We describe an algorithm based on a logarithmic barrier function, Newton's method, and linear conjugate gradients, that obtains an approximate minimizer of a smooth function over the nonnegative orthant. We develop a bound on the complexity of the approach, stated in terms of the required accuracy and the cost of a single gradient evaluation of the objective function and or a matrix-vector multiplication involving the Hessian of the objective. The approach can be implemented without explicit calculation or storage of the Hessian.
In this paper, we adapt the Newton-CG method of @cite_9 for unconstrained optimization to the problem of minimizing the primal log-barrier function , for a small, fixed value of @math . We target the optimality conditions ), which avoid enforcing tighter conditions on Hessian and gradient components that correspond to components of @math that are far from zero at optimality. This change allows us to solve a preconditioned Newton system of linear equations at each iteration in which the norm of the matrix can be bounded by a constant independent of iteration number. The Capped CG method developed in @cite_9 is used to solve this system, returning a useful search direction in a reasonable number of iterations. Our algorithm finds a point satisfying ) in @math iterations and @math gradient evaluations Hessian vector products when @math , making our algorithm the first method for bound-constrained optimization with this overall computational complexity. Further, our algorithm has the appealing practical feature that it puts minimal restrictions on the step size, allowing the line search to take steps that are much closer to the boundary than the current iterate.
@cite_9
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We study how to use the BFGS quasi-Newton matrices to precondition minimization methods for problems where the storage is critical. We give an update formula which generates matrices using information from the last m iterations, where m is any number supplied by the user. The quasi-Newton matrix is updated at every iteration by dropping the oldest information and replacing it by the newest informa- tion. It is shown that the matrices generated have some desirable properties. The resulting algorithms are tested numerically and compared with several well- known methods. 1. Introduction. For the problem of minimizing an unconstrained function of n variables, quasi-Newton methods are widely employed (4). They construct a se- quence of matrices which in some way approximate the hessian of (or its inverse). These matrices are symmetric; therefore, it is necessary to have n(n + l) 2 storage locations for each one. For large dimensional problems it will not be possible to re- tain the matrices in the high speed storage of a computer, and one has to resort to other kinds of algorithms. For example, one could use the methods (Toint (15), Shanno (12)) which preserve the sparsity structure of the hessian, or conjugate gradient methods (CG) which only have to store 3 or 4 vectors. Recently, some CG algorithms have been developed which use a variable amount of storage and which do not require knowledge about the sparsity structure of the problem (2), (7), (8). A disadvantage of these methods is that after a certain number of iterations the quasi-Newton matrix is discarded, and the algorithm is restarted using an initial matrix (usually a diagonal matrix). We describe an algorithm which uses a limited amount of storage and where the quasi-Newton matrix is updated continuously. At every step the oldest information contained in the matrix is discarded and replaced by new one. In this way we hope to have a more up to date model of our function. We will concentrate on the BFGS method since it is considered to be the most efficient. We believe that similar algo- rithms cannot be developed for the other members of the Broyden 0-class (1). Let be the function to be nnnimized, g its gradient and h its hessian. We define In the Newton log-barrier method, Newton steps are taken for the log-barrier function for a fixed value of the barrier parameter until a certain convergence criterion is satisfied. The barrier parameter is then decreased and the Newton process is repeated. A naive analysis indicates that Newton’s method does not exhibit superlinear convergence to the minimizer of each instance of the log-barrier function until it reaches a very small neighborhood, namely within O(μ2) of the minimizer, where μ is the barrier parameter. By analyzing the structure of the barrier Hessian and gradient in terms of the subspace of active constraint gradients and the associated null space, we show that this neighborhood is in fact much larger –O(μσ) for any σ∈(1,2] – thus explaining why reasonably fast local convergence can be attained in practice. Moreover, we show that the overall convergence rate of the Newton log-barrier algorithm is superlinear in the number of function derivative evaluations, provided that the nonlinear program is formulated with a linear objective and that the schedule for decreasing the barrier parameter is related in a certain way to the step length and convergence criteria for each Newton process. We consider minimization of a smooth nonconvex objective function using an iterative algorithm based on Newton’s method and the linear conjugate gradient algorithm, with explicit detection and use of negative curvature directions for the Hessian of the objective function. The algorithm tracks Newton-conjugate gradient procedures developed in the 1980s closely, but includes enhancements that allow worst-case complexity results to be proved for convergence to points that satisfy approximate first-order and second-order optimality conditions. The complexity results match the best known results in the literature for second-order methods. This paper describes how to incorporate sampled curvature information in a Newton-CG method and in a limited memory quasi-Newton method for statistical learning. The motivation for this work stems from supervised machine learning applications involving a very large number of training points. We follow a batch approach, also known in the stochastic optimization literature as a sample average approximation approach. Curvature information is incorporated in two subsampled Hessian algorithms, one based on a matrix-free inexact Newton iteration and one on a preconditioned limited memory BFGS iteration. A crucial feature of our technique is that Hessian-vector multiplications are carried out with a significantly smaller sample size than is used for the function and gradient. The efficiency of the proposed methods is illustrated using a machine learning application involving speech recognition.