, and the potential V is coercive at infinity. For such systems, we prove a result of global convergence toward bistable fronts which states that invasion of a stable homogeneous equilibrium (a local minimum of the potential) necessarily occurs via a traveling front connecting to another (lower) equilibrium. This provides, for instance, a generalization of the global convergence result obtained by Fife and McLeod [P. Fife, J.B. McLeod, The approach of solutions of nonlinear diffusion equations to traveling front solutions, Arch. Rat. Mech. Anal. 65 (1977) 335–361] in the case n=1. The proof is based purely on energy methods, it does not make use of comparison principles, which do not hold any more when n>1.

Affine scaling interior Levenberg–Marquardt method for bound-constrained semismooth equations under local error bound conditions

We develop and analyze a new affine scaling Levenberg–Marquardt method with nonmonotonic interior backtracking line search technique for solving bound-constrained semismooth equations under local error bound conditions. The affine scaling Levenberg–Marquardt equation is based on a minimization of the squared Euclidean norm of linear model adding a quadratic affine scaling matrix to find a solution that belongs to the bounded constraints on variable. The global convergence results are developed in a very general setting of computing trial directions by a semismooth Levenberg–Marquardt method where a backtracking line search technique projects trial steps onto the feasible interior set. We establish that close to the solution set the affine scaling interior Levenberg–Marquardt algorithm is shown to converge locally Q-superlinearly depending on the quality of the semismooth and Levenberg–Marquardt parameter under an error bound assumption that is much weaker than the standard nonsingularity condition, that is, BD-regular condition under nonsmooth case. A nonmonotonic criterion should bring about speed up the convergence progress in the contours of objective function with large curvature.

Global convergence of an SQP method without boundedness assumptions on any of the iterative sequences

Usual global convergence results for sequential quadratic programming (SQP) algorithms with linesearch rely on some a priori assumptions about the generated sequences, such as boundedness of the primal sequence and/or of the dual sequence and/or of the sequence of values of a penalty function used in the linesearch procedure. Different convergence statements use different combinations of assumptions, but they all assume boundedness of at least one of the sequences mentioned above. In the given context boundedness assumptions are particularly undesirable, because even for non-pathological and well-behaved problems the associated penalty functions (whose descent is used to produce primal iterates) may not be bounded below for any value of the penalty parameter. Consequently, boundedness assumptions on the iterates are not easily justifiable. By introducing a very simple and computationally cheap safeguard in the linesearch procedure, we prove boundedness of the primal sequence in the case when the feasible set is nonempty, convex, and bounded. If, in addition, the Slater condition holds, we obtain a complete global convergence result without any a priori assumptions on the iterative sequences. The safeguard consists of not accepting a further increase of constraints violation at iterates which are infeasible beyond a chosen threshold, which can always be ensured by the proposed modified SQP linesearch criterion.

A Dai–Yuan conjugate gradient algorithm with sufficient descent and conjugacy conditions for unconstrained optimization

A modification of the Dai–Yuan conjugate gradient algorithm is proposed. Using exact line search, the algorithm reduces to the original version of the Dai and Yuan computational scheme. For inexact line search the algorithm satisfies both sufficient descent and conjugacy conditions. A global convergence result is proved when the Wolfe line search conditions are used. Computational results, for a set consisting of 750 unconstrained optimization test problems, show that this new conjugate gradient algorithm substantially outperforms the Dai–Yuan conjugate gradient algorithm and comes close to its hybrid variants.

Convergence of supermemory gradient method

In this paper we consider the global convergence of a new supermemory gradient method for unconstrained optimization problems. New trust region radius is proposed to make the new method converge stably and averagely, and it will be suitable to solve large scale minimization problems. Some global convergence results are obtained under some mild conditions. Numerical results show that this new method is effective and stable in practical computation.

A practical update criterion for SQP method

Sequential quadratic programming (SQP) algorithms have proven to be efficient for solving nonlinearly constrained optimization problems under some conditions. In analysis of global convergence for SQP algorithms, it is usually assumed that the quasi-Newton matrices, approximating the Hessian of Lagrangian function, are uniformly positive definite. However, it is not known whether this condition is satisfied for quasi-Newton matrices. In this paper, we present a new update criterion for the quasi-Newton matrix and a new update method for the penalty parameter. We establish the global convergence result of a modifying SQP algorithm for equality constrained optimization problems without any assumption on quasi-Newton matrix sequence. Moreover, if the second-order sufficient condition holds at an optimal solution, the new update criterion reduces to the ordinary BFGS update. The superlinear rate of convergence of the algorithm can be obtained if an additional step is performed.

