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A new proof of Balinski's theorem on the connectivity of polytopes

- Pineda-Villavicencio, Guillermo

  • Authors: Pineda-Villavicencio, Guillermo
  • Date: 2021
  • Type: Text , Journal article
  • Relation: Discrete Mathematics Vol. 344, no. 7 (2021), p.
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  • Description: Balinski (1961) proved that the graph of a d-dimensional convex polytope is d-connected. We provide a new proof of this result. Our proof provides details on the nature of a separating set with exactly d vertices; some of which appear to be new. © 2021 Elsevier B.V.

Aggregate subgradient method for nonsmooth DC optimization

- Bagirov, Adil, Taheri, Sona, Joki, Kaisa, Karmitsa, Napsu, Mäkelä, Marko

  • Authors: Bagirov, Adil , Taheri, Sona , Joki, Kaisa , Karmitsa, Napsu , Mäkelä, Marko
  • Date: 2021
  • Type: Text , Journal article
  • Relation: Optimization Letters Vol. 15, no. 1 (2021), p. 83-96
  • Relation: http://purl.org/au-research/grants/arc/DP190100580
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  • Description: The aggregate subgradient method is developed for solving unconstrained nonsmooth difference of convex (DC) optimization problems. The proposed method shares some similarities with both the subgradient and the bundle methods. Aggregate subgradients are defined as a convex combination of subgradients computed at null steps between two serious steps. At each iteration search directions are found using only two subgradients: the aggregate subgradient and a subgradient computed at the current null step. It is proved that the proposed method converges to a critical point of the DC optimization problem and also that the number of null steps between two serious steps is finite. The new method is tested using some academic test problems and compared with several other nonsmooth DC optimization solvers. © 2020, Springer-Verlag GmbH Germany, part of Springer Nature.

Chebyshev multivariate polynomial approximation and point reduction procedure

- Sukhorukova, Nadezda, Ugon, Julien, Yost, David

  • Authors: Sukhorukova, Nadezda , Ugon, Julien , Yost, David
  • Date: 2021
  • Type: Text , Journal article
  • Relation: Constructive Approximation Vol. 53, no. 3 (2021), p. 529-544
  • Relation: http://purl.org/au-research/grants/arc/DP180100602
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  • Description: We apply the methods of nonsmooth and convex analysis to extend the study of Chebyshev (uniform) approximation for univariate polynomial functions to the case of general multivariate functions (not just polynomials). First of all, we give new necessary and sufficient optimality conditions for multivariate approximation, and a geometrical interpretation of them which reduces to the classical alternating sequence condition in the univariate case. Then, we present a procedure for verification of necessary and sufficient optimality conditions that is based on our generalization of the notion of alternating sequence to the case of multivariate polynomials. Finally, we develop an algorithm for fast verification of necessary optimality conditions in the multivariate polynomial case. © 2019, Springer Science+Business Media, LLC, part of Springer Nature.

Generalised rational approximation and its application to improve deep learning classifiers

- Peiris, V, Sharon, Nir, Sukhorukova, Nadezda, Ugon, Julien

  • Authors: Peiris, V , Sharon, Nir , Sukhorukova, Nadezda , Ugon, Julien
  • Date: 2021
  • Type: Text , Journal article
  • Relation: Applied Mathematics and Computation Vol. 389, no. (2021), p.
  • Relation: http://purl.org/au-research/grants/arc/DP180100602
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  • Description: A rational approximation (that is, approximation by a ratio of two polynomials) is a flexible alternative to polynomial approximation. In particular, rational functions exhibit accurate estimations to nonsmooth and non-Lipschitz functions, where polynomial approximations are not efficient. We prove that the optimisation problems appearing in the best uniform rational approximation and its generalisation to a ratio of linear combinations of basis functions are quasiconvex even when the basis functions are not restricted to monomials. Then we show how this fact can be used in the development of computational methods. This paper presents a theoretical study of the arising optimisation problems and provides results of several numerical experiments. We apply our approximation as a preprocessing step to deep learning classifiers and demonstrate that the classification accuracy is significantly improved compared to the classification of the raw signals. © 2020
  • Description: This research was supported by the Australian Research Council (ARC), Solving hard Chebyshev approximation problems through nonsmooth analysis (Discovery Project DP180100602 ).  This research was partially sponsored by Tel Aviv-Swinburne Research Collaboration Grant (2019).

