Parameter tuning of PBIL and CHC evolutionary algorithms applied to solve the Root Identification Problem

Abstract

Evolutionary algorithms are among the most successful approaches for solving a number of problems where systematic searches in huge domains must be performed. One problem of practical interest that falls into this category is known as The Root Identification Problem in Geometric Constraint Solving, where one solution to the geometric problem must be selected among a number of possible solutions bounded by an exponential number. In previous works we have shown that applying genetic algorithms, a category of evolutionary algorithms, to solve the Root Identification Problem is both feasible and effective.

In this work, we report on an empirical statistical study conducted to establish the influence of the driving parameters in the PBIL and CHC evolutionary algorithms when they are used to solve the Root Identification Problem. We identify a set of values that optimize algorithms performance. The driving parameters considered for the PBIL algorithm are population size, mutation probability, mutation shift and learning rate. For the CHC algorithm we studied population size, divergence rate, differential threshold and the set of best individuals. In both cases we applied unifactorial and multifactorial analysis, post hoc tests and best parameter level selection. Experimental results show that CHC outperforms PBIL when applied to solve the Root Identification Problem.

Introduction

Modern computer aided design and manufacturing systems are built on top of parametric geometric modeling engines. The field has developed sketching systems that automatically instantiate geometric objects from a rough sketch, annotated with dimensions and constraints input by the user. The sketch has only to be topologically correct and constraints are usually not yet satisfied.

Geometric problems defined by constraints have an exponential number of solution instances in the number of geometric elements involved. Generally, the user is only interested in one solution instance such that besides fulfilling the geometric constraints, it exhibits some additional properties. This solution instance is called the intended solution.

Selecting a solution instance amounts to selecting one among a number of different roots of a nonlinear equation or system of equations. The problem of selecting a given root was named the Root Identification Problem [1].

Several approaches to solve the Root Identification Problem have been reported in the literature. Examples are: selectively moving the geometric elements, conducting a dialogue with the constraint solver that identifies interactively the intended solution, and preserving the topology of the sketch input by the user. For a discussion of these approaches see, for example [1], [2] and references therein. Not only it is important to have in mind the interactive nature of computer aided design and manufacturing systems [3], [4], but also the high difficulty of solving the problem [5].

A new technique to automatically solve the Root Identification Problem for constructive solvers [6], was reported in [2], [7], [8]. The technique overconstrains the geometric problem by defining two different categories of constraints. One category includes the set of constraints specifically needed to solve the geometric constraint problem. The other category includes a set of extra constraints or predicates on the geometric elements which identify the intended solution instance.

Once the constructive solver has generated the space of solution instances, the extra constraints are used to drive an automatic search of the solution instances space using genetic algorithms [9]. The search outputs a solution instance that maximizes the number of extra constraints fulfilled.

Genetic algorithms are a category of evolutionary algorithms. Evolutionary algorithms (or, as a matter of fact, any heuristic method) are characterized by a set of parameters that determine the evolution of the algorithms for which specific values must be chosen. One of the main difficulties of applying an evolutionary algorithm to a given problem is to decide on an appropriate set of parameter values. These values are commonly chosen in practice by trial and error, tuned by hand, or taken from other fields [10].

Unfortunately, these methods do not guarantee the obtention of optimal parameter settings and since experimental results are always problem dependent, values that proved to be useful in other applications may not perform well on untried tasks [11]. Furthermore, when parameters are chosen in these ways, the relationship between the parameter values and the performance of the algorithm cannot be established.

In a previous work, see [12], two algorithms, among a large set of different population-based and single-point metaheuristics [13], [14], have been demonstrated to yield the best results when they are applied to the Root Identification Problem: Population-Based Incremental Learning (PBIL) [15], and Cross generational elitist selection Heterogeneous recombination and Cataclismic mutation (CHC) [16].

PBIL and CHC are two evolutionary algorithms that have received a large amount of attention as general purpose function optimizers. The PBIL algorithm is a method that combines generational mechanisms with simple competitive learning. It is argued that this algorithm is simple and outperforms genetic algorithms on a large set of optimization problems. CHC is a nontraditional genetic algorithm whose crossover operation is highly disruptive. This results in a search ability more effective than that of traditional genetic algorithms by balancing diversity and convergence.

In this work, we report on an empirical statistical study conducted to establish the impact of the driving parameters in the PBIL and CHC evolutionary algorithms when they are used to solve the Root Identification Problem in the field of Geometric Constraint Solving. Then we identify a set of values that optimize algorithms performance. The driving parameters considered for the PBIL algorithm are the population size, mutation probability, mutation shift and learning rate. For the CHC algorithm we studied population size, divergence rate, differential threshold and the set of best individuals. In both cases we applied unifactorial and multifactorial analysis, post hoc tests and best parameter level selection. Experimental results show that CHC outperforms PBIL when applied to solve the Root Identification Problem.

The remainder of this work is organized as follows. Section 2 briefly describes the main concepts involved in the Root Identification Problem and how evolutionary algorithms can be applied to solve it. Section 3 briefly describes the PBIL and CHC algorithms. Section 4 describes the experimental set up. Experimental results are discussed in Sections 5 and 6. Section 7 briefly compares PBIL and CHC performances, leaving Section 8 to draw some conclusions and to suggest future work.