Some recent developments in Lagrangian mean curvature flows

We review some recent results on the mean curvature flows of Lagrangian submanifolds from the perspective of geometric partial differential equations. These include global existence and convergence results, characterizations of first-time singularities, and constructions of self-similar solutions.

A Global Convergence Result for a Higher Order Difference Equation

Letf(z1,…,zk)∈C(Ik,I)be a given function, whereIis (bounded or unbounded) subinterval ofℝ, andk∈ℕ. Assume thatf(y1,y2,…,yk)≥f(y2,…,yk,y1)ify1≥max{y2,…,yk},f(y1,y2,…,yk)≤f(y2,…,yk,y1)ify1≤min{y2,…,yk}, andfis non- decreasing in the last variablezk. We then prove that every bounded solution of an autonomous difference equation of orderk, namely,xn=f(xn−1,…,xn−k),n=0,1,2,…,with initial valuesx−k,…,x−1∈I, is convergent, and every unbounded solution tends either to+∞or to−∞.

Translation-invariant monotone systems, and a global convergence result for enzymatic futile cycles

Strongly monotone systems of ordinary differential equations which have a certain translation-invariance property are shown to have the property that all projected solutions converge to a unique equilibrium. This result may be seen as a dual of a well-known theorem of Mierczyński for systems that satisfy a conservation law. As an application, it is shown that enzymatic futile cycles have a global convergence property.

Penalty Algorithms in Hilbert Spaces

We analyze the classical penalty algorithm for nonlinear programming in Hilbert spaces and obtain global convergence results, as well as asymptotic superlinear convergence order. These convergence results generalize similar results obtained for finite-dimensional problems. Moreover, the nature of the algorithms allows us to solve the unconstrained subproblems in finite-dimensional spaces.

Augmented penalty algorithms based on BFGS secant approximations and trust regions

An iterative method for solving the nonlinear minimization problem with equality constraints is presented. The method is based on the sequential minimization of the augmented Lagrangian function. The unconstrained minimization subproblems are solved by using a technique based on the conjugate gradients combined with the trust region approach as globalization strategy. The Hessian matrix of the augmented Lagrangian is updated by using a BFGS-like structured secant approximation. Global convergence results are shown. The method is applied to solve the equality constrained nonlinear least squares problem. Numerical results are presented not only with well-known test problems but also with ill-posed problems involving a regularization. Even though the tested problems are small and medium, features like the conjugate gradient/trust region strategy and the structured secant approximation make the proposed algorithm specially efficient for large scale problems.

Convergence Properties of the Dependent PRP Conjugate Gradient Methods

In this paper, a new region of β k with respect to β k PRP is given. With two Armijo-type line searches, the authors investigate the global convergence properties of the dependent PRP conjugate gradient methods, which extend the global convergence results of PRP conjugate gradient method proved by Grippo and Lucidi (1997) and Dai and Yuan (2002).

Global convergence properties of the two new dependent Fletcher–Reeves conjugate gradient methods

In this paper, we propose two new dependent Fletcher–Reeves conjugate gradient methods arising from different choice for the scalar β k . We make two different kinds of estimations of upper bounds of ∣ β k ∣ with respect to β k FR , which are based on Abel Theorem of non-convergent series of positive items. With several different line searches, global convergence results are established for the two new methods which extend the previous dependent Fletcher–Reeves conjugate gradient methods.

Interior Gradient and Proximal Methods for Convex and Conic Optimization

Interior gradient (subgradient) and proximal methods for convex constrained minimization have been much studied, in particular for optimization problems over the nonnegative octant. These methods are using non-Euclidean projections and proximal distance functions to exploit the geometry of the constraints. In this paper, we identify a simple mechanism that allows us to derive global convergence results of the produced iterates as well as improved global rates of convergence estimates for a wide class of such methods, and with more general convex constraints. Our results are illustrated with many applications and examples, including some new explicit and simple algorithms for conic optimization problems. In particular, we derive a class of interior gradient algorithms which exhibits an $O(k^{-2})$ global convergence rate estimate.

