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Minimizing nonsmooth DC functions via successive DC piecewise-affine approximations

- Gaudioso, Manlio, Giallombardo, Giovanni, Miglionico, Giovanna, Bagirov, Adil

**Authors:**Gaudioso, Manlio , Giallombardo, Giovanni , Miglionico, Giovanna , Bagirov, Adil**Date:**2018**Type:**Text , Journal article**Relation:**Journal of Global Optimization Vol. 71, no. 1 (2018), p. 37-55**Full Text:**false**Reviewed:****Description:**We introduce a proximal bundle method for the numerical minimization of a nonsmooth difference-of-convex (DC) function. Exploiting some classic ideas coming from cutting-plane approaches for the convex case, we iteratively build two separate piecewise-affine approximations of the component functions, grouping the corresponding information in two separate bundles. In the bundle of the first component, only information related to points close to the current iterate are maintained, while the second bundle only refers to a global model of the corresponding component function. We combine the two convex piecewise-affine approximations, and generate a DC piecewise-affine model, which can also be seen as the pointwise maximum of several concave piecewise-affine functions. Such a nonconvex model is locally approximated by means of an auxiliary quadratic program, whose solution is used to certify approximate criticality or to generate a descent search-direction, along with a predicted reduction, that is next explored in a line-search setting. To improve the approximation properties at points that are far from the current iterate a supplementary quadratic program is also introduced to generate an alternative more promising search-direction. We discuss the main convergence issues of the line-search based proximal bundle method, and provide computational results on a set of academic benchmark test problems. © 2017, Springer Science+Business Media, LLC.

An algorithm for the estimation of a regression function by continuous piecewise linear functions

- Bagirov, Adil, Clausen, Conny, Kohler, Michael

**Authors:**Bagirov, Adil , Clausen, Conny , Kohler, Michael**Date:**2008**Type:**Text , Journal article**Relation:**Computational Optimization and Applications Vol. 45, no. (2008), p. 159-179**Relation:**http://purl.org/au-research/grants/arc/DP0666061**Full Text:****Reviewed:****Description:**The problem of the estimation of a regression function by continuous piecewise linear functions is formulated as a nonconvex, nonsmooth optimization problem. Estimates are defined by minimization of the empirical L 2 risk over a class of functions, which are defined as maxima of minima of linear functions. An algorithm for finding continuous piecewise linear functions is presented. We observe that the objective function in the optimization problem is semismooth, quasidifferentiable and piecewise partially separable. The use of these properties allow us to design an efficient algorithm for approximation of subgradients of the objective function and to apply the discrete gradient method for its minimization. We present computational results with some simulated data and compare the new estimator with a number of existing ones.**Description:**The problem of the estimation of a regression function by continuous piecewise linear functions is formulated as a nonconvex, nonsmooth optimization problem. Estimates are defined by minimization of the empirical L 2 risk over a class of functions, which are defined as maxima of minima of linear functions. An algorithm for finding continuous piecewise linear functions is presented. We observe that the objective function in the optimization problem is semismooth, quasidifferentiable and piecewise partially separable. The use of these properties allow us to design an efficient algorithm for approximation of subgradients of the objective function and to apply the discrete gradient method for its minimization. We present computational results with some simulated data and compare the new estimator with a number of existing ones. © 2008 Springer Science+Business Media, LLC.

New constructions of A-magic graphs using labeling matrices

- Sugeng, Kiki Ariyanti, Miller, Mirka

**Authors:**Sugeng, Kiki Ariyanti , Miller, Mirka**Date:**2008**Type:**Text , Journal article**Relation:**Journal of combinatorial mathematics and combinatorial computing Vol. 65, no. (May 2008), p. 147-151**Full Text:**false**Reviewed:**

An optimization approach to the study of drug-drug interactions

- Mammadov, Musa, Banerjee, Arunava

**Authors:**Mammadov, Musa , Banerjee, Arunava**Date:**2005**Type:**Text , Conference paper**Relation:**Paper pesented at Sixteenth Australasian Workshop on Combinatorial Algorithms, AWOCA 2005, Ballarat, Victoria : 18th-21st September 2005 p. 201-216**Full Text:****Description:**Drug-drug interaction is one of the important problems of Adverse Drug Reaction (ADR). In this paper we develop an optimization approach for the study of this problem. This approach is based on drug-reaction relationships represented in the form of a vector of weights, which can be defined as a solution to some global optimization problem. Although this approach can be used for solving many ADR problems, we concentrate here only on drug-drug interactions. Based on drug-reaction relationships, we formulate this problem as an optimization problem. The approach is applied to different classes of reactions from the Australian Adverse Drug Reaction Advisory Committee (ADRAC) database.**Description:**2003001384