Incremental DC optimization algorithm for large-scale clusterwise linear regression

- Bagirov, Adil, Taheri, Sona, Cimen, Emre

  • Authors: Bagirov, Adil , Taheri, Sona , Cimen, Emre
  • Date: 2021
  • Type: Text , Journal article
  • Relation: Journal of Computational and Applied Mathematics Vol. 389, no. (2021), p. 1-17
  • Relation: https://purl.org/au-research/grants/arc/DP190100580
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  • Description: The objective function in the nonsmooth optimization model of the clusterwise linear regression (CLR) problem with the squared regression error is represented as a difference of two convex functions. Then using the difference of convex algorithm (DCA) approach the CLR problem is replaced by the sequence of smooth unconstrained optimization subproblems. A new algorithm based on the DCA and the incremental approach is designed to solve the CLR problem. We apply the Quasi-Newton method to solve the subproblems. The proposed algorithm is evaluated using several synthetic and real-world data sets for regression and compared with other algorithms for CLR. Results demonstrate that the DCA based algorithm is efficient for solving CLR problems with the large number of data points and in particular, outperforms other algorithms when the number of input variables is small. © 2020 Elsevier B.V.

Necessary and sufficient optimality conditions in DC semi-infinite programming

- Correa, Rafael, López, Marco, Pérez-Aros, Pedro

  • Authors: Correa, Rafael , López, Marco , Pérez-Aros, Pedro
  • Date: 2021
  • Type: Text , Journal article
  • Relation: SIAM Journal on Optimization Vol. 31, no. 1 (2021), p. 837-865
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  • Description: This paper deals with particular families of DC optimization problems involving suprema of convex functions. We show that the specific structure of this type of function allows us to cover a variety of problems in nonconvex programming. Necessary and sufficient optimality conditions for these families of DC optimization problems are established, where some of these structural features are conveniently exploited. More precisely, we derive necessary and sufficient conditions for (global and local) optimality in DC semi-infinite programming and DC cone-constrained optimization, under natural constraint qualifications. Finally, a penalty approach to DC abstract programming problems is developed in the last section. © 2021 Society for Industrial and Applied Mathematics

On a semismooth* Newton method for solving generalized equations

- Gfrerer, Helmut, Outrata, Jiri

  • Authors: Gfrerer, Helmut , Outrata, Jiri
  • Date: 2021
  • Type: Text , Journal article
  • Relation: SIAM Journal on Optimization Vol. 31, no. 1 (2021), p. 489-517
  • Relation: https://doi.org/10.1137/19M1257408
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  • Description: In the paper, a Newton-type method for the solution of generalized equations (GEs) is derived, where the linearization concerns both the single-valued and the multivalued part of the considered GE. The method is based on the new notion of semismoothness\ast, which, together with a suitable regularity condition, ensures the local superlinear convergence. An implementable version of the new method is derived for a class of GEs, frequently arising in optimization and equilibrium models. © 2021 Society for Industrial and Applied Mathematics

Primal necessary characterizations of transversality properties

- Cuong, Nguyen, Kruger, Alexander

  • Authors: Cuong, Nguyen , Kruger, Alexander
  • Date: 2021
  • Type: Text , Journal article
  • Relation: Positivity Vol. 25, no. 2 (2021), p. 531-558
  • Relation: http://purl.org/au-research/grants/arc/DP160100854
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  • Description: This paper continues the study of general nonlinear transversality properties of collections of sets and focuses on primal necessary (in some cases also sufficient) characterizations of the properties. We formulate geometric, metric and slope characterizations, particularly in the convex setting. The Hölder case is given a special attention. Quantitative relations between the nonlinear transversality properties of collections of sets and the corresponding regularity properties of set-valued mappings as well as two nonlinear transversality properties of a convex set-valued mapping to a convex set in the range space are discussed. © 2020, Springer Nature Switzerland AG.

