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

Global optimization using the branch‐and‐bound algorithm with a combination of Lipschitz bounds over simplices

Many problems in economy may be formulated as global optimization problems. Most numerically promising methods for solution of multivariate unconstrained Lipschitz optimization problems of dimension greater than 2 use rectangular or simplicial branch‐and‐bound techniques with computationally cheap, but rather crude lower bounds. The proposed branch‐and‐bound algorithm with simplicial partitions for global optimization uses a combination of 2 types of Lipschitz bounds. One is an improved Lipschitz bound with the first norm. The other is a combination of simple bounds with different norms. The efficiency of the proposed global optimization algorithm is evaluated experimentally and compared with the results of other well‐known algorithms. The proposed algorithm often outperforms the comparable branch‐and‐bound algorithms. Santrauka Daug įvairių ekonomikos uždavinių yra formuluojami kaip globaliojo optimizavimo uždaviniai. Didžioji dalis Lipšico globaliojo optimizavimo metodų, tinkamų spręsti didesnės dimensijos, t. y. n > 2, uždavinius, naudoja stačiakampį arba simpleksinį šakų ir rėžių metodus bei paprastesnius rėžius. Šiame darbe pasiūlytas simpleksinis šakų ir rėžių algoritmas, naudojantis dviejų tipų viršutinių rėžių junginį. Pirmasis yra pagerintas rėžis su pirmąja norma, kitas – trijų paprastesnių rėžių su skirtingomis normomis junginys. Gautieji eksperimentiniai pasiūlyto algoritmo rezultatai yra palyginti su kitų gerai žinomų Lipšico optimizavimo algoritmų rezultatais.  First published online: 21 Oct 2010 Reikšminiai žodžiai: šakų ir rėžių algoritmas, globalusis optimizavimas, Lipšico optimizavimas, Lipšico rėžis

Improved Lipschitz bounds with the first norm for function values over multidimensional simplex

A branch and bound algorithm for global optimization is proposed, where the maximum of an upper bounding function based on Lipschitz condition and the first norm over a simplex is used as the upper bound of function. In this case the graph of bounding function is intersection of n‐dimensional pyramids and its maximum point is found solving a system of linear equations. The efficiency of the proposed global optimization algorithm is evaluated experimentally. First Published Online: 14 Oct 201

Influence of Lipschitz bounds on the speed of global optimization

Global optimization methods based on Lipschitz bounds have been analyzed and applied widely to solve various optimization problems. In this paper a bound for Lipschitz function is proposed, which is computed using function values at the vertices of a simplex and the radius of the circumscribed sphere. The efficiency of a branch and bound algorithm with proposed bound and combinations of bounds is evaluated experimentally while solving a number of multidimensional test problems for global optimization. The influence of different bounds on the performance of a branch and bound algorithm has been investigated

Improving the implementation of quantum blockchain based on hypergraphs

In recent years, there has been a growing interest in the potential of quantum computing for enhancing the security and efficiency of blockchain technology. While quantum blockchain protocols offer improved security over their classical counterparts, implementing such protocols on present-day quantum computers poses difficulties due to the limited number of qubits and quantum gates and the significant effects of noise. In this paper, we propose a set of improvements for implementing a quantum blockchain protocol based on hypergraphs that aim to reduce the required resources and operations and increase noise tolerance. Specifically, we focus on enhancing the state-of-the-art quantum circuits that underpin the quantum blockchain by optimizing the so-called T-count and T-depth, which represent the number of quantum gates and the circuit depth, respectively. Our proposed implementations also leverage proven error detection and correction codes to improve noise tolerance. To evaluate the effectiveness of our proposed improvements, we tested them on real quantum devices. Our results demonstrate a significant reduction in the T-count and T-depth. Overall, our proposed improvements provide a promising direction for the practical implementation of quantum blockchain protocols on current quantum computers and lay a foundation for further research in this area

Apie redukuotą Patero-Lipšico optimizavimą

A well-known example of global optimization that provides solutions within fixed error limits is optimization of functions with a known Lipschitz constant. In many real-life problems this constant is unknown. To address that a method called Pareto-Lipschitzian Optimization (PLO) was described that provides solutions within fixed error limits for functions with unknown Lipschitz constants. In this approach, a set of all unknown Lipschitz constants is regarded as multiple criteria using the concept of Pareto Optimality (PO). In this paper, a new version of the Pareto-Lipschitzian Optimization method (PLOR) is proposed where a set of unknown Lipschitzian constants is reduced just to the minimal and maximal ones. In the both methods, partition patterns are similar to those of DIRECT. The difference is in the rules of sequential partitions defining non-dominated sets. In PLO, it includes all Pareto-Optimal sets defined by all Lipschitz constants. In PLOR, it considers just two elements corresponding to the maximal and minimal Lipschitz constant. in DIRECT, it selects a part of the Pareto-Optimal set which is determined by some heuristic parameter