**Authors:**Mammadov, Musa , Banerjee, Arunava**Date:**2005**Type:**Text , Conference paper**Relation:**Paper pesented at Sixteenth Australasian Workshop on Combinatorial Algorithms, AWOCA 2005, Ballarat, Victoria : 18th-21st September 2005 p. 201-216**Full Text:****Description:**Drug-drug interaction is one of the important problems of Adverse Drug Reaction (ADR). In this paper we develop an optimization approach for the study of this problem. This approach is based on drug-reaction relationships represented in the form of a vector of weights, which can be defined as a solution to some global optimization problem. Although this approach can be used for solving many ADR problems, we concentrate here only on drug-drug interactions. Based on drug-reaction relationships, we formulate this problem as an optimization problem. The approach is applied to different classes of reactions from the Australian Adverse Drug Reaction Advisory Committee (ADRAC) database.**Description:**2003001384

Solving second-order conic systems with variable precision

- Cucker, Felipe, Peña, Javier, Roshchina, Vera

**Authors:**Cucker, Felipe , Peña, Javier , Roshchina, Vera**Date:**2014**Type:**Text , Journal article**Relation:**Mathematical Programming Vol. 150, no. 2 (2014), p. 217-250**Full Text:**false**Reviewed:****Description:**We describe and analyze an interior-point method to decide feasibility problems of second-order conic systems. A main feature of our algorithm is that arithmetic operations are performed with finite precision. Bounds for both the number of arithmetic operations and the finest precision required are exhibited. © 2014, Springer-Verlag Berlin Heidelberg and Mathematical Optimization Society.

Gradient-free method for nonsmooth distributed optimization

- Li, Jueyou, Wu, Changzhi, Wu, Zhiyou, Long, Qiang

**Authors:**Li, Jueyou , Wu, Changzhi , Wu, Zhiyou , Long, Qiang**Date:**2014**Type:**Text , Journal article**Relation:**Journal of Global Optimization Vol.61, no.2 (March 2014), p.325-340**Full Text:****Reviewed:****Description:**In this paper, we consider a distributed nonsmooth optimization problem over a computational multi-agent network. We first extend the (centralized) Nesterov’s random gradient-free algorithm and Gaussian smoothing technique to the distributed case. Then, the convergence of the algorithm is proved. Furthermore, an explicit convergence rate is given in terms of the network size and topology. Our proposed method is free of gradient, which may be preferred by practical engineers. Since only the cost function value is required, our method may suffer a factor up to d (the dimension of the agent) in convergence rate over that of the distributed subgradient-based methods in theory. However, our numerical simulations show that for some nonsmooth problems, our method can even achieve better performance than that of subgradient-based methods, which may be caused by the slow convergence in the presence of subgradient.

**Authors:**Li, Jueyou , Wu, Changzhi , Wu, Zhiyou , Long, Qiang**Date:**2014**Type:**Text , Journal article**Relation:**Journal of Global Optimization Vol.61, no.2 (March 2014), p.325-340**Full Text:****Reviewed:****Description:**In this paper, we consider a distributed nonsmooth optimization problem over a computational multi-agent network. We first extend the (centralized) Nesterov’s random gradient-free algorithm and Gaussian smoothing technique to the distributed case. Then, the convergence of the algorithm is proved. Furthermore, an explicit convergence rate is given in terms of the network size and topology. Our proposed method is free of gradient, which may be preferred by practical engineers. Since only the cost function value is required, our method may suffer a factor up to d (the dimension of the agent) in convergence rate over that of the distributed subgradient-based methods in theory. However, our numerical simulations show that for some nonsmooth problems, our method can even achieve better performance than that of subgradient-based methods, which may be caused by the slow convergence in the presence of subgradient.

