Global Convergence Theory Research Articles

Inexact free derivative quasi-Newton method for large-scale nonlinear system of equations

In this work, we propose a free derivative quasi-Newton method for solving large-scale nonlinear systems of equation. We introduce a two-stage linear search direction and develop its global convergence theory. Besides, we prove that the method enjoys superlinear convergence rate. Finally, numerical experiments illustrate that the proposed method is competitive with respect to Newton-Krylov methods and other well-known methods for solving large-scale nonlinear systems of equations.

On maximum residual nonlinear Kaczmarz-type algorithms for large nonlinear systems of equations

Recently, a class of nonlinear Kaczmarz (NK) algorithms has been proposed to solve large-scale nonlinear systems of equations. The NK algorithm is a generalization of the Newton–Raphson (NR) method and does not need to compute the entire Jacobian matrix. In this paper, we present a maximum residual nonlinear Kaczmarz (MRNK) algorithm for solving large-scale nonlinear systems of equations, which employs a maximum violation row selection and acts only on single rows of the entire Jacobian matrix at a time. Furthermore, we also establish the convergence theory of MRNK. In addition, inspired by the effectiveness of block Kaczmarz algorithms for solving linear systems, we further present a block MRNK (MRBNK) algorithm based on an approximate maximum residual criterion. Based on sketch-and-project technique and sketched Newton–Raphson method, we propose the deterministic sketched Newton–Raphson (DSNR) method which is equivalent to MRNBK, and then the global convergence theory of DSNR is established based on some assumptions and μ-strongly quasi-convex condition. Furthermore, the convergence theory of DSNR is provided under star-convex assumption. Finally, some numerical examples are tested to show the effectiveness of our new technique.

Evolution Strategies under the 1/5 Success Rule

For large space dimensions, the log-linear convergence of the elitist evolution strategy with a 1/5 success rule on the sphere fitness function has been observed, experimentally, from the very beginning. Finding a mathematical proof took considerably more time. This paper presents a review and comparison of the most consistent theories developed so far, in the critical interpretation of the author, concerning both global convergence and the estimation of convergence rates. I discuss the local theory of the one-step expected progress and success probability for the (1+1) ES with a normal/uniform distribution inside the sphere mutation, thereby minimizing the SPHERE function, but also the adjacent global convergence and convergence rate theory, essentially based on the 1/5 rule. Small digressions into complementary theories (martingale, irreducible Markov chain, drift analysis) and different types of algorithms (population based, recombination, covariance matrix adaptation and self-adaptive ES) complete the review.

An Active-Set Fischer–Burmeister Trust-Region Algorithm to Solve a Nonlinear Bilevel Optimization Problem

In this paper, the Fischer–Burmeister active-set trust-region (FBACTR) algorithm is introduced to solve the nonlinear bilevel programming problems. In FBACTR algorithm, a Karush–Kuhn–Tucker (KKT) condition is used with the Fischer–Burmeister function to transform a nonlinear bilevel programming (NBLP) problem into an equivalent smooth single objective nonlinear programming problem. To ensure global convergence for the FBACTR algorithm, an active-set strategy is used with a trust-region globalization strategy. The theory of global convergence for the FBACTR algorithm is presented. To clarify the effectiveness of the proposed FBACTR algorithm, applications of mathematical programs with equilibrium constraints are tested.

Sketched Newton--Raphson

We propose a new globally convergent stochastic second-order method. Our starting point is the development of a new sketched Newton--Raphson (SNR) method for solving large scale nonlinear equations of the form $F(x)=0$ with $F:\mathbb{R}^p \rightarrow \mathbb{R}^m$. We then show how to design several stochastic second-order optimization methods by rewriting the optimization problem of interest as a system of nonlinear equations and applying SNR. For instance, by applying SNR to find a stationary point of a generalized linear model, we derive completely new and scalable stochastic second-order methods. We show that the resulting method is very competitive as compared to state-of-the-art variance reduced methods. Furthermore, using a variable splitting trick, we also show that the stochastic Newton method (SNM) is a special case of SNR and use this connection to establish the first global convergence theory of SNM. We establish the global convergence of SNR by showing that it is a variant of the online stochastic gradient descent (SGD) method, and then leveraging proof techniques of \textttSGD. As a special case, our theory also provides a new global convergence theory for the original Newton--Raphson method under strictly weaker assumptions as compared to the classic monotone convergence theory.

Accelerated derivative-free nonlinear least-squares applied to the estimation of Manning coefficients

A general framework for solving nonlinear least squares problems without the employment of derivatives is proposed in the present paper together with a new general global convergence theory. With the aim to cope with the case in which the number of variables is big (for the standards of derivative-free optimization), two dimension-reduction procedures are introduced. One of them is based on iterative subspace minimization and the other one is based on spline interpolation with variable nodes. Each iteration based on those procedures is followed by an acceleration step inspired in the Sequential Secant Method. The practical motivation for this work is the estimation of parameters in Hydraulic models applied to dam breaking problems. Numerical examples of the application of the new method to those problems are given.

