An asymptotically superlinearly convergent semismooth Newton augmented Lagrangian method for Linear Programming

Powerful interior-point methods (IPM) based commercial solvers, such as Gurobi and Mosek, have been hugely successful in solving large-scale linear programming (LP) problems. The high efficiency of these solvers depends critically on the sparsity of the problem data and advanced matrix factorization techniques. For a large scale LP problem with data matrix $A$ that is dense (possibly structured) or whose corresponding normal matrix $AA^T$ has a dense Cholesky factor (even with re-ordering), these solvers may require excessive computational cost and/or extremely heavy memory usage in each interior-point iteration. Unfortunately, the natural remedy, i.e., the use of iterative methods based IPM solvers, although can avoid the explicit computation of the coefficient matrix and its factorization, is not practically viable due to the inherent extreme ill-conditioning of the large scale normal equation arising in each interior-point iteration. To provide a better alternative choice for solving large scale LPs with dense data or requiring expensive factorization of its normal equation, we propose a semismooth Newton based inexact proximal augmented Lagrangian ({\sc Snipal}) method. Different from classical IPMs, in each iteration of {\sc Snipal}, iterative methods can efficiently be used to solve simpler yet better conditioned semismooth Newton linear systems. Moreover, {\sc Snipal} not only enjoys a fast asymptotic superlinear convergence but is also proven to enjoy a finite termination property. Numerical comparisons with Gurobi have demonstrated encouraging potential of {\sc Snipal} for handling large-scale LP problems where the constraint matrix $A$ has a dense representation or $AA^T$ has a dense factorization even with an appropriate re-ordering.Comment: Due to the limitation "The abstract field cannot be longer than 1,920 characters", the abstract appearing here is slightly shorter than that in the PDF fil

Algebraische Zahlentheorie

The workshop brought together researchers from Europe, the US and Japan, who reported on various recent developments in algebraic number theory and related fields. Dominant themes were p-adic methods, L-functions and automorphic forms but other topics covered a very wide range of algebraic number theory

On the Artin formalism for pp-adic Garrett--Rankin LL-functions

Our main objective in the present article is to study the factorization problem for triple-product $p$-adic $L$-functions, particularly in the scenarios when the defining properties of the $p$-adic $L$-functions involved have no bearing on this problem, although Artin formalism would suggest such a factorization. Our analysis is guided by the ETNC philosophy and it involves a comparison of diagonal cycles, Beilinson--Flach elements, and Beilinson--Kato elements, much in the spirit of the work of Gross (that is based on a comparison of elliptic units and cyclotomic units) and Dasgupta (that dwells on a comparison of Beilinson--Flach elements and cyclotomic units) for smaller-rank motives.Comment: 92 page

High-resolution regional gravity field recovery from Poisson wavelets using heterogeneous observational techniques

2016-2017 > Academic research: refereed > Publication in refereed journal201804_a bcmaVersion of RecordPublishe

Differential operators on rational projective curves

Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/30118/1/0000494.pd

Graded hypoellipticity of BGG sequences

In this paper we introduce the notion of Rockland sequences of differential operators on filtered manifolds. This concept generalizes the notion of elliptic sequences and is formulated in terms of a Rockland condition. We show that these sequences are hypoelliptic and have analytic properties similar to elliptic sequences, providing regularity, maximal hypoelliptic estimates, and Hodge decomposition. These analytic properties follow from a generalization of the Rockland theorem using the Heisenberg tangent groupoid construction. The main motivation lies in the fact that Bernstein-Gelfand-Gelfand sequences over regular parabolic geometries are Rockland sequences in a graded sense. We use the BGG machinery to construct similar sequences for a large class of filtered manifolds, and illustrate these results in some explicit examples.Comment: Changed the proof of Proposition 4.17; added details to the proof of Lemma 4.15; few minor adjustments; fixed some typos. Now 102 page

Usability of structured lattices for a post-quantum cryptography: practical computations, and a study of some real Kummer extensions

Lattice-based cryptography is an excellent candidate for post-quantum cryptography, i.e. cryptosystems which are resistant to attacks run on quantum computers. For efficiency reason, most of the constructions explored nowadays are based on structured lattices, such as module lattices or ideal lattices. The security of most constructions can be related to the hardness of retrieving a short element in such lattices, and one does not know yet to what extent these additional structures weaken the cryptosystems. A related problem – which is an extension of a classical problem in computational number theory – called the Short Principal Ideal Problem (or SPIP), consists of finding a short generator of a principal ideal. Its assumed hardness has been used to build some cryptographic schemes. However it has been shown to be solvable in quantum polynomial time over cyclotomic fields, through an attack which uses the Log-unit lattice of the field considered. Later, practical results showed that multiquadratic fields were also weak to this strategy. The main general question that we study in this thesis is To what extent can structured lattices be used to build a post-quantum cryptography