A Robust SQP Method for Mathematical Programs with Linear Complementarity Constraints

The relationship between the mathematical program with linear complementarity constraints (MPLCC) and its inequality relaxation is studied. Based on this relationship, a new sequential quadratic programming (SQP) method is presented for solving the MPLCC. A certain SQP technique is introduced to deal with the possible infeasibility of quadratic programming subproblems. Global convergence results are derived without assuming the linear independence constraint qualification for MPEC, the nondegeneracy condition, and any feasibility condition of the quadratic programming subproblems. Preliminary numerical results are reported.

Importance Sampling and MM-Algorithms with Applications to Options Pricing

In this paper we study the problem of finding unbiased variance reducing Monte Carlo estimators. An optimal estimator is characterized by a minimization problem. Our novel approach to solve this problem is based on maximization/minimization algorithms (MM-algorithms). First, we prove a general global convergence result for this type of algorithm. Then we construct a specific MM-algorithm for the minimization problem associated with variance reducing Monte Carlo estimators. In general, it is not possible to evaluate the objective function. Therefore, we construct large sample approximations of the objective function and associate minimization problems with these approximations. The minimization problems determine M-estimators which we use to approximate the minimum point of the original minimization problem. We prove consistency and asymptotic normality of these M-estimators. Furthermore, we modify the MM-algorithm of the original minimization problem to obtain a MM-algorithm for calculating approximations to the M-estimators. These approximations are approximations to optimal unbiased variance reducing Monte Carlo estimators. We perform some numerical experiments in the context of derivatives pricing which show that computing these optimal estimators by a MM-algorithm is efficient.

Affine scaling inexact generalized Newton algorithm with interior backtracking technique for solving bound-constrained semismooth equations

We develop and analyze an affine scaling inexact generalized Newton algorithm in association with nonmonotone interior backtracking line technique for solving systems of semismooth equations subject to bounds on variables. By combining inexact affine scaling generalized Newton with interior backtracking line search technique, each iterate switches to inexact generalized Newton backtracking step to strict interior point feasibility. The global convergence results are developed in a very general setting of computing trial steps by the affine scaling generalized Newton-like method that is augmented by an interior backtracking line search technique projection onto the feasible set. Under some reasonable conditions we establish that close to a regular solution the inexact generalized Newton method is shown to converge locally p-order q-superlinearly. We characterize the order of local convergence based on convergence behavior of the quality of the approximate subdifferentials and indicate how to choose an inexact forcing sequence which preserves the rapid convergence of the proposed algorithm. A nonmonotonic criterion should bring about speeding up the convergence progress in some ill-conditioned cases.

A new smoothing quasi-Newton method for nonlinear complementarity problems

The nonlinear complementarity problem can be reformulated as a nonsmooth equation. In this paper we propose a new smoothing quasi-Newton method for the solution of the nonlinear complementarity problems by constructing a new smoothing approximation function. Global and local superlinear convergence results of the proposed algorithm are established under suitable conditions. Some numerical results are reported in the paper.

A critical analysis on global convergence of Hopfield-type neural networks

This paper deals with the global convergence and stability of the Hopfield-type neural networks under the critical condition that M/sub 1/(/spl Gamma/)=L/sup -1/D/spl Gamma/-(/spl Gamma/W+W/sup T//spl Gamma/)/(2) is nonnegative for any diagonal matrix /spl Gamma/, where W is the weight matrix of the network, L=diag{L/sub 1/,L/sub 2/,...,L/sub N/} with L/sub i/ being the Lipschitz constant of g/sub i/ and G(u)=(g/sub 1/(u/sub 1/),g/sub 2/(u/sub 2/),...,g/sub N/(u/sub N/))/sup T/ is the activation mapping of the network. Many stability results have been obtained for the Hopfield-type neural networks in the noncritical case that M/sub 1/(/spl Gamma/) is positive definite for some positive definite diagonal matrix /spl Gamma/. However, very few results are available on the global convergence and stability of the networks in the critical case. In this paper, by exploring two intrinsic features of the activation mapping, two generic global convergence results are established in the critical case for the Hopfield-type neural networks, which extend most of the previously known globally asymptotic stability criteria to the critical case. The results obtained discriminate the critical dynamics of the networks, and can be applied directly to a group of Hopfield-type neural network models. An example has also been presented to demonstrate both theoretical importance and practical significance of the critical results obtained.

An adaptive approach of conic trust-region method for unconstrained optimization problems

In this paper, an adaptive trust region method based on the conic model for unconstrained optimization problems is proposed and analyzed. We establish the global and superlinear convergence results of the method. Numerical tests are reported that confirm the efficiency of the new method.