Evaluation complexity bounds for smooth constrained nonlinear optimization using scaled KKT conditions and high-order models

Evaluation complexity for convexly constrained optimization is considered and it is shown first that the complexity bound of $O(\epsilon^{-3/2})$ proved by Cartis, Gould and Toint (IMAJNA 32(4) 2012, pp.1662-1695) for computing an $\epsilon$-approximate first-order critical point can be obtained under significantly weaker assumptions. Moreover, the result is generalized to the case where high-order derivatives are … Read more

Quantitative recovery conditions for tree-based compressed sensing

As shown in [9, 1], signals whose wavelet coefficients exhibit a rooted tree structure can be recovered — using specially-adapted compressed sensing algorithms — from just $n=\mathcal{O}(k)$ measurements, where $k$ is the sparsity of the signal. Motivated by these results, we introduce a simplified proportional-dimensional asymptotic framework which enables the quantitative evaluation of recovery guarantees … Read more

Global convergence rate analysis of unconstrained optimization methods based on probabilistic models

We present global convergence rates for a line-search method which is based on random first-order models and directions whose quality is ensured only with certain probability. We show that in terms of the order of the accuracy, the evaluation complexity of such a method is the same as its counterparts that use deterministic accurate models; … Read more

Corrigendum: On the complexity of finding first-order critical points in constrained nonlinear optimization

In a recent paper (Cartis, Gould and Toint, Math. Prog. A 144(1-2) 93–106, 2014), the evaluation complexity of an algorithm to find an approximate first-order critical point for the general smooth constrained optimization problem was examined. Unfortunately, the proof of Lemma 3.5 in that paper uses a result from an earlier paper in an incorrect … Read more

Worst-case evaluation complexity of regularization methods for smooth unconstrained optimization using Hölder continuous gradients

The worst-case behaviour of a general class of regularization algorithms is considered in the case where only objective function values and associated gradient vectors are evaluated. Upper bounds are derived on the number of such evaluations that are needed for the algorithm to produce an approximate first-order critical point whose accuracy is within a user-defined … Read more

Active-set prediction for interior point methods using controlled perturbations

We propose the use of controlled perturbations to address the challenging question of optimal active-set prediction for interior point methods. Namely, in the context of linear programming, we consider perturbing the inequality constraints/bounds so as to enlarge the feasible set. We show that if the perturbations are chosen appropriately, the solution of the original problem … Read more

A new and improved quantitative recovery analysis for iterative hard thresholding algorithms in compressed sensing

We present a new recovery analysis for a standard compressed sensing algorithm, Iterative Hard Thresholding (IHT) (Blumensath and Davies, 2008), which considers the fixed points of the algorithm. In the context of arbitrary measurement matrices, we derive a sufficient condition for convergence of IHT to a fixed point and a necessary condition for the existence … Read more

Branching and Bounding Improvements for Global Optimization Algorithms with Lipschitz Continuity Properties

We present improvements to branch and bound techniques for globally optimizing functions with Lipschitz continuity properties by developing novel bounding procedures and parallelisation strategies. The bounding procedures involve nonconvex quadratic or cubic lower bounds on the objective and use estimates of the spectrum of the Hessian or derivative tensor, respectively. As the nonconvex lower bounds … Read more

An example of slow convergence for Newton’s method on a function with globally Lipschitz continuous Hessian

An example is presented where Newton’s method for unconstrained minimization is applied to find an $\epsilon$-approximate first-order critical point of a smooth function and takes a multiple of $\epsilon^{-2}$ iterations and function evaluations to terminate, which is as many as the steepest-descent method in its worst-case. The novel feature of the proposed example is that … Read more

An exact tree projection algorithm for wavelets

We propose a dynamic programming algorithm for projection onto wavelet tree structures. In contrast to other recently proposed algorithms which only give approximate tree projections for a given sparsity, our algorithm is guaranteed to calculate the projection exactly. We also prove that our algorithm has O(Nk) complexity, where N is the signal dimension and k … Read more