Robust modelling of implicit interfaces by the scaled boundary finite element method

- Dsouza, Shaima, Pramod, A. L. N., Ooi, Ean Tat, Song, Chongming, Natarajan, Sundarajan

  • Authors: Dsouza, Shaima , Pramod, A. L. N. , Ooi, Ean Tat , Song, Chongming , Natarajan, Sundarajan
  • Date: 2021
  • Type: Text , Journal article
  • Relation: Engineering Analysis with Boundary Elements Vol. 124, no. (2021), p. 266-286
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  • Description: In this paper, we propose a robust framework based on the scaled boundary finite element method to model implicit interfaces in two-dimensional differential equations in nonhomegeneous media. The salient features of the proposed work are: (a) interfaces can be implicitly defined and need not conform to the background mesh; (b) Dirichlet boundary conditions can be imposed directly along the interface; (c) does not require special numerical integration technique to compute the bilinear and the linear forms and (d) can work with an efficient local mesh refinement using hierarchical background meshes. Numerical examples involving straight interface, circular interface and moving interface problems are solved to validate the proposed technique. Further, the presented technique is compared with conforming finite element method in terms of accuracy and convergence. From the numerical studies, it is seen that the proposed framework yields solutions whose error is O(h2) in L2 norm and O(h) in the H1 semi-norm. Further the condition number increases with the mesh size similar to the FEM. © 2021 Elsevier Ltd

The generalized bregman distance

- Burachik, Regina, Dao, Minh, Lindstrom, Scott

  • Authors: Burachik, Regina , Dao, Minh , Lindstrom, Scott
  • Date: 2021
  • Type: Text , Journal article
  • Relation: SIAM Journal on Optimization Vol. 31, no. 1 (2021), p. 404-424
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  • Description: Recently, a new kind of distance has been introduced for the graphs of two point-to-set operators, one of which is maximally monotone. When both operators are the subdifferential of a proper lower semicontinuous convex function, this kind of distance specializes under modest assumptions to the classical Bregman distance. We name this new kind of distance the generalized Bregman distance, and we shed light on it with examples that utilize the other two most natural representative functions: the Fitzpatrick function and its conjugate. We provide sufficient conditions for convexity, coercivity, and supercoercivity: properties which are essential for implementation in proximal point type algorithms. We establish these results for both the left and right variants of this new kind of distance. We construct examples closely related to the Kullback-Leibler divergence, which was previously considered in the context of Bregman distances and whose importance in information theory is well known. In so doing, we demonstrate how to compute a difficult Fitzpatrick conjugate function, and we discover natural occurrences of the Lambert \scrW function, whose importance in optimization is of growing interest. © 2021 Society for Industrial and Applied Mathematics

A note on primal-dual stability in infinite linear programming

- Goberna, Miguel, López, Marco, Ridolfi, Andrea, Vera de Serio, Virginia

  • Authors: Goberna, Miguel , López, Marco , Ridolfi, Andrea , Vera de Serio, Virginia
  • Date: 2020
  • Type: Text , Journal article
  • Relation: Optimization Letters Vol. 14, no. 8 (2020), p. 2247-2263
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  • Description: In this note we analyze the simultaneous preservation of the consistency (and of the inconsistency) of linear programming problems posed in infinite dimensional Banach spaces, and their corresponding dual problems, under sufficiently small perturbations of the data. We consider seven different scenarios associated with the different possibilities of perturbations of the data (the objective functional, the constraint functionals, and the right hand-side function), i.e., which of them are known, and remain fixed, and which ones can be perturbed because of their uncertainty. The obtained results allow us to give sufficient and necessary conditions for the coincidence of the optimal values of both problems and for the stability of the duality gap under the same type of perturbations. There appear substantial differences with the finite dimensional case due to the distinct topological properties of cones in finite and infinite dimensional Banach spaces. © 2020, Springer-Verlag GmbH Germany, part of Springer Nature.
  • Description: Funding details: Australian Research Council, ARC, DP180100602: http://purl.org/au-research/grants/arc/DP180100602