An interior-point trust-region algorithm to solve a nonlinear bilevel programming problem

<abstract><p>In this paper, a nonlinear bilevel programming (NBLP) problem is transformed into an equivalent smooth single objective nonlinear programming (SONP) problem utilized slack variable with a Karush-Kuhn-Tucker (KKT) condition. To solve the equivalent smooth SONP problem effectively, an interior-point Newton's method with Das scaling matrix is used. This method is locally method and to guarantee convergence from any starting point, a trust-region strategy is used. The proposed algorithm is proved to be stable and capable of generating approximal optimal solution to the nonlinear bilevel programming problem.</p> <p>A global convergence theory of the proposed algorithm is introduced and applications to mathematical programs with equilibrium constraints are given to clarify the effectiveness of the proposed approach.</p></abstract>

An active-set with barrier method and trust-region mechanism to solve a nonlinear Bilevel programming problem

<abstract><p>Nonlinear Bilevel programming (NBLP) problem is a hard problem and very difficult to be resolved by using the classical method. In this paper, Karush-Kuhn-Tucker (KKT) condition is used with Fischer-Burmeister function to convert NBLP problem to an equivalent smooth single objective nonlinear programming (SONP) problem. An active-set strategy is used with Barrier method and trust-region technique to solve the smooth SONP problem effectively and guarantee a convergence to optimal solution from any starting point. A global convergence theory for the active-set barrier trust-region (ACBTR) algorithm is studied under five standard assumptions. An applications to mathematical programs are introduced to clarify the effectiveness of ACBTR algorithm. The results show that ACBTR algorithm is stable and capable of generating approximal optimal solution to the NBLP problem.</p></abstract>

A global Jacobian smoothing algorithm for nonlinear complementarity problems

In this paper, we use the smoothing Jacobian strategy to propose a new algorithm for solving complementarity problems based on its reformulation as a nonsmooth system of equations. This algorithm can be seen as a generalization of the one proposed in [18]. We develop its global convergence theory and under certain assumptions, we demonstrate that the proposed algorithm converges locally and, q-superlinearly or q-quadratically to a solution of the problem. Some numerical experiments show a good performance of this algorithm.

Trust-Region Based Penalty Barrier Algorithm for Constrained Nonlinear Programming Problems: An Application of Design of Minimum Cost Canal Sections

In this paper, a penalty method is used together with a barrier method to transform a constrained nonlinear programming problem into an unconstrained nonlinear programming problem. In the proposed approach, Newton’s method is applied to the barrier Karush–Kuhn–Tucker conditions. To ensure global convergence from any starting point, a trust-region globalization strategy is used. A global convergence theory of the penalty–barrier trust-region (PBTR) algorithm is studied under four standard assumptions. The PBTR has new features; it is simpler, has rapid convergerce, and is easy to implement. Numerical simulation was performed on some benchmark problems. The proposed algorithm was implemented to find the optimal design of a canal section for minimum water loss for a triangle cross-section application. The results are promising when compared with well-known algorithms.

Sample Efficient Reinforcement Learning with REINFORCE

Policy gradient methods are among the most effective methods for large-scale reinforcement learning, and their empirical success has prompted several works that develop the foundation of their global convergence theory. However, prior works have either required exact gradients or state-action visitation measure based mini-batch stochastic gradients with a diverging batch size, which limit their applicability in practical scenarios. In this paper, we consider classical policy gradient methods that compute an approximate gradient with a single trajectory or a fixed size mini-batch of trajectories under soft-max parametrization and log-barrier regularization, along with the widely-used REINFORCE gradient estimation procedure. By controlling the number of "bad" episodes and resorting to the classical doubling trick, we establish an anytime sub-linear high probability regret bound as well as almost sure global convergence of the average regret with an asymptotically sub-linear rate. These provide the first set of global convergence and sample efficiency results for the well-known REINFORCE algorithm and contribute to a better understanding of its performance in practice.

On the use of Jordan Algebras for improving global convergence of an Augmented Lagrangian method in nonlinear semidefinite programming

Jordan Algebras are an important tool for dealing with semidefinite programming and optimization over symmetric cones in general. In this paper, a judicious use of Jordan Algebras in the context of sequential optimality conditions is done in order to generalize the global convergence theory of an Augmented Lagrangian method for nonlinear semidefinite programming. An approximate complementarity measure in this context is typically defined in terms of the eigenvalues of the constraint matrix and the eigenvalues of an approximate Lagrange multiplier. By exploiting the Jordan Algebra structure of the problem, we show that a simpler complementarity measure, defined in terms of the Jordan product, is stronger than the one defined in terms of eigenvalues. Thus, besides avoiding a tricky analysis of eigenvalues, a stronger necessary optimality condition is presented. We then prove the global convergence of an Augmented Lagrangian algorithm to this improved necessary optimality condition. The results are also extended to an interior point method. The optimality conditions we present are sequential ones, and no constraint qualification is employed; in particular, a global convergence result is available even when Lagrange multipliers are unbounded.

