Interior Point Methods and Kernel Functions of a Linear Programming Problem

In this thesis the Interior -- Point Method (IPM) for Linear Programming problem (LP) that is based on the generic kernel function is considered. The complexity (in terms of iteration bounds) of the algorithm is first analyzed for a class of kernel functions defined by (3-1). This class is fairly general; it includes classical logarithmic kernel function, prototype self-regular kernel function as well as non-self-regular functions, thus it serves as a unifying frame for the analysis of IPM. Historically, most results in the theory of IPM are based on logarithmic kernel functions while other two classes are more recent. They were considered with the intention to improve theoretical and practical performance of IPMs. The complexity results that are obtained match the best known complexity results for these methods. Next, the analysis of the IPM was summarized and performed for three more kernel functions. For two of them we again matched the best known complexity results. The theoretical concepts of IPM were illustrated by basic implementation for the classical logarithmic kernel function and for the parametric kernel function both described in (3-1). Even this basic implementation shows potential for a good performance. Better implementation and more numerical testing would be necessary to draw more definite conclusions. However, that was not the goal of the thesis, the goal was to show that IPM with kernel functions different than classical logarithmic kernel function can have best known theoretical complexity

Kernel-Based Interior-Point Algorithms for the Linear Complementarity Problem

In this thesis, we consider the Linear Complementarity Problem (LCP), which is a well-known mathematical problem with many practical applications. The objective of the LCP is to find a certain vector that will satisfy a set of linear inequalities and (non-linear) complementary equation. A kernel-based primal-dual Interior-Point Method (IPM) for solving LCP was introduced and analyzed. The class of kernel functions used in this thesis is a class of so-called eligible kernel functions that are fairly general. We have shown for a positive semi-definite matrix M, that the algorithm is globally convergent and has very good convergence properties. For some instances of the eligible kernel functions, the complexity of the algorithm, in terms of the number of iterations, considered in this thesis matches the best complexity results obtained in the literature for these types of methods. This is the main emphasis of the thesis. The theoretical concepts were illustrated by basic implementation in MATLAB for the classical kernel function and for the parametric kernel function (Table 3.3). A series of numerical tests were conducted that shows that even these basic implementations have a potential for good performance. Better implementation and more numerical testing would be necessary to draw more definite conclusions

Interior-point methods for P∗(κ)-linear complementarity problem based on generalized trigonometric barrier function

Recently, M.~Bouafoa, et al. investigated a new kernel function which differs from the self-regular kernel functions. The kernel function has a trigonometric Barrier Term. In this paper we generalize the analysis presented in the above paper for $P_{*}(\kappa)$ Linear Complementarity Problems (LCPs). It is shown that the iteration bound for primal-dual large-update and small-update interior-point methods based on this function is as good as the currently best known iteration bounds for these type methods. The analysis for LCPs deviates significantly from the analysis for linear optimization. Several new tools and techniques are derived in this paper.publishedVersio

On a smoothed penalty-based algorithm for global optimization

This paper presents a coercive smoothed penalty framework for nonsmooth and nonconvex constrained global optimization problems. The properties of the smoothed penalty function are derived. Convergence to an ε -global minimizer is proved. At each iteration k, the framework requires the ε(k) -global minimizer of a subproblem, where ε(k)→ε . We show that the subproblem may be solved by well-known stochastic metaheuristics, as well as by the artificial fish swarm (AFS) algorithm. In the limit, the AFS algorithm convergence to an ε(k) -global minimum of the real-valued smoothed penalty function is guaranteed with probability one, using the limiting behavior of Markov chains. In this context, we show that the transition probability of the Markov chain produced by the AFS algorithm, when generating a population where the best fitness is in the ε(k)-neighborhood of the global minimum, is one when this property holds in the current population, and is strictly bounded from zero when the property does not hold. Preliminary numerical experiments show that the presented penalty algorithm based on the coercive smoothed penalty gives very competitive results when compared with other penalty-based methods.The authors would like to thank two anonymous referees for their valuable comments and suggestions to improve the paper. This work has been supported by COMPETE: POCI-01-0145-FEDER-007043 and FCT - Fundac¸ao para a Ci ˜ encia e Tecnologia within the projects UID/CEC/00319/2013 and ˆ UID/MAT/00013/2013.info:eu-repo/semantics/publishedVersio

A local dynamic conditional correlation model

This paper introduces the idea that the variances or correlations in financial returns may all change conditionally and slowly over time. A multi-step local dynamic conditional correlation model is proposed for simultaneously modelling these components. In particular, the local and conditional correlations are jointly estimated by multivariate kernel regression. A multivariate k-NN method with variable bandwidths is developed to solve the curse of dimension problem. Asymptotic properties of the estimators are discussed in detail. Practical performance of the model is illustrated by applications to foreign exchange rates.Local and conditional correlations; multivariate nonparametric ARCH; multivariate kernel regression; multivariate k-NN method