/pdf/convex-analysisnoer-san-nojin-zhan-nituite-5dl2rbmoyr.pdf

Convex Analysisの二,三の進展について

Economics and Information Theory

This paper is devoted to bilevel optimization, a branch of mathematical programming of both practical and theoretical interest. Starting with a simple example, we proceed towards a general formulation. We then present fields of application, focus on solution approaches, and make the connection with MPECs (Mathematical Programs with Equilibrium Constraints).

/pdf/an-overview-of-bilevel-optimization-4x7xo28qzy.pdf

An overview of bilevel optimization

tions. Bootstrap has found many applications in engineering field, including artificial neural networks, biomedical engineering, environmental engineering, image processing, and radar and sonar signal processing. Basic concepts of the bootstrap are summarized in each section as a step-by-step algorithm for ease of implementation. Most of the applications are taken from the signal processing literature. The principles of the bootstrap are introduced in Chapter 2. Both the nonparametric and parametric bootstrap procedures are explained. Babu and Singh (1984) have demonstrated that in general, these two procedures behave similarly for pivotal (Studentized) statistics. The fact that the bootstrap is not the solution for all of the problems has been known to statistics community for a long time; however, this fact is rarely touched on in the manuscripts meant for practitioners. It was first observed by Babu (1984) that the bootstrap does not work in the infinite variance case. Bootstrap Techniques for Signal Processing explains the limitations of bootstrap method with an example. I especially liked the presentation style. The basic results are stated without proofs; however, the application of each result is presented as a simple step-by-step process, easy for nonstatisticians to follow. The bootstrap procedures, such as moving block bootstrap for dependent data, along with applications to autoregressive models and for estimation of power spectral density, are also presented in Chapter 2. Signal detection in the presence of noise is generally formulated as a testing of hypothesis problem. Chapter 3 introduces principles of bootstrap hypothesis testing. The topics are introduced with interesting real life examples. Flow charts, typical in engineering literature, are used to aid explanations of the bootstrap hypothesis testing procedures. The bootstrap leads to second-order correction due to pivoting; this improvement in the results due to pivoting is also explained. In the second part of Chapter 3, signal processing is treated as a regression problem. The performance of the bootstrap for matched filters as well as constant false-alarm rate matched filters is also illustrated. Chapters 2 and 3 focus on estimation problems. Chapter 4 introduces bootstrap methods used in model selection. Due to the inherent structure of the subject matter, this chapter may be difficult for nonstatisticians to follow. Chapter 5 is the most impressive chapter in the book, especially from the standpoint of statisticians. It provides real data bootstrap applications to illustrate the theory covered in the earlier chapters. These include applications to optimal sensor placement for knock detection and land-mine detection. The authors also provide a MATLAB toolbox comprising frequently used routines. Overall, this is a very useful handbook for engineers, especially those working in signal processing.

Probability and Random Processes

We provide stronger and more general primal-dual convergence results for Frank-Wolfe-type algorithms (a.k.a. conditional gradient) for constrained convex optimization, enabled by a simple framework of duality gap certificates. Our analysis also holds if the linear subproblems are only solved approximately (as well as if the gradients are inexact), and is proven to be worst-case optimal in the sparsity of the obtained solutions.

On the application side, this allows us to unify a large variety of existing sparse greedy methods, in particular for optimization over convex hulls of an atomic set, even if those sets can only be approximated, including sparse (or structured sparse) vectors or matrices, low-rank matrices, permutation matrices, or max-norm bounded matrices.

We present a new general framework for convex optimization over matrix factorizations, where every Frank-Wolfe iteration will consist of a low-rank update, and discuss the broad application areas of this approach.

/pdf/revisiting-frank-wolfe-projection-free-sparse-convex-4tk23q5q3e.pdf

Revisiting Frank-Wolfe: Projection-Free Sparse Convex Optimization

Part 1 Models: urban traffic planning - the transportation planning process, organization and goal definition, base-year inventory, model analysis, travel forecast, network evaluation, discussion the basic equilibrium model and extensions - the Wardrop conditions, the mathematical program for user equilibrium, properties of equilibrium solutions, user equilibrium versus system optimum, non-separable costs and multiclass-user transportation networks, related network problems, discussion, some extentions general traffic equilibrium models - traffic equilibrium models, properties of equilibrium solutions. Part 2 Methods: algorithms for the basic model and its extensions - the Frank-Wolfe algorithm and its extensions, algorithm concepts, algorithms for the basic model, algorithms for elastic demand problems, algorithms for stochastic assignment models, algorithms for side-constrained assignment models, discussion algorithms for general traffic equilibria - algorithm concepts, algorithms for general traffic equilibria, discussion.

The Traffic Assignment Problem: Models and Methods

The class of simplicial decomposition (SD) schemes has shown to provide efficient tools for nonlinear network flows. When applied to the traffic assignment problem, shortest route subproblems are solved in order to generate extreme points of the polyhedron of feasible flows, and, alternately, master problems are solved over the convex hull of the generated extreme points. We review the development of simplicial decomposition and the closely related column generation methods for the traffic assignment problem; we then present a modified, disaggregated, representation of feasible solutions in SD algorithms for convex problems over Cartesian product sets, with application to the symmetric traffic assignment problem. The new algorithm, which is referred to as disaggregate simplicial decomposition (DSD), is given along with a specialized solution method for the disaggregate master problem. Numerical results for several well known test problems and a new one are presented. These experimentations indicate that o...

Simplicial decomposition with disaggregated representation for the traffic assignment problem

http://www.apmath.spbu.ru/cnsa/pdf/obzor/PatrikssonM2008.pdf

A survey on the continuous nonlinear resource allocation problem

http://publications.lib.chalmers.se/records/fulltext/local_141623.pdf

Stochastic mathematical programs with equilibrium constraints

Preface. 1. Introduction. 2. Technical Preliminaries. 3. Instances of the Cost Approximation Algorithm. 4. Merit Functions for Variational Inequality Problems. 5. Convergence of the CA Algorithm for Nonlinear Programs. 6. Convergence of the CA Algorithm for Variational Inequality Problems. 7. Finite Identification of Active Constraints and of Solutions. 8. Parallel and Sequential Decomposition CA Algorithms. 9. A Column Generation/Simplicial Decomposition Algorithm. A. Definitions. References. Index.

Michael Patriksson

Papers

The Traffic Assignment Problem: Models and Methods

Simplicial decomposition with disaggregated representation for the traffic assignment problem

A survey on the continuous nonlinear resource allocation problem

Stochastic mathematical programs with equilibrium constraints

Nonlinear Programming and Variational Inequality Problems: A Unified Approach