Applying the canonical dual theory in optimal control problems

- Zhu, Jinghao, Wu, Dan, Gao, David

**Authors:**Zhu, Jinghao , Wu, Dan , Gao, David**Date:**2012**Type:**Text , Journal article**Relation:**Journal of global optimization Vol. 54, no. 2 (2012), p. 221-233**Full Text:**false**Reviewed:****Description:**This paper presents some applications of the canonical dual theory in optimal control problems. The analytic solutions of several nonlinear and nonconvex problems are investigated by global optimizations. It turns out that the backward differential flow defined by the KKT equation may reach the globally optimal solution. The analytic solution to an optimal control problem is obtained via the expression of the co-state. Some examples are illustrated.

Analysis of design structure matrix methods in design process improvement

**Authors:**Gunawan, Indra**Date:**2012**Type:**Text , Journal article**Relation:**International Journal of Modelling and Simulation Vol. 32, no. 2 (2012), p. 95-103**Full Text:**false**Reviewed:****Description:**In this paper, design structure matrix (DSM) methods are presented to manage design iteration or rework which is inherent in the design process. Three DSM methods: path searching, powers of the adjacency matrix, and reachability matrix methods are discussed. Their advantages and disadvantages with respect to the project scope are summarized. As a case study, DSM methods are implemented to reduce the design iteration or rework in a complex engineering project. The main advantage of the DSM methods over traditional project management tools such as CPM or Gantt chart is in compactness and ability to present an organized and eﬃcient mapping among tasks that is clear and easy to read regardless of size.

A proximal bundle method for nonsmooth DC optimization utilizing nonconvex cutting planes

- Joki, Kaisa, Bagirov, Adil, Karmitsa, Napsu, Makela, Marko

**Authors:**Joki, Kaisa , Bagirov, Adil , Karmitsa, Napsu , Makela, Marko**Date:**2017**Type:**Text , Journal article**Relation:**Journal of Global Optimization Vol. 68, no. 3 (2017), p. 501-535**Relation:**http://purl.org/au-research/grants/arc/DP140103213**Full Text:**false**Reviewed:****Description:**In this paper, we develop a version of the bundle method to solve unconstrained difference of convex (DC) programming problems. It is assumed that a DC representation of the objective function is available. Our main idea is to utilize subgradients of both the first and second components in the DC representation. This subgradient information is gathered from some neighborhood of the current iteration point and it is used to build separately an approximation for each component in the DC representation. By combining these approximations we obtain a new nonconvex cutting plane model of the original objective function, which takes into account explicitly both the convex and the concave behavior of the objective function. We design the proximal bundle method for DC programming based on this new approach and prove the convergence of the method to an -critical point. The algorithm is tested using some academic test problems and the preliminary numerical results have shown the good performance of the new bundle method. An interesting fact is that the new algorithm finds nearly always the global solution in our test problems.

Nonmeasurable subgroups of compact groups

- Hernández, Salvador, Hofmann, Karl, Morris, Sidney

**Authors:**Hernández, Salvador , Hofmann, Karl , Morris, Sidney**Date:**2016**Type:**Text , Journal article**Relation:**Journal of Group Theory Vol. 19, no. 1 (2016), p. 179-189**Full Text:****Reviewed:****Description:**In 1985 S. Saeki and K. Stromberg published the following question: Does every infinite compact group have a subgroup which is not Haar measurable? An affirmative answer is given for all compact groups with the exception of some metric profinite groups which are almost perfect and strongly complete. In this spirit it is also shown that every compact group contains a non-Borel subgroup. © 2016 by De Gruyter 2016 Generalitat Valenciana PROMETEO/2014/062 We are grateful for our referee's useful comments. In particular, the suggestion that originally we had overlooked [Pacific J. Math. 116 (1985), 217-241] shortened the proof of Theorem 4.3 considerably.

**Authors:**Hernández, Salvador , Hofmann, Karl , Morris, Sidney**Date:**2016**Type:**Text , Journal article**Relation:**Journal of Group Theory Vol. 19, no. 1 (2016), p. 179-189**Full Text:****Reviewed:****Description:**In 1985 S. Saeki and K. Stromberg published the following question: Does every infinite compact group have a subgroup which is not Haar measurable? An affirmative answer is given for all compact groups with the exception of some metric profinite groups which are almost perfect and strongly complete. In this spirit it is also shown that every compact group contains a non-Borel subgroup. © 2016 by De Gruyter 2016 Generalitat Valenciana PROMETEO/2014/062 We are grateful for our referee's useful comments. In particular, the suggestion that originally we had overlooked [Pacific J. Math. 116 (1985), 217-241] shortened the proof of Theorem 4.3 considerably.

