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26Bagirov, Adil
24Gao, David
22Kruger, Alexander
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11Outrata, Jiri
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A counterexample to De Pierro's conjecture on the convergence of under-relaxed cyclic projections

- Cominetti, Roberto, Roshchina, Vera, Williamson, Andrew

**Authors:**Cominetti, Roberto , Roshchina, Vera , Williamson, Andrew**Date:**2019**Type:**Text , Journal article**Relation:**Optimization Vol. 68, no. 1 (2019), p. 3-12**Full Text:****Reviewed:****Description:**The convex feasibility problem consists in finding a point in the intersection of a finite family of closed convex sets. When the intersection is empty, a best compromise is to search for a point that minimizes the sum of the squared distances to the sets. In 2001, de Pierro conjectured that the limit cycles generated by the ε-under-relaxed cyclic projection method converge when ε ↓ 0 towards a least squares solution. While the conjecture has been confirmed under fairly general conditions, we show that it is false in general by constructing a system of three compact convex sets in R3 for which the ε-under-relaxed cycles do not converge. © 2018 Informa UK Limited, trading as Taylor & Francis Group.

**Authors:**Cominetti, Roberto , Roshchina, Vera , Williamson, Andrew**Date:**2019**Type:**Text , Journal article**Relation:**Optimization Vol. 68, no. 1 (2019), p. 3-12**Full Text:****Reviewed:****Description:**The convex feasibility problem consists in finding a point in the intersection of a finite family of closed convex sets. When the intersection is empty, a best compromise is to search for a point that minimizes the sum of the squared distances to the sets. In 2001, de Pierro conjectured that the limit cycles generated by the ε-under-relaxed cyclic projection method converge when ε ↓ 0 towards a least squares solution. While the conjecture has been confirmed under fairly general conditions, we show that it is false in general by constructing a system of three compact convex sets in R3 for which the ε-under-relaxed cycles do not converge. © 2018 Informa UK Limited, trading as Taylor & Francis Group.

Characterizations of nonsmooth robustly quasiconvex functions

- Bui, Hoa, Khanh, Pham, Tran, Thi

**Authors:**Bui, Hoa , Khanh, Pham , Tran, Thi**Date:**2019**Type:**Text , Journal article**Relation:**Journal of Optimization Theory and Applications Vol. 180, no. 3 (2019), p. 775-786**Full Text:****Reviewed:****Description:**Two criteria for the robust quasiconvexity of lower semicontinuous functions are established in terms of Fréchet subdifferentials in Asplund spaces. The first criterion extends to such spaces a result established by Barron et al. (Discrete Contin Dyn Syst Ser B 17:1693–1706, 2012). The second criterion is totally new even if it is applied to lower semicontinuous functions on finite-dimensional spaces. © 2018, Springer Science+Business Media, LLC, part of Springer Nature.

**Authors:**Bui, Hoa , Khanh, Pham , Tran, Thi**Date:**2019**Type:**Text , Journal article**Relation:**Journal of Optimization Theory and Applications Vol. 180, no. 3 (2019), p. 775-786**Full Text:****Reviewed:****Description:**Two criteria for the robust quasiconvexity of lower semicontinuous functions are established in terms of Fréchet subdifferentials in Asplund spaces. The first criterion extends to such spaces a result established by Barron et al. (Discrete Contin Dyn Syst Ser B 17:1693–1706, 2012). The second criterion is totally new even if it is applied to lower semicontinuous functions on finite-dimensional spaces. © 2018, Springer Science+Business Media, LLC, part of Springer Nature.

- Cibulka, Radek, Fabian, Marian, Kruger, Alexander

**Authors:**Cibulka, Radek , Fabian, Marian , Kruger, Alexander**Date:**2019**Type:**Text , Journal article**Relation:**Journal of Mathematical Analysis and Applications Vol. 473, no. 2 (2019), p. 811-836**Full Text:**false**Reviewed:****Description:**There are two basic ways of weakening the definition of the well-known metric regularity property by fixing one of the points involved in the definition. The first resulting property is called metric subregularity and has attracted a lot of attention during the last decades. On the other hand, the latter property which we call semiregularity can be found under several names and the corresponding results are scattered in the literature. We provide a self-contained material gathering and extending the existing theory on the topic. We demonstrate a clear relationship with other regularity properties, for example, the equivalence with the so-called openness with a linear rate at the reference point is shown. In particular cases, we derive necessary and/or sufficient conditions of both primal and dual type. We illustrate the importance of semiregularity in the convergence analysis of an inexact Newton-type scheme for generalized equations with not necessarily differentiable single-valued part. © 2019 Elsevier Inc.

The Demyanov–Ryabova conjecture is false

**Authors:**Roshchina, Vera**Date:**2019**Type:**Text , Journal article**Relation:**Optimization Letters Vol. 13, no. 1 (2019), p. 227-234**Full Text:**false**Reviewed:****Description:**It was conjectured by Demyanov and Ryabova (Discrete Contin Dyn Syst 31(4):1273–1292, 2011) that the minimal cycle in the sequence obtained via repeated application of the Demyanov converter to a finite family of polytopes is at most two. We construct a counterexample for which the minimal cycle has length 4.

