Global optimization based on bisection of rectangles, function values at diagonals, and a set of Lipschitz constants

Paulavičius, Remigijus; Chiter, Lakhdar; Žilinskas, Julius

doi:10.1007/s10898-016-0485-6

Cited by 34 publications

(42 citation statements)

References 35 publications

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“…On the other hand, because this approach starts by sampling all 2 n corners of the search space, it is limited in practice to low-dimensional problems. The papers by Paulavičius et al [52,53] consider rectangular partitions, but subdivide rectangles using bisection instead of trisection and sample two points within each rectangle instead of one.…”

Section: Resultsmentioning

confidence: 99%

“…The diagonal along which the two points are located may change, as in does in Fig. 25, but Paulavičius et al [52] show that this scheme works not only in this simple example but also more generally, for any number of dimensions. Bisection is always applied to one of the long sides of a rectangle and, if there is a tie for longest, they select the side with the lower index (as we did in Fig.…”

Section: Direct Using Bisection Of Rectangles (Birect Birect-l Gb-bmentioning

confidence: 95%

“…Paulavičius et al [52] have developed an innovative variant of DIRECT called BIRECT that samples two points per rectangle and, furthermore, employs rectangle bisection instead of trisection. The key idea is shown in Fig.…”

Section: Direct Using Bisection Of Rectangles (Birect Birect-l Gb-bmentioning

confidence: 99%

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The DIRECT algorithm: 25 years Later

Jones

Martins

2020

J Glob Optim

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Introduced in 1993, the DIRECT global optimization algorithm provided a fresh approach to minimizing a black-box function subject to lower and upper bounds on the variables. In contrast to the plethora of nature-inspired heuristics, DIRECT was deterministic and had only one hyperparameter (the desired accuracy). Moreover, the algorithm was simple, easy to implement, and usually performed well on low-dimensional problems (up to six variables). Most importantly, DIRECT balanced local and global search (exploitation vs. exploration) in a unique way: in each iteration, several points were sampled, some for global and some for local search. This approach eliminated the need for “tuning parameters” that set the balance between local and global search. However, the very same features that made DIRECT simple and conceptually attractive also created weaknesses. For example, it was commonly observed that, while DIRECT is often fast to find the basin of the global optimum, it can be slow to fine-tune the solution to high accuracy. In this paper, we identify several such weaknesses and survey the work of various researchers to extend DIRECT so that it performs better. All of the extensions show substantial improvement over DIRECT on various test functions. An outstanding challenge is to improve performance robustly across problems of different degrees of difficulty, ranging from simple (unimodal, few variables) to very hard (multimodal, sharply peaked, many variables). Opportunities for further improvement may lie in combining the best features of the different extensions.

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Section: Resultsmentioning

confidence: 99%

Section: Direct Using Bisection Of Rectangles (Birect Birect-l Gb-bmentioning

confidence: 95%

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The DIRECT algorithm: 25 years Later

Jones

Martins

2020

J Glob Optim

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“…They also have some advantages over the others in terms of easy implementations [38,35]. There exist important methods such as Branch and Bound [54,27], Cutting Plane [39] and space filling curves based methods [17] and other important methods [13,28,29].…”

mentioning

confidence: 99%

On a new smoothing technique for non-smooth, non-convex optimization

Yılmaz¹,

Şahiner²

2020

NACO

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In many global optimization techniques, the local search methods are used for different issues such as to obtain a new initial point and to find the local solution rapidly. Most of these local search methods base on the smoothness of the problem. In this study, we propose a new smoothing approach in order to smooth out non-smooth and non-Lipschitz functions playing a very important role in global optimization problems. We illustrate our smoothing approach on well-known test problems in the literature. The numerical results show the efficiency of our method.

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“…• The paper [1] is dedicated to a Lipschitz global optimization problem using the well-known DIRECT algorithm and the diagonal partitioning strategy. One of the main advantages of the diagonal partitioning scheme is that the objective function is evaluated at two points at each hyper-rectangle and, therefore, more comprehensive information about the objective function is considered with respect to the central sampling strategy used in most DIRECTtype algorithms.…”

mentioning

confidence: 99%