The new robust conic GPLM method with an application to finance : prediction of credit default

- Özmen, Ay, Weber, Gerhard-Wilhelm, Çavu, Defterli, Özlem

**Authors:**Özmen, Ay , Weber, Gerhard-Wilhelm , Çavu , Defterli, Özlem**Date:**2012**Type:**Text , Journal article**Relation:**Journal of Global Optimization Vol.56, no. 2 (2012), p. 233–249**Full Text:**false**Reviewed:****Description:**This paper contributes to classification and identification in modern finance through advanced optimization. In the last few decades, financial misalignments and, thereby, financial crises have been increasing in numbers due to the rearrangement of the financial world. In this study, as one of the most remarkable of these, countries' debt crises, which result from illiquidity, are tried to predict with some macroeconomic variables. The methodology consists of a combination of two predictive regression models, logistic regression and robust conic multivariate adaptive regression splines (RCMARS), as linear and nonlinear parts of a generalized partial linear model. RCMARS has an advantage of coping with the noise in both input and output data and of obtaining more consistent optimization results than CMARS. An advanced version of conic generalized partial linear model which includes robustification of the data set is introduced: robust conic generalized partial linear model (RCGPLM). This new model is applied on a data set that belongs to 45 emerging markets with 1,019 observations between the years 1980 and 2005. © 2012 Springer Science+Business Media, LLC.

On antimagic labelings of disjoint union of complete s-partite graphs

- Dafik, Miller, Mirka, Ryan, Joe, Baca, Martin

**Authors:**Dafik , Miller, Mirka , Ryan, Joe , Baca, Martin**Date:**2008**Type:**Text , Journal article**Relation:**Journal of combinatorial mathematics and combinatorial computing Vol. 65, no. (May 2008 2008), p. 41-49**Full Text:****Reviewed:****Description:**By an (a, d)-edge-antimagic total labeling of a graph G(V, E) we mean a bijective function f from V(G) u E(G) onto the set. { 1, 2, ... ,ǀV(C)ǀ+IE(G)I} such that the set of all the edge-weights, w(uv) ,.... f(u) + f(uv) + f(v), uv C E (G), is {a, a+ d, a+ 2d, . . . , a + (lE(G)I-1)d}, for two integers a > 0 and d

**Authors:**Dafik , Miller, Mirka , Ryan, Joe , Baca, Martin**Date:**2008**Type:**Text , Journal article**Relation:**Journal of combinatorial mathematics and combinatorial computing Vol. 65, no. (May 2008 2008), p. 41-49**Full Text:****Reviewed:****Description:**By an (a, d)-edge-antimagic total labeling of a graph G(V, E) we mean a bijective function f from V(G) u E(G) onto the set. { 1, 2, ... ,ǀV(C)ǀ+IE(G)I} such that the set of all the edge-weights, w(uv) ,.... f(u) + f(uv) + f(v), uv C E (G), is {a, a+ d, a+ 2d, . . . , a + (lE(G)I-1)d}, for two integers a > 0 and d

- 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**Full Text:**false**Reviewed:****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.

Proceedings of the Sixteenth Australasian Workshop on Combinatorial Algorithms (AWOCA 2005)

- Ryan, Joe, Manyem, Prabhu, Sugeng, Kiki Ariyanti, Miller, Mirka

**Authors:**Ryan, Joe , Manyem, Prabhu , Sugeng, Kiki Ariyanti , Miller, Mirka**Date:**2005**Type:**Text , Conference proceedings**Full Text:**false

All (k;g)-cages are edge-superconnected

- Lin, Yuqing, Miller, Mirka, Balbuena, Camino, Marcote, Xavier

**Authors:**Lin, Yuqing , Miller, Mirka , Balbuena, Camino , Marcote, Xavier**Date:**2006**Type:**Text , Journal article**Relation:**Networks Vol. 47, no. 2 (2006), p. 102-110**Full Text:**false**Reviewed:****Description:**A (k;g)-cage is k-regular graph with girth g and with the least possible number of vertices. In this article we prove that (k;g)-cages are edge-superconnected if g is even. Earlier, Marcote and Balbuena proved that (k;g)-cages are edge-superconnected if g is odd [Networks 43 (2004), 54-59]. Combining our results, we conclude that all (k;g)-cages are edge-superconnected. © 2005 Wiley Periodicals, Inc.**Description:**C1**Description:**2003001830

