Distance-based exponential probability models on constrained combinatorial optimization problems

Estimation of distribution algorithms have already demonstrated their utility when solving a broad range of combinatorial problems. However, there is still room for methodological improvements when approaching constrained type problems. The great majority of works in the literature implement external repairing or penalty schemes, or use ad-hoc sampling methods in order to avoid unfeasible solutions. In this work, we present a new way to develop EDAs for this type of problems by implementing distance-based exponential probability models defined exclusively on the set of feasible solutions. In order to illustrate this procedure, we take the 2-partition balanced Graph Partitioning Problem as a case of study, and design efficient learning and sampling methods in order to use these distance-based probability models in EDAs

A note on the Boltzmann distribution and the linear ordering problem

The Boltzmann distribution plays a key role in the field of optimization as it directly connects this field with that of probability. Basically, given a function to optimize, the Boltzmann distribution associated to this function assigns higher probability to the candidate solutions with better quality. Therefore, an efficient sampling of the Boltzmann distribution would turn optimization into an easy task. However, inference tasks on this distribution imply performing operations over an exponential number of terms, which hinders its applicability. As a result, the scientific community has investigated how the structure of objective functions is translated to probabilistic properties in order to simplify the corresponding Boltzmann distribution. In this paper, we elaborate on the properties induced in the Boltzmann distribution associated to permutation-based combinatorial optimization problems. Particularly, we prove that certain characteristics of the linear ordering problem are translated as conditional independence relations to the Boltzmann distribution in the form of L − decomposability

Characterising the rankings produced by combinatorial optimisation problems and finding their intersections

The aim of this paper is to introduce the concept of intersection between combinatorial optimisation problems. We take into account that most algorithms, in their machinery, do not consider the exact objective function values of the solutions, but only a comparison between them. In this sense, if the solutions of an instance of a combinatorial optimisation problem are sorted into their objective function values, we can see the instances as (partial) rankings of the solutions of the search space. Working with specific problems, particularly, the linear ordering problem and the symmetric and asymmetric traveling salesman problem, we show that they can not generate the whole set of (partial) rankings of the solutions of the search space, but just a subset. First, we characterise the set of (partial) rankings each problem can generate. Secondly, we study the intersections between these problems: those rankings which can be generated by both the linear ordering problem and the symmetric/asymmetric traveling salesman problem, respectively. The fact of finding large intersections between problems can be useful in order to transfer heuristics from one problem to another, or to define heuristics that can be useful for more than one problem

Anatomy of the attraction basins: Breaking with the intuition

olving combinatorial optimization problems efficiently requires the development of algorithms that consider the specific properties of the problems. In this sense, local search algorithms are designed over a neighborhood structure that partially accounts for these properties. Considering a neighborhood, the space is usually interpreted as a natural landscape, with valleys and mountains. Under this perception, it is commonly believed that, if maximizing, the solutions located in the slopes of the same mountain belong to the same attraction basin, with the peaks of the mountains being the local optima. Unfortunately, this is a widespread erroneous visualization of a combinatorial landscape. Thus, our aim is to clarify this aspect, providing a detailed analysis of, first, the existence of plateaus where the local optima are involved, and second, the properties that define the topology of the attraction basins, picturing a reliable visualization of the landscapes. Some of the features explored in this article have never been examined before. Hence, new findings about the structure of the attraction basins are shown. The study is focused on instances of permutation-based combinatorial optimization problems considering the 2-exchange and the insert neighborhoods. As a consequence of this work, we break away from the extended belief about the anatomy of attraction basins

ANALYTICAL SOLUTION FOR TRANSIENT ONEDIMENSIONAL COUETTE FLOW CONSIDERING CONSTANT AND TIME-DEPENDENT PRESSURE GRADIENTS

This paperaims to determine the velocity profile, in transient state, for a parallel incompressible flow known as Couette flow. The Navier-Stokes equations were applied upon this flow. Analytical solutions, based in Fourier series and integral transforms, were obtained for the one-dimensional transient Couette flow, taking into account constant and time-dependent pressure gradients acting on the fluid since the same instant when the plate starts it´s movement. Taking advantage of the orthogonality and superposition properties solutions were foundfor both considered cases. Considering a time-dependent pressure gradient, it was found a general solution for the Couette flow for a particular time function. It was found that the solution for a time-dependent pressure gradient includes the solutions for a zero pressure gradient and for a constant pressure gradient

