BBOB-Benchmarking the DIRECT Global Optimization Algorithm Petr Pošík Czech Technical University in Prague, Faculty of Electrical Engineering, Dept. of Cybernetics Technická 2, 166 27 Prague 6 posik@labe.felk.cvut.cz ABSTRACT The DIRECT global optimization algorithm is tested on the BBOB 2009 testbed. The algorithm is rather time and space consuming since it does not forget any point it samples dur- ing the optimization. Furthermore, all the sampled points are considered when deciding where to sample next. The re- sults suggest that the algorithm is a viable alternative only for low-dimensional search spaces (5D at most). Categories and Subject Descriptors G.1.6 [Numerical Analysis]: Optimization—Global Opti- mization, Unconstrained Optimization ; F.2.1 [Analysis of Algorithms and Problem Complexity]: Numerical Al- gorithms and Problems General Terms Algorithms, Experimentation, Performance, Reliability Keywords Benchmarking, Black-box optimization, Evolutionary com- putation, Global optimization, DIRECT 1. INTRODUCTION The DIRECT algorithm was introduced in [5]. The name of the algorithm not only expresses that it belongs to the class of direct search algorithms, it also describes the basic principle of the algorithm: the DIRECT acronym stands for DIviding RECTangles. 2. ALGORITHM DESCRIPTION In this paper, a MATLAB implementation [2, 6] of the DIRECT algorithm is used. Only the basic algorithm design principles are described here; for the detailed description see [5] or [2]. The algorithm divides the search space to non-overlaping hyperrectangles; in each time instant, the whole search space Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. To copy otherwise, to republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. GECCO’09, July 8–12, 2009, Montréal Québec, Canada. Copyright 2009 ACM 978-1-60558-505-5/09/07 ...$5.00. is completely covered by all the hyperrectangles. The point in the middle of each hyperrectangle (the base point) is eval- uated. Each hyperrectangle thus has two (among others) important characteristics: 1. the fitness of its base point, and 2. the size of the hyperrectangle. There are many possible definitions of the hyperrectangle size, here the distance from the basepoint to the hyperect- angle corner is used. In each iteartion the algorithm decides which of the ex- isting hyperrectangles should be split, i.e. where the next points should be sampled. The decision is based on the fol- lowing 2 assumptions: • the better fitness of the base point, the higher the chance of finding an improvement, and • the larger the rectangle, the higher the chance of find- ing an improvement. The algorithm thus selects for division 1 large rectangles and rectangles with highly fit base points, while it ignores small rectangles, or rectangles with non-fit base points. The division process then proceeds by splitting the rect- angle into thirds by evaluating 2D points (D is the search space dimensionality) lying in the distance ± 1 3 of the respec- tive side length of the hyperrectangle from the base point, and deciding the order in which the individual dimensions should be divided into thirds. The demonstration of the DIRECT sampling process can be seen in Fig. 1. 2.1 Implementation Modifications The MATLAB implementation [2] was modified to better suit the needs of the BBOB-benchmarking. 1. The original implementation allowed the user to use it in the test mode when the value of global optimum is known. However, in that case all other termination criteria were ignored, i.e. the algorithm runs until it finds the global optimum. The modification takes into account also other termination criteria, the maximum number of evaluations in particular. 1 The selection of the next rectangles to divide is basically a multi-objective problem. The algorithm does not use the usual definition of non-dominated solutions; instead it se- lects (the right subset of) the rectangles that lie on the boundary of the convex hull of all rectangles described by their size and the base point fitness. 2315