Solving DC programs using the cutting angle method

- Ferrer, Albert, Bagirov, Adil, Beliakov, Gleb

**Authors:**Ferrer, Albert , Bagirov, Adil , Beliakov, Gleb**Date:**2015**Type:**Text , Journal article**Relation:**Journal of Global Optimization Vol. 61, no. 1 (2015), p. 71-89**Relation:**http://purl.org/au-research/grants/arc/DP140103213**Full Text:**false**Reviewed:****Description:**In this paper, we propose a new algorithm for global minimization of functions represented as a difference of two convex functions. The proposed method is a derivative free method and it is designed by adapting the extended cutting angle method. We present preliminary results of numerical experiments using test problems with difference of convex objective functions and box-constraints. We also compare the proposed algorithm with a classical one that uses prismatical subdivisions.

On the nonexistence of graphs of diameter 2 and defect 2

- Miller, Mirka, Nguyen, Minh Hoang, Pineda-Villavicencio, Guillermo

**Authors:**Miller, Mirka , Nguyen, Minh Hoang , Pineda-Villavicencio, Guillermo**Date:**2009**Type:**Text , Journal article**Relation:**The Journal of Combinatorial Mathematics and Combinatorial Computing Vol. 71, no. (2009), p. 5-20**Full Text:**false**Reviewed:****Description:**In 1960, Hoffman and Singleton investigated the existence of Moore graphs of diameter 2 (graphs of maximum degree d and d² + 1 vertices), and found that such graphs exist only for d = 2; 3; 7 and possibly 57. In 1980, Erdös et al., using eigenvalue analysis, showed that, with the exception of C4, there are no graphs of diameter 2, maximum degree d and d² vertices. In this paper, we show that graphs of diameter 2, maximum degree d and d² - 1 vertices do not exist for most values of d with d ≥ 6, and conjecture that they do not exist for any d ≥ 6.**Description:**2003007893

**Authors:**Marshall, Kim , Ryan, Joe**Date:**2008**Type:**Text , Journal article**Relation:**Journal of Combinatorial Mathematics and Combinatorial Computing Vol. 65, no. (May 2008 2008), p. 51-60**Full Text:**false**Reviewed:****Description:**The term mode graph was introduced by Boland, Kauffman and Panroug [2] to defiue a connected graph G such that, for every pair of vertices v, w in G, the number of vertices with eccentricity e(v) is equal to the number of vertices with eccentricity e(w). As a natural extension to this work, the concept of an antimode graph was introduced to describe a graph for which if e(v) ≠ e(w) then the number of vertices with eccentricity e(v) is not equal to the number of vertices with eccentricity e(w). ln this paper we determine the existence of some classes of antimode graphs, namely equisequential and (a, d)-antimode graphs.

Non-convex quadratic minimization problems with quadratic constraints: Global optimality conditions

- Jeyakumar, Vaithilingam, Rubinov, Alex, Wu, Zhiyou

**Authors:**Jeyakumar, Vaithilingam , Rubinov, Alex , Wu, Zhiyou**Date:**2007**Type:**Text , Journal article**Relation:**Mathematical Programming Vol. 110, no. 3 (2007), p. 521-541**Full Text:**false**Reviewed:****Description:**In this paper, we first examine how global optimality of non-convex constrained optimization problems is related to Lagrange multiplier conditions. We then establish Lagrange multiplier conditions for global optimality of general quadratic minimization problems with quadratic constraints. We also obtain necessary global optimality conditions, which are different from the Lagrange multiplier conditions for special classes of quadratic optimization problems. These classes include weighted least squares with ellipsoidal constraints, and quadratic minimization with binary constraints. We discuss examples which demonstrate that our optimality conditions can effectively be used for identifying global minimizers of certain multi-extremal non-convex quadratic optimization problems. © Springer-Verlag 2007.**Description:**C1

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.**Full Text:**false**Reviewed:****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.

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