Probability Distributions.- Linear Models for Regression.- Linear Models for Classification.- Neural Networks.- Kernel Methods.- Sparse Kernel Machines.- Graphical Models.- Mixture Models and EM.- Approximate Inference.- Sampling Methods.- Continuous Latent Variables.- Sequential Data.- Combining Models.

Pattern Recognition and Machine Learning

Over the past decade, the field of finite-dimensional variational inequality and complementarity problems has seen a rapid development in its theory of existence, uniqueness and sensitivity of solution(s), in the theory of algorithms, and in the application of these techniques to transportation planning, regional science, socio-economic analysis, energy modeling, and game theory. This paper provides a state-of-the-art review of these developments as well as a summary of some open research topics in this growing field.

Finite-dimensional variational inequality and nonlinear complementarity problems: a survey of theory, algorithms and applications

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).

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An overview of bilevel optimization

http://inis.jinr.ru/sl/m_mathematics/mn_numerical methods/mnd_numerical calculus/allgower e.l., georg k. introduction to numerical continuation methods (colorado, 1990)(397s).pdf

Numerical Continuation Methods

From the Publisher:
Introduction to Numerical Continuation Methods continues to be useful for researchers and graduate students in mathematics, sciences, engineering, economics, and business looking for an introduction to computational methods for solving a large variety of nonlinear systems of equations. A background in elementary analysis and linear algebra is adequate preparation for reading this book; some knowledge from a first course in numerical analysis may also be helpful.

/pdf/introduction-to-numerical-continuation-methods-ily9kpadk5.pdf

Introduction to Numerical Continuation Methods

Adomian's decomposition method (ADM) is a nonnumerical method which can be adapted for solving nonlinear ordinary differential equations. In this paper, the principle of the decomposition method is described, and its advantages as well as drawbacks are discussed. Then an aftertreatment technique (AT) is proposed, which yields the analytic approximate solution with fast convergence rate and high accuracy through the application of Pade approximation to the series solution derived from ADM. Some concrete examples are also studied to show with numerical results how the AT works efficiently.

An aftertreatment technique for improving the accuracy of Adomian's decomposition method

Over the past several decades, the optimization over the efficient set has seen a substantial development. The aim of this paper is to provide a state-of-the-art survey of the development. Given p linear criteria c1x,ccc,cp x and a feasible region X of Rn, the linear multicriteria problem is to find a point x of X such that no point x' of X satisfies (c1 x',ccc,cp x')≥(c1 x,ccc,cp x) and (c1x',ccc,cp x')≠q (c1 x ,ccc,cp x). Such a point is called an efficient point. The optimization over the efficient set is the maximization of a given function φ over the set of efficient points. The difficulty of this problem is mainly due to the nonconvexity of this set. The existing algorithms for solving this problem could be classified into several groups such as adjacent vertex search algorithm, nonadjacent vertex search algorithm, branch-and-bound based algorithm, Lagrangian relaxation based algorithm, dual approach and bisection algorithm. In this paper we review a typical algorithm from each group and compare them from the computational point of view.

Optimization over the efficient set: overview

The existence and computation of competitive equilibria in markets with an indivisible commodity

In this paper we establish a basic theory for variable dimension algorithms which were originally developed for computing fixed points by Van der Laan and Talman. We introduce a new concept ‘primal—dual pair of subdivided manifolds’ and by utilizing it we propose a basic model which will serve as a foundation for constructing a wide class of variable dimension algorithms. Our basic model furnishes interpretations to several existing methods: Lemke's algorithm for the linear complementarity problem, its extension to the nonlinear complementarity problem, a variable dimension algorithm on conical subdivisions and Merrill's algorithm. We shall present a method for solving systems of equations as an application of the second method and an efficient implementation of the fourth method to which our interpretation leads us. A method for constructing triangulations with an arbitrary refinement factor of mesh size is also proposed.

Variable dimension algorithms: Basic theory, interpretations and extensions of some existing methods

The linear multiplicative programming is the minimization of the product of affine functions over a polyhedral set. The problem with two affine functions reduces to a parametric linear program and can be solved efficiently. For the objective function with more than two affine functions multiplied, no efficient algorithms that solve the problem to optimality have been proposed, however Benson and Boger have proposed a heuristic algorithm that exploits links of the problem with concave minimization and multicriteria optimization. We will propose a heuristic method for the problem as well as its modification to enhance the accuracy of approximation. Computational experiments demonstrate that the method and its modification solve randomly generated problems within a few percent of relative error.

Yoshitsugu Yamamoto

Papers

An aftertreatment technique for improving the accuracy of Adomian's decomposition method

Optimization over the efficient set: overview

The existence and computation of competitive equilibria in markets with an indivisible commodity

Variable dimension algorithms: Basic theory, interpretations and extensions of some existing methods

Heuristic Methods for Linear Multiplicative Programming