The Global Convergence of Self-Scaling BFGS Algorithm with Nonmonotone Line Search for Unconstrained Nonconvex Optimization Problems

The self-scaling quasi-Newton method solves an unconstrained optimization problem by scaling the Hessian approximation matrix before it is updated at each iteration to avoid the possible large eigenvalues in the Hessian approximation matrices of the objective function. It has been proved in the literature that this method has the global and superlinear convergence when the objective function is convex （or even uniformly convex）. We propose to solve unconstrained nonconvex optimization problems by a self-scaling BFGS algorithm with nonmonotone linear search. Nonmonotone line search has been recognized in numerical practices as a competitive approach for solving large-scale nonlinear problems. We consider two different nonmonotone line search forms and study the global convergence of these nonmonotone self-scale BFGS algorithms. We prove that, under some weaker condition than that in the literature, both forms of the self-scaling BFGS algorithm are globally convergent for unconstrained nonconvex optimization problems.     

On diagonally preconditioning the truncated Newton method for super-scale linearly constrained nonlinear programming

We present an algorithm for super-scale linearly constrained nonlinear programming (LCNP) based on Newton's method. In large-scale programming solving the Newton equation at each iteration can be expensive and may not be justified when far from a local solution. For super-scale problems, the truncated Newton method (where an inaccurate solution is computed by using the conjugate-gradient method) is recommended; a diagonal BFGS preconditioning of the gradient is used, so that the number of iterations to solve the equation is reduced. The procedure for updating that preconditioning is described for LCNP when the set of active constraints or the partition of basic, superbasic and nonbasic (structural) variables have been changed.     

Optimization of unconstrained functions with sparse hessian matrices-newton-type methods

Newton-type methods for unconstrained optimization problems have been very successful when coupled with a modified Cholesky factorization to take into account the possible lack of positive-definiteness in the Hessian matrix. In this paper we discuss the application of these method to large problems that have a sparse Hessian matrix whose sparsity is known a priori. Quite often it is difficult, if not impossible, to obtain an analytic representation of the Hessian matrix. Determining the Hessian matrix by the standard method of finite-differences is costly in terms of gradient evaluations for large problems. Automatic procedures that reduce the number of gradient evaluations by exploiting sparsity are examined and a new procedure is suggested. Once a sparse approximation to the Hessian matrix has been obtained, there still remains the problem of solving a sparse linear system of equations at each iteration. A modified Cholesky factorization can be used. However, many additional nonzeros (fill-in) may be created in the factors, and storage problems may arise. One way of approaching this problem is to ignore fill-in in a systematic manner. Such technique are calledpartial factorization schemes. Various existing partial factorization are analyzed and three new ones are developed. The above algorithms were tested on a set of problems. The overall conclusions were that these methods perfom well in practice.     

Limited memory quasi-newton method for large-scale linearly equality-constrained minimization

1. IntroductionConsider the following linearly constrained nonlinear programming problemwhere x e R", A E Rmxn and f E C2. We are interested in the case when n and m arelarge and when the Hessian matrix of f is difficult to compute or is dense. It is ajssumed thatA is a matrix of full row rank and that the level set S(xo) = {x: f(x) 5 f(xo), Ax ~ b} isnonempty and compact.In the past few years j there were two kinds of methods for solving the large-scaleproblem (1.1). FOr the one kind, pr…     

A truncated aggregate smoothing Newton method for minimax problems

Aggregate function is a useful smoothing function to the max-function of some smooth functions and has been used to solve minimax problems, linear and nonlinear programming, generalized complementarity problems, etc. The aggregate function is a single smooth but complex function, its gradient and Hessian calculations are time-consuming. In this paper, a truncated aggregate smoothing stabilized Newton method for solving minimax problems is presented. At each iteration, only a small subset of the components in the max-function are aggregated, hence the number of gradient and Hessian calculations is reduced dramatically. The subset is adaptively updated with some truncating criterions, concerning only with computation of function values and not their gradients or Hessians, to guarantee the global convergence and, for the inner iteration, locally quadratic convergence with as few computational cost as possible. Numerical results show the efficiency of the proposed algorithm.     