An existence result for quasi-equilibrium problems via Ekeland’s variational principle

- Cotrina, John, Théra, Michel, Zúñiga, Javier

  • Authors: Cotrina, John , Théra, Michel , Zúñiga, Javier
  • Date: 2020
  • Type: Text , Journal article
  • Relation: Journal of Optimization Theory and Applications Vol. 187, no. 2 (2020), p. 336-355
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  • Description: This paper deals with the existence of solutions to equilibrium and quasi-equilibrium problems without any convexity assumption. Coverage includes some equivalences to the Ekeland variational principle for bifunctions and basic facts about transfer lower continuity. An application is given to systems of quasi-equilibrium problems. © 2020, Springer Science+Business Media, LLC, part of Springer Nature.
  • Description: Research of M. Théra is supported by the Australian Research Council (ARC) Grant DP160100854 and benefited from the support of the FMJH Program PGMO and from the support of EDF. http://purl.org/au-research/grants/arc/DP160100854

An incremental nonsmooth optimization algorithm for clustering using L1 and L∞ norms

- Ordin, Burak, Bagirov, Adil, Mohebi, Ehsam

Assessing cohesion of the rocks proposing a new intelligent technique namely group method of data handling

- Chen, Wusi, Khandelwal, Manoj, Murlidhar, Bhatawdekar, Bui, Dieu, Tahir, Mahmood, Katebi, Javad

  • Authors: Chen, Wusi , Khandelwal, Manoj , Murlidhar, Bhatawdekar , Bui, Dieu , Tahir, Mahmood , Katebi, Javad
  • Date: 2020
  • Type: Text , Journal article
  • Relation: Engineering with Computers Vol. 36, no. 2 (2020), p. 783-793
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  • Description: In this study, evaluation and prediction of rock cohesion is assessed using multiple regression as well as group method of data handling (GMDH). It is a well-known fact that cohesion is the most crucial rock shear strength parameter, which is a key parameter for the stability evaluation of some geotechnical structures such as rock slope. To fulfill the aim of this study, a database of three model input parameters, i.e., p wave velocity, uniaxial compressive strength and Brazilian tensile strength and one model output, which is cohesion of limestone samples was prepared and utilized by GMDH. Different GMDH models with neurons and layers and selection pressure were tested and assessed. It was found that GMDH model number 4 (with 8 layers) shows the best performance among all of tested models between the input and output parameters for the prediction and assessment of rock cohesion with coefficient of determination (R2) values of 0.928 and 0.929, root mean square error values of 0.3545 and 0.3154 for training and testing datasets, respectively. Multiple regression analysis was also performed on the same database and R2 values were obtained as 0.8173 and 0.8313 between input and output parameters for the training and testing of the models, respectively. The GMDH technique developed in this study is introduced as a new model in field of rock shear strength parameters. © 2019, Springer-Verlag London Ltd., part of Springer Nature.

Comparative approaches to probabilistic finite element methods for slope stability analysis

- Dyson, Ashley, Tolooiyan, Ali

  • Authors: Dyson, Ashley , Tolooiyan, Ali
  • Date: 2020
  • Type: Text , Journal article
  • Relation: Simulation Modelling Practice and Theory Vol. 100, no. (2020), p.
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  • Description: Probabilistic slope stability analyses are often preferable to deterministic methods when soils are inherently heterogeneous, or the reliability of geotechnical parameters is largely unknown. These methods are suitable for evaluating the risk of slope failure by producing a range of potential scenarios for the slope stability factor of safety. Several probabilistic methods including the Point Estimate Method, Monte Carlo Method and Random Finite Element Method, can be combined with the Finite Element technique. In this study, various shear strength distributions are considered for three different probabilistic Finite Element Methods to determine Factor of Safety and Probability of Failure distributions, based on the associated method of slope stability analysis. Results are presented for a case study of an Australian open-cut coal mine, with a range of shear strength parameter distributions for coal and interseam cohesive materials considered. Coal and interseam shear strength parameters are varied independently, to determine the effects of each material on the slope Factor of Safety. © 2019 Elsevier B.V.