An Experimental Study in Adaptive Kernel Selection for Bayesian Optimization

Bayesian Optimization has been widely used along with Gaussian Processes for solving expensive-to-evaluate black-box optimization problems. Overall, this approach has shown good results, and particularly for parameter tuning of machine learning algorithms. Nonetheless, Bayesian Optimization has to be also configured to achieve the best possible performance, being the selection of the kernel function a crucial choice. This paper investigates the convenience of adaptively changing the kernel function during the optimization process, instead of fixing it a priori. Six adaptive kernel selection strategies are introduced and tested in well-known synthetic and real-world optimization problems. In order to provide a more complete evaluation of the proposed kernel selection variants, two major kernel parameter setting approaches have been tested. According to our results, apart from having the advantage of removing the selection of the kernel out of the equation, adaptive kernel selection criteria show a better performance than fixed-kernel approaches

in-depth analysis of SVM kernel learning and its components

The performance of support vector machines in non-linearly-separable classification problems strongly relies on the kernel function. Towards an automatic machine learning approach for this technique, many research outputs have been produced dealing with the challenge of automatic learn- ing of good-performing kernels for support vector machines. However, these works have been carried out without a thorough analysis of the set of components that influence the behavior of support vector machines and their interaction with the kernel. These components are related in an in- tricate way and it is difficult to provide a comprehensible analysis of their joint effect. In this paper we try to fill this gap introducing the necessary steps in order to understand these interactions and provide clues for the research community to know where to place the emphasis. First of all, we identify all the factors that affect the final performance of support vector machines in relation to the elicitation of kernels. Next, we analyze the factors independently or in pairs and study the influence each component has on the final classification performance, providing recommendations and insights into the kernel setting for support vector machines.IT1244-19 PID2019-104966GB-I0

Early classification of time series using multi-objective optimization techniques

In early classification of time series the objective is to build models which are able to make class-predictions for time series as accurately and as early as possible, when only a part of the series is available. It is logical to think that accuracy and earliness are conflicting objectives, since the more we wait, more data points from the series are available, and it is easier to make accurate class-predictions. Con- sidering this, the problem can be very naturally formulated as a multi-objective optimization problem, and solved as such. However, the solutions proposed in the literature up to now, reduce the problem into a single-objective problem by com- bining both objectives somehow. In this paper, we present a novel multi-objective formulation of the problem of early classification, and we design a solution us- ing multi-objective optimization techniques. This method will provide a variety of solutions which find different trade-offs between both objectives, allowing the user to select the most suitable solution a-posteriori, depending on the accuracy and earliness requirements of the problem at hand. To prove the usefulness of our proposal, we carry out an extensive experimentation process using 45 benchmark databases and we present a case study in the financial domain

VELOCITY PROFILE MODELING FOR NON-ISOTHERMAL FLOWS INSIDE A CIRCULAR TUBE

This research proposes a new method to establish the velocity field and the dimensionless velocity profile for Newtonian and non-Newtonian flows inside a circular tube. Several studies developed regarding different fluid types (such as potency law fluid, Bingham and Herschel-Bulkley, among others) observed that a rational or irrational polynomial was used for the dependent velocity field variable. Thus, a rational polynomial was established as a starting point for this research as the dependent velocity field variable. Dimensionless velocity profiles obtained from the proposed fluid-dynamics model were experimentally compared only with dimensionless velocity profiles for non-isothermal Newtonian flows of glycerol, in cooling as well as heating. On the other hand, it was possible to calculate that RMS errors found using relative dimensionless velocity data obtained from the proposed fluid-dynamics model creates very small errors, which are comparable to RMS errors found using data obtained from application of a numerical method. Finally, the proposed fluid-dynamics model was validated with a dimensionless velocity profile obtained from the flow of a cooling process, resulting in the validity of the proposed model