A class of conjugate gradient methods for convex constrained monotone equations

The recent designed non-linear conjugate gradient method of Dai and Kou [SIAM J Optim. 2013;23:296–320] is very efficient currently in solving large-scale unconstrained minimization problems due to its simpler iterative form, lower storage requirement and its closeness to the scaled memoryless BFGS method. Just because of these attractive properties, this method was extended successfully to solve higher dimensional symmetric non-linear equations in recent years. Nevertheless, its numerical performance in solving convex constrained monotone equations has never been explored. In this paper, combining with the projection method of Solodov and Svaiter, we develop a family of non-linear conjugate gradient methods for convex constrained monotone equations. The proposed methods do not require the Jacobian information of equations, and even they do not store any matrix in each iteration. They are potential to solve non-smooth problems with higher dimensions. We prove the global convergence of the class of the proposed methods and establish its R-linear convergence rate under some reasonable conditions. Finally, we also do some numerical experiments to show that the proposed methods are efficient and promising.     

Parallel quasi-Newton methods for unconstrained optimization

We discuss methods for solving the unconstrained optimization problem on parallel computers, when the number of variables is sufficiently small that quasi-Newton methods can be used. We concentrate mainly, but not exclusively, on problems where function evaluation is expensive. First we discuss ways to parallelize both the function evaluation costs and the linear algebra calculations in the standard sequential secant method, the BFGS method. Then we discuss new methods that are appropriate when there are enough processors to evaluate the function, gradient, and part but not all of the Hessian at each iteration. We develop new algorithms that utilize this information and analyze their convergence properties. We present computational experiments showing that they are superior to parallelization either the BFGS methods or Newton's method under our assumptions on the number of processors and cost of function evaluation. Finally we discuss ways to effectively utilize the gradient values at unsuccessful trial points that are available in our parallel methods and also in some sequential software packages.Research supported by AFOSR grant AFOSR-85-0251, ARO contract DAAG 29-84-K-0140, NSF grants DCR-8403483 and CCR-8702403, and NSF cooperative agreement DCR-8420944.     

Inexact trust region PGC method for large sparse unconstrained optimization

We present an algorithm, partitioning group correction (PGC) algorithm based on trust region and conjugate gradient method, for large-scale sparse unconstrained optimization. In large sparse optimization, computing the whole Hessian matrix and solving the Newton-like equations at each iteration can be considerably expensive when a trust region method is adopted. The method depends on a symmetric consistent partition of the columns of the Hessian matrix and an inaccurate solution to the Newton-like equations by conjugate gradient method. And we allow that the current direction exceeds the trust region bound if it is a good descent direction. Besides, we studies a method dealing with some sparse matrices having a dense structure part. Some good convergence properties are kept and we contrast the computational behavior of our method with that of other algorithms. Our numerical tests show that the algorithm is promising and quite effective, and that its performance is comparable to or better than that of other algorithms available.     

Adaptive Algorithm for Constrained Least-Squares Problems

This paper is concerned with the implementation and testing of an algorithm for solving constrained least-squares problems. The algorithm is an adaptation to the least-squares case of sequential quadratic programming (SQP) trust-region methods for solving general constrained optimization problems. At each iteration, our local quadratic subproblem includes the use of the Gauss–Newton approximation but also encompasses a structured secant approximation along with tests of when to use this approximation. This method has been tested on a selection of standard problems. The results indicate that, for least-squares problems, the approach taken here is a viable alternative to standard general optimization methods such as the Byrd–Omojokun trust-region method and the Powell damped BFGS line search method.     