Design and optimisation of drainage systems for fractured slopes using the XFEM and FEM

- Shaghaghi, Tahereh, Ghadrdan, Mohsen, Tolooiyan, Ali

  • Authors: Shaghaghi, Tahereh , Ghadrdan, Mohsen , Tolooiyan, Ali
  • Date: 2020
  • Type: Text , Journal article
  • Relation: Simulation Modelling Practice and Theory Vol. 103, no. (2020), p.
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  • Description: The reliable and optimised design of a drainage system for saturated slopes is often a challenging geotechnical task. Such a task becomes even more challenging when a slope contains pre-existing joints and discontinuities. In saturated and semi-saturated conditions, the existence of joints may lead to a complex distribution of pore water pressure within the slope, affecting the effective stress distribution and the stability of the slope. This paper aims to study the effect of horizontal borehole drainage systems with different arrangements on pore water pressure distributions within a saturated fractured slope. In this study, several coupled pore fluid diffusion and stress-strain analyses were conducted using the e-Xtended Finite Element Method (XFEM) in conjunction with the Finite Element Method (FEM) to simulate the efficiency of a drainage system of a deep slope at the second largest open-cut mine in Australia. As one of the objectives of this study, the effect of water flow inside a joint and normal to the joint surface (normal flow) is considered as an essential simulation component. The results show that the pore water pressure distribution at the vicinity of the joint is considerably influenced by the magnitude of normal flow. Such influence should be taken into account when designing a drainage system, as the magnitude of normal flow and the performance of the drainage system may affect each other directly. © 2020 Elsevier B.V.

Dual sufficient characterizations of transversality properties

- Cuong, Nguyen, Kruger, Alexander

  • Authors: Cuong, Nguyen , Kruger, Alexander
  • Date: 2020
  • Type: Text , Journal article
  • Relation: Positivity Vol. 24, no. 5 (2020), p. 1313-1359
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  • Description: This paper continues the study of ‘good arrangements’ of collections of sets near a point in their intersection. Our aim is to develop a general scheme for quantitative analysis of several transversality properties within the same framework. We consider a general nonlinear setting and establish dual (subdifferential and normal cone) sufficient characterizations of transversality properties of collections of sets in Banach/Asplund spaces. Besides quantitative estimates for the rates/moduli of the corresponding properties, we establish here also estimates for the other parameters involved in the definitions, particularly the size of the neighbourhood where a property holds. Interpretations of the main general nonlinear characterizations for the case of Hölder transversality are provided. Some characterizations are new even in the linear setting. As an application, we provide dual sufficient conditions for nonlinear extensions of the new transversality properties of a set-valued mapping to a set in the range space due to Ioffe. © 2020, Springer Nature Switzerland AG.
  • Description: The research was supported by the Australian Research Council, Project DP160100854, and the European Union’s Horizon 2020 research and innovation programme under the Marie Sk

Early-stage reciprocity in sustainable scientific collaboration

- Wang, Wei, Ren, Jing, Alrashoud, Mubarak, Xia, Feng, Mao, Mengyi, Tolba, Amr

  • Authors: Wang, Wei , Ren, Jing , Alrashoud, Mubarak , Xia, Feng , Mao, Mengyi , Tolba, Amr
  • Date: 2020
  • Type: Text , Journal article
  • Relation: Journal of Informetrics Vol. 14, no. 3 (2020), p.
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  • Description: Scientific collaboration is of significant importance in tackling grand challenges and breeding innovations. Despite the increasing interest in investigating and promoting scientific collaborations, we know little about the collaboration sustainability as well as mechanisms behind it. In this paper, we set out to study the relationships between early-stage reciprocity and collaboration sustainability. By proposing and defining h-index reciprocity, we give a comprehensive statistical analysis on how reciprocity influences scientific collaboration sustainability, and find that scholars are not altruism and the key to sustainable collaboration is fairness. The unfair h-index reciprocity has an obvious negative impact on collaboration sustainability. The bigger the reciprocity difference, the less sustainable in collaboration. This work facilitates understanding sustainable collaborations and thus will benefit both individual scholar in optimizing collaboration strategies and the whole academic society in improving teamwork efficiency. © 2020 Elsevier Ltd.
  • Description: The authors extend their appreciation to the International Scientific Partnership Program ISPP at King Saud University for funding this research work through ISPP-78. This work is partially supported by China Postdoctoral Science Foundation ( 2019M651115 ).

