We present a multilevel numerical algorithm for the exact solution of the Euclidean trust-region subproblem. This particular subproblem typically arises when optimizing a nonlinear (possibly non-convex) objective function whose variables are discretized continuous functions, in which case the different levels of discretization provide a natural multilevel context. The trust-region problem is considered at the highest … Read more
An Adaptive Cubic Overestimation (ACO) algorithm for unconstrained optimization is proposed, generalizing at the same time an unpublished method due to Griewank (Technical Report NA/12, 1981, DAMTP, Univ. of Cambridge), an algorithm by Nesterov & Polyak (Math. Programming 108(1), 2006, pp 177-205) and a proposal by Weiser, Deuflhard & Erdmann (Optim. Methods Softw. 22(3), 2007, … Read more
It is known that quasi-Newton updates can be characterized by variational means, sometimes in more than one way. This paper has two main goals. We first formulate variational problems appearing in quasi-Newton methods within the space of symmetric matrices. This simplies both their formulations and their subsequent solutions. We then construct, for the first time, … Read more
We consider the problem of locating a single radiating source from several noisy measurements using a maximum likelihood (ML) criteria. The resulting optimization problem is nonconvex and nonsmooth and thus finding its global solution is in principal a hard task. Exploiting the special structure of the objective function, we introduce and analyze two iterative schemes … Read more
We present a new matrix-free method for the trust region subproblem, assuming that the approximate Hessian is updated by the limited memory BFGS formula with m = 2. The resulting updating scheme, called 2-BFGS, give us the ability to determine via simple formulas the eigenvalues of the resulting approximation. Thus, at each iteration, we can … Read more
The NEWUOA software is described briefly, with some numerical results that show good efficiency and accuracy in the unconstrained minimization without derivatives of functions of up to 320 variables. Some preliminary work on an extension of NEWUOA that allows simple bounds on the variables is also described. It suggests a variation of a technique in … Read more
Let the least value of a function of many variables be required. If its gradient is available, then one can tell whether search directions are downhill, and first order conditions help to identify the solution. It seems in practice, however, that the vast majority of unconstrained calculations do not employ any derivatives. A view of … Read more
Shor’s r-algorithm is an iterative method for unconstrained optimization, designed for minimizing nonsmooth functions, for which its reported success has been considerable. Although some limited convergence results are known, nothing seems to be known about the algorithm’s rate of convergence, even in the smooth case. We study how the method behaves on convex quadratics, proving … Read more
A recursive trust-region method is introduced for the solution of bound-constrained nonlinear nonconvex optimization problems for which a hierarchy of descriptions exists. Typical cases are infinite-dimensional problems for which the levels of the hierarchy correspond to discretization levels, from coarse to fine. The new method uses the infinity norm to define the shape of the … Read more
Third order methods will in most cases use fewer iterations than a second order method to reach the same accuracy. However, the number of arithmetic operations per iteration is higher for third order methods than a second order method. Newton’s method is the most commonly used second order method and Halley’s method is the most … Read more