We propose data profiles as a tool for analyzing the performance of derivative-free optimization solvers when there are constraints on the computational budget. We use performance and data profiles, together with a convergence test that measures the decrease in function value, to analyze the performance of three solvers on sets of smooth, noisy, and piecewise-smooth … Read more
In this report, we propose a primal interior-point method for large sparse generalized minimax optimization. After a short introduction, where the problem is stated, we introduce the basic equations of the Newton method applied to the KKT conditions and propose a primal interior-point method. Next we describe the basic algorithm and give more details concerning … Read more
We consider the problem of finding an approximate minimizer of a general quadratic function subject to a two-norm constraint. The Steihaug-Toint method minimizes the quadratic over a sequence of expanding subspaces until the iterates either converge to an interior point or cross the constraint boundary. The benefit of this approach is that an approximate solution … Read more
New approximate secant equations are shown to result from the knowledge of (problem dependent) invariant subspace information, which in turn suggests improvements in quasi-Newton methods for unconstrained minimization. It is also shown that this type of information may often be extracted from the multigrid structure of discretized infinite dimensional problems. A new limited-memory BFGS using … Read more
We introduce a new trust-region method for unconstrained optimization where the radius update is computed using the model information at the current iterate rather than at the preceding one. The update is then performed according to how well the current model retrospectively predicts the value of the objective function at last iterate. Global convergence to … Read more
In this article, we consider some problems of bracketing an optimum point for a real-valued, single variable function. We show that, no method, satisfying certain assumptions and requiring a bounded number of function evaluations, can exist to bracket the minimum point of a unimodal function. A similar result is given also for the problem of … Read more
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