Effect of rock mass permeability and rock fracture leak-off coefficient on the pore water pressure distribution in a fractured slope

- Shaghaghi, Tahereh, Ghadrdan, Mohsen, Tolooiyan, Ali

  • Authors: Shaghaghi, Tahereh , Ghadrdan, Mohsen , Tolooiyan, Ali
  • Date: 2020
  • Type: Text , Journal article
  • Relation: Simulation Modelling Practice and Theory Vol. 105, no. (2020), p. 1-13
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  • Description: The reliable assessment of the stability of saturated slopes becomes a challenging task when slopes are consisting of discontinuous materials and containing pre-existing joints. The discontinuous nature of the slopes' material could increase the overall permeability of the slope, while existing joints facilitate groundwater leakage through the joint surfaces into the slope which subsequently exerts a major impact on deformation and the effective stress distribution. This paper aims to study the Pore Water Pressure (PWP) distribution changes in a saturated fractured slope by conducting advanced coupled pore fluid diffusion and stress-strain analyses, while investigating the sensitivity of results to the variation of permeability and leakage properties of fracture surfaces. Modelling of jointed slopes is carried out using the e-Xtended Finite Element Method (XFEM) in conjunction with the Finite Element Method (FEM). In this study, the fluid flow inside the joint is the major focus at which the constitutive response of the fluid inside the joint considers both tangential and normal flows. To demonstrate the state-of-the-art simulation technique presented in this paper, simulation of a fractured slope at the second largest open-pit mine in Australia is performed as a case study. This study shows the effect of a variable leak-off coefficient of the joint surfaces and the permeability magnitude on the pore water pressure distribution.
  • Description: This research has been supported financially by the Earth Resources Regulation of the Victorian State Government Department of Economic Development, Jobs, Transport and Resources. The first and second authors are funded by the GHERG LV Batter Stability Project Scholarship and Faculty Tuition Scholarship of Federation University Australia.

Effects of a proper feature selection on prediction and optimization of drilling rate using intelligent techniques

- Liao, Xiufeng, Khandelwal, Manoj, Yang, Haiqing, Koopialipoor, Mohammadreza, Murlidhar, Bhatawdekar

  • Authors: Liao, Xiufeng , Khandelwal, Manoj , Yang, Haiqing , Koopialipoor, Mohammadreza , Murlidhar, Bhatawdekar
  • Date: 2020
  • Type: Text , Journal article
  • Relation: Engineering with Computers Vol. 36, no. 2 (Apr 2020), p. 499-510
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  • Description: One of the important factors during drilling times is the rate of penetration (ROP), which is controlled based on different variables. Factors affecting different drillings are of paramount importance. In the current research, an attempt was made to better recognize drilling parameters and optimize them based on an optimization algorithm. For this purpose, 618 data sets, including RPM, flushing media, and compressive strength parameters, were measured and collected. After an initial investigation, the compressive strength feature of samples, which is an important parameter from the rocks, was used as a proper criterion for classification. Then using intelligent systems, three different levels of the rock strength and all data were modeled. The results showed that systems which were classified based on compressive strength showed a better performance for ROP assessment due to the proximity of features. Therefore, these three levels were used for classification. A new artificial bee colony algorithm was used to solve this problem. Optimizations were applied to the selected models under different optimization conditions, and optimal states were determined. As determining drilling machine parameters is important, these parameters were determined based on optimal conditions. The obtained results showed that this intelligent system can well improve drilling conditions and increase the ROP value for three strength levels of the rocks. This modeling system can be used in different drilling operations.