Some modified Hestenes-Stiefel conjugate gradient algorithms with application in image restoration

It is efficient to use the Hestenes–Stiefe (HS) conjugate gradient algorithm in solving large-scale complex smooth optimization problems because of its simplicity and low calculation requirements. Additionally, this algorithm has been used to simultaneously solve large-scale nonsmooth problems, nonlinear equations, and practical application. In this paper, the authors propose some modified HS conjugate gradient algorithms that not only address large-scale nonlinear equations and nonsmooth convex problems but also have the following properties. i) The algorithms portray a decreasing trait and trust-region property without any additional condition. ii) It combines the steepest descent algorithm with the conjugate gradient algorithm. iii) They employ the global convergence theory and iv) can be successfully used to solve nonlinear optimization problems and image restoration.

Global inexact quasi-Newton method for nonlinear system of equations with constraints

In this paper, we propose a Projected Inexact Quasi-Newton Method (PIQN) to solve nonlinear systems of equations with nonnegative constraints and we develop its respective global convergence theory. Particularly, we show that the method enjoys superlinear convergence rate. In addition, we show the good numerical performance of the method when applied to problems that arise in chemistry, engineering and mathematics. We highlight the efficiency of the method to solve Nonlinear Complementarity Problems (NCP), which appear in many application fields and are recognized as hard problems.

An Augmented Lagrangian algorithm for nonlinear semidefinite programming applied to the covering problem

In this work, we present an Augmented Lagrangian algorithm for nonlinear semidefinite problems (NLSDPs), which is a natural extension of its consolidated counterpart in nonlinear programming. This method works with two levels of constraints; one that is penalized and other that is kept within the subproblems. This is done to allow exploiting the subproblem structure while solving it. The global convergence theory is based on recent results regarding approximate Karush–Kuhn–Tucker optimality conditions for NLSDPs, which are stronger than the usually employed Fritz John optimality conditions. Additionally, we approach the problem of covering a given object with a fixed number of balls with a minimum radius, where we employ some convex algebraic geometry tools, such as Stengle’s Positivstellensatz and its variations, which allows for a much more general model. Preliminary numerical experiments are presented.

Critical Analysis of the International Standards on Corporate Governance and Its Compliance

Critical Analysis of the International Standards on Corporate Governance and Its Compliance

A Modified FR Conjugate Gradient Method with Strong Wolfe Line Search

In this paper, a modified FR conjugate gradient method(MFR) is presented for solving unconstrained optimization problems. The sufficient descent property of the MFR method is satisfied without any line search. Moreover, the MFR method reduces to the standard FR method if line search is exact. Under some mild conditions, the theory of global convergence is established. Numerical results are also given to show the efficiency of the MFR method.

Augmented Lagrangians with constrained subproblems and convergence to second-order stationary points

Augmented Lagrangian methods with convergence to second-order stationary points in which any constraint can be penalized or carried out to the subproblems are considered in this work. The resolution of each subproblem can be done by any numerical algorithm able to return approximate second-order stationary points. The developed global convergence theory is stronger than the ones known for current algorithms with convergence to second-order points in the sense that, besides the flexibility introduced by the general lower-level approach, it includes a loose requirement for the resolution of subproblems. The proposed approach relies on a weak constraint qualification, that allows Lagrange multipliers to be unbounded at the solution. It is also shown that second-order resolution of subproblems increases the chances of finding a feasible point, in the sense that limit points are second-order stationary for the problem of minimizing the squared infeasibility. The applicability of the proposed method is illustrated in numerical examples with ball-constrained subproblems.

An active-set algorithm and a trust-region approach in constrained minimax problem

In this paper, we propose an algorithm to solve a finite minimax problem with side constraints. An active-set strategy is used in the algorithm to transform inequality constraints to equality constraints. This allows the use of the well-developed techniques for solving the equality constrained optimization problems. A trust-region globalization strategy is added to the proposed algorithm to ensure global convergence. A projected Hessian technique is used in the algorithm to overcome the difficulty of having an infeasible trust-region subproblem. A global convergence theory for the proposed algorithm is presented under standard assumptions. Finally, numerical experiments are reported to indicate that the new algorithm performs efficiently in practice.

The Convergence Analysis of Parallel Alternating Two-stage Iterative Algorithm for Linear Complementarity Problem

In this paper, the authors first present parallel alternating two-stage iterative Algorithm some new relaxation algorithms for solving the linear complementarity problem. And then, when the coefficient matrices are monotone or H-matrices, they establish the global convergence theory of the algorithm. The algorithm has less computational complexity and quicker velocity and is especially suitable for parallel computation of large-scale problem.