1 
Accelerated algorithms for composite saddlepoint problems and applicationsHe, Yunlong 12 January 2015 (has links)
This dissertation considers the composite saddlepoint (CSP) problem which is motivated by realworld applications in the areas of machine learning and image processing. Two new accelerated algorithms for solving composite saddlepoint problems are introduced.
Due to the twoblock structure of the CSP problem, it can be solved by any algorithm belonging to the blockdecomposition hybrid proximal extragradient (BDHPE) framework. The framework consists of a family of inexact proximal point methods for solving a general twoblock structured monotone inclusion problem which, at every iteration, solves two prox subinclusions according to a certain relative error criterion. By exploiting the fact that the two prox subinclusions in the context of the CSP problem are equivalent to two composite convex programs, the first part of this dissertation proposes a new instance of the BDHPE framework that approximately solves them using an accelerated gradient method. It is shown that this new instance has better iterationcomplexity than the previous ones.
The second part of this dissertation introduces a new algorithm for solving a special class of CSP problems. The new algorithm is a special instance of the hybrid proximal extragradient (HPE) framework in which a Nesterov's accelerated variant is used to approximately solve the prox subproblems. One of the advantages of the this method is that it works for any constant choice of proximal stepsize. Moreover, a suitable choice of the latter stepsize yields a method with the best known (accelerated inner) iteration complexity for the aforementioned class of saddlepoint problems.
Experiment results on both synthetic CSP problems and realworld problems show that the two method significantly outperform several stateoftheart algorithms.

2 
Algorithm Design and Analysis for LargeScale Semidefinite Programming and Nonlinear ProgrammingLu, Zhaosong 24 June 2005 (has links)
The limiting behavior of weighted paths associated with the semidefinite program (SDP) map $X^{1/2}SX^{1/2}$ was studied and some applications to error bound analysis and superlinear convergence of a class of
primaldual interiorpoint methods were provided. A new approach for solving largescale wellstructured sparse SDPs via a saddle point mirrorprox algorithm with ${cal O}(epsilon^{1})$ efficiency was developed based on exploiting sparsity structure and reformulating SDPs into smooth convexconcave saddle point problems. An iterative solverbased
longstep primaldual infeasible pathfollowing algorithm for convex quadratic programming (CQP) was developed. The search directions of
this algorithm were computed by means of a preconditioned iterative linear solver. A uniform bound, depending only on the CQP data, on
the number of iterations performed by a preconditioned iterative linear solver was established. A polynomial bound on the number of
iterations of this algorithm was also obtained. One efficient ``nearly exact' type of method for solving largescale ``lowrank' trust region
subproblems was proposed by completely avoiding the computations of Cholesky or partial Cholesky factorizations. A computational study of this method was also provided by applying it to solve some largescale nonlinear programming problems.

3 
Fast simulation of (nearly) incompressible nonlinear elastic material at large strain via adaptive mixed FEMBalg, Martina, Meyer, Arnd 19 October 2012 (has links) (PDF)
The main focus of this work lies in the simulation of the deformation of mechanical components which consist of nonlinear elastic, incompressible material and that are subject to large deformations. Starting from a nonlinear formulation one can derive a discrete problem by using linearisation techniques and an adaptive mixed finite element method. This turns out to be a saddle point problem that can be solved via a BramblePasciak conjugate gradient method. With some modifications the simulation can be improved.

4 
Fast simulation of (nearly) incompressible nonlinear elastic material at large strain via adaptive mixed FEMBalg, Martina, Meyer, Arnd 19 October 2012 (has links)
The main focus of this work lies in the simulation of the deformation of mechanical components which consist of nonlinear elastic, incompressible material and that are subject to large deformations. Starting from a nonlinear formulation one can derive a discrete problem by using linearisation techniques and an adaptive mixed finite element method. This turns out to be a saddle point problem that can be solved via a BramblePasciak conjugate gradient method. With some modifications the simulation can be improved.:1. Introduction
2. Basics
3. Mixed variational formulation
4. Solution method
5. Error estimation
6. LBB conditions
7. Improvement suggestions

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