An Improved Gradient Projection-based Decomposition Technique for Support Vector Machines

In this paper we propose some improvements to a recent decomposition technique for the large quadratic program arising in training support vector machines. As standard decomposition approaches, the technique we consider is based on the idea to optimize, at each iteration, a subset of the variables through the solution of a quadratic programming subproblem. The innovative features of this approach consist in using a very effective gradient projection method for the inner subproblems and a special rule for selecting the variables to be optimized at each step. These features allow to obtain promising performance by decomposing the problem into few large subproblems instead of many small subproblems as usually done by other decomposition schemes. We improve this technique by introducing a new inner solver and a simple strategy for reducing the computational cost of each iteration. We evaluate the effectiveness of these improvements by solving large-scale benchmark problems and by comparison with a widely used decomposition package.     

A curvilinear optimization method based upon iterative estimation of the eigensystem of the Hessian matrix

An algorithm was recently presented that minimizes a nonlinear function in several variables using a Newton-type curvilinear search path. In order to determine this curvilinear search path the eigenvalue problem of the Hessian matrix of the objective function has to be solved at each iteration of the algorithm. In this paper an iterative procedure requiring gradient information only is developed for the approximation of the eigensystem of the Hessian matrix. It is shown that for a quadratic function the approximated eigenvalues and eigenvectors tend rapidly to the actual eigenvalues and eigenvectors of its Hessian matrix. The numerical tests indicate that the resulting algorithm is very fast and stable. Moreover, the fact that some approximations to the eigenvectors of the Hessian matrix are available is used to get past saddle points and accelerate the rate of convergence on flat functions.     

A sparse sequential quadratic programming algorithm

Described here is the structure and theory for a sequential quadratic programming algorithm for solving sparse nonlinear optimization problems. Also provided are the details of a computer implementation of the algorithm along with test results. The algorithm maintains a sparse approximation to the Cholesky factor of the Hessian of the Lagrangian. The solution to the quadratic program generated at each step is obtained by solving a dual quadratic program using a projected conjugate gradient algorithm. An updating procedure is employed that does not destroy sparsity.     

A New Diagonal Quasi-Newton Updating Method With Scaled Forward Finite Differences Directional Derivative for Unconstrained Optimization

A new diagonal quasi-Newton updating algorithm for unconstrained optimization is presented. The elements of the diagonal matrix approximating the Hessian are determined as scaled forward finite differences directional derivatives of the components of the gradient. Under mild classical assumptions, the convergence of the algorithm is proved to be linear. Numerical experiments with 80 unconstrained optimization test problems, of different structures and complexities, as well as five applications from MINPACK-2 collection, prove that the suggested algorithm is more efficient and more robust than the quasi-Newton diagonal algorithm retaining only the diagonal elements of the BFGS update, than the weak quasi-Newton diagonal algorithm, than the quasi-Cauchy diagonal algorithm, than the diagonal approximation of the Hessian by the least-change secant updating strategy and minimizing the trace of the matrix, than the Cauchy with Oren and Luenberger scaling algorithm in its complementary form (i.e. the Barzilai-Borwein algorithm), than the steepest descent algorithm, and than the classical BFGS algorithm. However, our algorithm is inferior to the limited memory BFGS algorithm (L-BFGS).     

A regularized limited memory BFGS method for nonconvex unconstrained minimization

The limited memory BFGS method (L-BFGS) is an adaptation of the BFGS method for large-scale unconstrained optimization. However, The L-BFGS method need not converge for nonconvex objective functions and it is inefficient on highly ill-conditioned problems. In this paper, we proposed a regularization strategy on the L-BFGS method, where the used regularization parameter may play a compensation role in some sense when the condition number of Hessian approximation tends to become ill-conditioned. Then we proposed a regularized L-BFGS method and established its global convergence even when the objective function is nonconvex. Numerical results show that the proposed method is efficient.     

