513 research outputs found
Lipschitz gradients for global optimization in a one-point-based partitioning scheme
A global optimization problem is studied where the objective function
is a multidimensional black-box function and its gradient satisfies the
Lipschitz condition over a hyperinterval with an unknown Lipschitz constant
. Different methods for solving this problem by using an a priori given
estimate of , 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 (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
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