Worst-case evaluation complexity and optimality of second-order methods for nonconvex smooth optimization

We establish or refute the optimality of inexact second-order methods for unconstrained nonconvex optimization from the point of view of worst-case evaluation complexity, improving and generalizing the results of Cartis, Gould and Toint (2010,2011). To this aim, we consider a new general class of inexact second-order algorithms for unconstrained optimization that includes regularization and trust-region … Read more

On the use of the saddle formulation in weakly-constrained 4D-VAR data assimilation

This paper discusses the practical use of the saddle variational formulation for the weakly-constrained 4D-VAR method in data assimilation. It is shown that the method, in its original form, may produce erratic results or diverge because of the inherent lack of monotonicity of the produced objective function values. Convergent, variationaly coherent variants of the algorithm … Read more

Maintaining a Basis Matrix in the Linear Programming Interior Point Method

To precondition the normal equation system from the linear programming (LP) interior point method, basis preconditioners choose a basis matrix dependent on column scaling factors. Two criteria for choosing the basis matrix are compared which yield a maximum volume or maximum weight basis. Finding a maximum volume basis requires a combinatorial effort, but it gives … Read more

Computation of exact bootstrap confidence intervals: complexity and deterministic algorithms

The bootstrap is a nonparametric approach for calculating quantities, such as confidence intervals, directly from data. Since calculating exact bootstrap quantities is believed to be intractable, randomized resampling algorithms are traditionally used. Motivated by the fact that the variability from randomization can lead to inaccurate outputs, we propose a deterministic approach. First, we establish several … Read more

Iteratively Linearized Reweighted Alternating Direction Method of Multipliers for a Class of Nonconvex Problems

In this paper, we consider solving a class of nonconvex and nonsmooth problems frequently appearing in signal processing and machine learning research. The traditional alternating direction method of multipliers encounters troubles in both mathematics and computations in solving the nonconvex and nonsmooth subproblem. In view of this, we propose a reweighted alternating direction method of … Read more

Inner Conditions for Error Bounds and Metric Subregulerity of Multifunctions

We introduce a new class of sets, functions and multifunctions which is shown to be large and to enjoy some nice common properties with the convex setting. Error bounds for objects attached to this class are characterized in terms of inner conditions of Abadie’s type, that is conditions bearing on normal cones and coderivatives at … Read more

Worst-case convergence analysis of gradient and Newton methods through semidefinite programming performance estimation

We provide new tools for worst-case performance analysis of the gradient (or steepest descent) method of Cauchy for smooth strongly convex functions, and Newton’s method for self-concordant functions. The analysis uses semidefinite programming performance estimation, as pioneered by Drori en Teboulle [Mathematical Programming, 145(1-2):451–482, 2014], and extends recent performance estimation results for the method of … Read more

A convergence frame for inexact nonconvex and nonsmooth algorithms and its applications to several iterations

In this paper, we consider the convergence of an abstract inexact nonconvex and nonsmooth algorithm. We promise a pseudo sufficient descent condition and a pseudo relative error condition, which both are related to an auxiliary sequence, for the algorithm; and a continuity condition is assumed to hold. In fact, a wide of classical inexact nonconvex … Read more

An incremental mirror descent subgradient algorithm with random sweeping and proximal step

We investigate the convergence properties of incremental mirror descent type subgradient algorithms for minimizing the sum of convex functions. In each step we only evaluate the subgradient of a single component function and mirror it back to the feasible domain, which makes iterations very cheap to compute. The analysis is made for a randomized selection … Read more

Same-Day Delivery with Drone Resupply

Unmanned Aerial Vehicles (UAVs), commonly referred to as drones, have recently seen an increased level of interest as their potential use in same-day home delivery has been promoted and advocated by large retailers and courier delivery companies. We introduce a novel way to exploit drones in same-day home delivery settings: drone resupply. We consider a … Read more