Lipschitz gradients for global optimization in a one-point-based partitioning scheme

A global optimization problem is studied where the objective function $f(x)$ is a multidimensional black-box function and its gradient $f'(x)$ satisfies the Lipschitz condition over a hyperinterval with an unknown Lipschitz constant $K$. Different methods for solving this problem by using an a priori given estimate of $K$, its adaptive estimates, and adaptive estimates of local Lipschitz constants are known in the literature. Recently, the authors have proposed a one-dimensional algorithm working with multiple estimates of the Lipschitz constant for $f'(x)$ (the existence of such an algorithm was a challenge for 15 years). In this paper, a new multidimensional geometric method evolving the ideas of this one-dimensional scheme and using an efficient one-point-based partitioning strategy is proposed. Numerical experiments executed on 800 multidimensional test functions demonstrate quite a promising performance in comparison with popular DIRECT-based methods.Comment: 25 pages, 4 figures, 5 tables. arXiv admin note: text overlap with arXiv:1103.205

An Efficient Global Optimization Algorithm with Adaptive Estimates of the Local Lipschitz Constants

In this work, we present a new deterministic partition-based Global Optimization (GO) algorithm that uses estimates of the local Lipschitz constants associated with different sub-regions of the domain of the objective function. The estimates of the local Lipschitz constants associated with each partition are the result of adaptively balancing the global and local information obtained so far from the algorithm, given in terms of absolute slopes. We motivate a coupling strategy with local optimization algorithms to accelerate the convergence speed of the proposed approach. In the end, we compare our approach HALO (Hybrid Adaptive Lipschitzian Optimization) with respect to popular GO algorithms using hundreds of test functions. From the numerical results, the performance of HALO is very promising and can extend our arsenal of efficient procedures for attacking challenging real-world GO problems. The Python code of HALO is publicly available on GitHub. https://github.com/dannyzx/HAL

Application of reduced-set pareto-lipschitzian optimization to truss optimization

In this paper, a recently proposed global Lipschitz optimization algorithm Pareto-Lipschitzian Optimization with Reduced-set (PLOR) is further developed, investigated and applied to truss optimization problems. Partition patterns of the PLOR algorithm are similar to those of DIviding RECTangles (DIRECT), which was widely applied to different real-life problems. However here a set of all Lipschitz constants is reduced to just two: the maximal and the minimal ones. In such a way the PLOR approach is independent of any user-defined parameters and balances equally local and global search during the optimization process. An expanded list of other well-known DIRECT-type algorithms is used in investigation and experimental comparison using the standard test problems and truss optimization problems. The experimental investigation shows that the PLOR algorithm gives very competitive results to other DIRECT-type algorithms using standard test problems and performs pretty well on real truss optimization problems

On Using the Decision Trees to Identify the Local Extrema in Parallel Global Optimization Algorithm

In the present work, the solving of the multidimensional global optimization problems using decision tree to reveal the attractor regions of the local minima is considered. The objective function of the problem is defined as a “black box”, may be non-differentiable, multi-extremal and computational costly. We assume that the function satisfies the Lipschitz condition with a priory unknown constant. Global search algorithm is applied for the search of global minimum in the problems of such type. It is well known that the solution complexity essentially depends on the presence of multiple local extrema. Within the framework of the global search algorithm, we propose a method for selecting the vicinity of local extrema of the objective function based on analysis of accumulated search information. Conducting such an analysis using machine learning techniques allows making a decision to run a local method, which can speed up the convergence of the algorithm. This suggestion was confirmed by the results of numerical experiments demonstrating the speedup when solving a series of test problems.In the present work, the solving of the multidimensional global optimization problems using decision tree to reveal the attractor regions of the local minima is considered. The objective function of the problem is defined as a “black box”, may be non-differentiable, multi-extremal and computational costly. We assume that the function satisfies the Lipschitz condition with a priory unknown constant. Global search algorithm is applied for the search of global minimum in the problems of such type. It is well known that the solution complexity essentially depends on the presence of multiple local extrema. Within the framework of the global search algorithm, we propose a method for selecting the vicinity of local extrema of the objective function based on analysis of accumulated search information. Conducting such an analysis using machine learning techniques allows making a decision to run a local method, which can speed up the convergence of the algorithm. This suggestion was confirmed by the results of numerical experiments demonstrating the speedup when solving a series of test problems

GENDIRECT: a GENeralized DIRECT-type algorithmic framework for derivative-free global optimization

Over the past three decades, numerous articles have been published discussing the renowned DIRECT algorithm (DIvididing RECTangles). These articles present innovative ideas to enhance its performance and adapt it to various types of optimization problems. A comprehensive collection of deterministic, derivative-free algorithmic implementations based on the DIRECT framework has recently been introduced as part of the DIRECTGO project. DIRECTGO empowers users to conveniently employ diverse DIRECT-type algorithms, enabling efficient solutions to practical optimization problems. Despite their variations, DIRECT-type algorithms share a common algorithmic structure and typically differ only at certain steps. Therefore, we propose GENDIRECT -- GENeralized DIRECT-type framework that encompasses and unifies DIRECT-type algorithms into a single, generalized framework within this paper. GENDIRECT offers a practical alternative to the creation of yet another ``new'' DIRECT-type algorithm that closely resembles existing ones. Instead, GENDIRECT allows the efficient generation of known or novel DIRECT-type optimization algorithms by assembling different algorithmic components. This approach provides considerably more flexibility compared to both the DIRECTGO toolbox and individual DIRECT-type algorithms. A few hundred thousand DIRECT-type algorithms can be combined using GENDIRECT, facilitating users' easy customization and the addition of new algorithmic components. By modifying specific components of five highly promising DIRECT-type algorithms found in the existing literature using GENDIRECT, the significant potential of GENDIRECT has been demonstrated. The resulting newly developed improved approaches exhibit greater efficiency and enhanced robustness in dealing with problems of varying complexity.Comment: 29 pages, 6 figures, 8 table