Using parallel function evaluations to improve hessian approximation for unconstrained optimization

This paper presents a new class of methods for solving unconstrained optimization problems on parallel computers. The methods are intended to solve small to moderate dimensional problems where function and derivative evaluation is the dominant cost. They utilize multiple processors to evaluate the function, (finite difference) gradient, and a portion of the finite difference Hessian simultaneously at each iterate. We introduce three types of new methods, which all utilize the new finite difference Hessian information in forming the new Hessian approximation at each iteration; they differ in whether and how they utilize the standard secant information from the current step as well. We present theoretical analyses of the rate of convergence of several of these methods. We also present computational results which illustrate their performance on parallel computers when function evaluation is expensive.Research supported by AFOSR grant AFOSR-85-0251, ARO contract DAAG 29-84-K-0140, NSF grant DCR-8403483, and NFS cooperative agreement DCR -8420944.     

Parallel algorithms for large-scale linearly constrained minimization problem

In this paper, two PVD-type algorithms are proposed for solving inseparable linear constraint optimization. Instead of computing the residual gradient function, the new algorithm uses the reduced gradients to construct the PVD directions in parallel computation, which can greatly reduce the computation amount each iteration and is closer to practical applications for solve large-scale nonlinear programming. Moreover, based on an active set computed by the coordinate rotation at each iteration, a feasible descent direction can be easily obtained by the extended reduced gradient method. The direction is then used as the PVD direction and a new PVD algorithm is proposed for the general linearly constrained optimization. And the global convergence is also proved.     

Variational Methods for Non-Linear Least-Squares

We consider Newton-like line search descent methods for solving non-linear least-squares problems. The basis of our approach is to choose a method, or parameters within a method, by minimizing a variational measure which estimates the error in an inverse Hessian approximation. In one approach we consider sizing methods and choose sizing parameters in an optimal way. In another approach we consider various possibilities for hybrid Gauss-Newton/BFGS methods. We conclude that a simple Gauss-Newton/BFGS hybrid is both efficient and robust and we illustrate this by a range of comparative tests with other methods. These experiments include not only many well known test problems but also some new classes of large residual problem.     

An SR1/BFGS SQP algorithm for nonconvex nonlinear programs with block-diagonal Hessian matrix

We present a quasi-Newton sequential quadratic programming (SQP) algorithm for nonlinear programs in which the Hessian of the Lagrangian function is block-diagonal. Problems with this characteristic frequently arise in the context of optimal control; for example, when a direct multiple shooting parametrization is used. In this article, we describe an implementation of a filter line-search SQP method that computes search directions using an active-set quadratic programming (QP) solver. To take advantage of the block-diagonal structure of the Hessian matrix, each block is approximated separately by quasi-Newton updates. For nonconvex instances, that arise, for example, in optimum experimental design control problems, these blocks are often found to be indefinite. In that case, the block-BFGS quasi-Newton update can lead to poor convergence. The novel aspect in this work is the use of SR1 updates in place of BFGS approximations whenever possible. The resulting indefinite QPs necessitate an inertia control mechanism within the sparse Schur-complement factorization that is carried out by the active-set QP solver. This permits an adaptive selection of the Hessian approximation that guarantees sufficient progress towards a stationary point of the problem. Numerical results demonstrate that the proposed approach reduces the number of SQP iterations and CPU time required for the solution of a set of optimal control problems.     

Computing hyperbolic choreographies

An algorithm is presented for numerical computation of choreographies in spaces of constant negative curvature in a hyperbolic cotangent potential, extending the ideas given in a companion paper [14] for computing choreographies in the plane in a Newtonian potential and on a sphere in a cotangent potential. Following an idea of Diacu, Pérez-Chavela and Reyes Victoria [9], we apply stereographic projection and study the problem in the Poincaré disk. Using approximation by trigonometric polynomials and optimization methods with exact gradient and exact Hessian matrix, we find new choreographies, hyperbolic analogues of the ones presented in [14]. The algorithm proceeds in two phases: first BFGS quasi-Newton iteration to get close to a solution, then Newton iteration for high accuracy.