- Song, Chongmin, Ooi, Ean Tat, Pramod, Aladurthi, Natarajan, Sundararajan

**Authors:**Song, Chongmin , Ooi, Ean Tat , Pramod, Aladurthi , Natarajan, Sundararajan**Date:**2018**Type:**Text , Journal article**Relation:**Engineering Analysis with Boundary Elements Vol. 94, no. (2018), p. 10-24**Full Text:**false**Reviewed:****Description:**In this paper, an adaptive refinement strategy based on the scaled boundary finite element method on quadtree meshes for linear elasticity problems is discussed. Within this framework, the elements with hanging nodes are treated as polygonal elements and thus does not require special treatment. The adaptive refinement is supplemented with a novel error indicator. The local error is estimated directly from the solution of the scaled boundary governing equations. The salient feature is that it does not require any stress recovery techniques. The efficacy and the robustness of the proposed approach are demonstrated with a few numerical examples.

- Chen, Xiaojun, Luo, Tao, Ooi, Ean Tat, Ooi, Ean Hin, Song, Chongmin

**Authors:**Chen, Xiaojun , Luo, Tao , Ooi, Ean Tat , Ooi, Ean Hin , Song, Chongmin**Date:**2018**Type:**Text , Journal article**Relation:**Theoretical and Applied Fracture Mechanics Vol. 94, no. (2018), p. 120-133**Full Text:**false**Reviewed:****Description:**This paper presents a method to improve the computational efficiency of the scaled boundary finite element formulation for functionally graded materials. Both isotropic and orthotropic functionally graded materials are considered. This is achieved using a combination of quadtree and polygon meshes. This hybrid meshing approach is particularly suitable to be used with the SBFEM for functionally graded materials because of the significant amount of calculations required to compute the stiffness matrices of the polygons/cells in the mesh. When a quadtree structure is adopted, most of the variables required for the numerical simulation can be pre-computed and stored in the memory, retrieved and scaled as required during the computations, leading to an efficient method for crack propagation modeling. The scaled boundary finite element formulation enables accurate computation of the stress intensity factors directly from the stress solutions without any special post-processing techniques or local mesh refinement in the vicinity of the crack tip. Numerical benchmarks demonstrate the efficiency of the proposed method as opposed to using a purely polygon-mesh based approach. © 2018 Elsevier Ltd

Double bundle method for finding clarke stationary points in nonsmooth dc programming

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

**Authors:**Joki, Kaisa , Bagirov, Adil , Karmitsa, Napsu , Makela, Marko , Taheri, Sona**Date:**2018**Type:**Text , Journal article**Relation:**SIAM Journal on Optimization Vol. 28, no. 2 (2018), p. 1892-1919**Relation:**http://purl.org/au-research/grants/arc/DP140103213**Full Text:****Reviewed:****Description:**The aim of this paper is to introduce a new proximal double bundle method for unconstrained nonsmooth optimization, where the objective function is presented as a difference of two convex (DC) functions. The novelty in our method is a new escape procedure which enables us to guarantee approximate Clarke stationarity for solutions by utilizing the DC components of the objective function. This optimality condition is stronger than the criticality condition typically used in DC programming. Moreover, if a candidate solution is not approximate Clarke stationary, then the escape procedure returns a descent direction. With this escape procedure, we can avoid some shortcomings encountered when criticality is used. The finite termination of the double bundle method to an approximate Clarke stationary point is proved by assuming that the subdifferentials of DC components are polytopes. Finally, some encouraging numerical results are presented.

**Authors:**Joki, Kaisa , Bagirov, Adil , Karmitsa, Napsu , Makela, Marko , Taheri, Sona**Date:**2018**Type:**Text , Journal article**Relation:**SIAM Journal on Optimization Vol. 28, no. 2 (2018), p. 1892-1919**Relation:**http://purl.org/au-research/grants/arc/DP140103213**Full Text:****Reviewed:****Description:**The aim of this paper is to introduce a new proximal double bundle method for unconstrained nonsmooth optimization, where the objective function is presented as a difference of two convex (DC) functions. The novelty in our method is a new escape procedure which enables us to guarantee approximate Clarke stationarity for solutions by utilizing the DC components of the objective function. This optimality condition is stronger than the criticality condition typically used in DC programming. Moreover, if a candidate solution is not approximate Clarke stationary, then the escape procedure returns a descent direction. With this escape procedure, we can avoid some shortcomings encountered when criticality is used. The finite termination of the double bundle method to an approximate Clarke stationary point is proved by assuming that the subdifferentials of DC components are polytopes. Finally, some encouraging numerical results are presented.

Elucidating the impact of micro-scale heterogeneous bacterial distribution on biodegradation

- Schmidt, Susanne, Kreft, Jan-Ulrich, Mackay, Rae, Picioreanu, Cristian, Thullner, Martin

**Authors:**Schmidt, Susanne , Kreft, Jan-Ulrich , Mackay, Rae , Picioreanu, Cristian , Thullner, Martin**Date:**2018**Type:**Text , Journal article**Relation:**Advances in Water Resources Vol. 116, no. (2018), p. 67-76**Full Text:**false**Reviewed:****Description:**Groundwater microorganisms hardly ever cover the solid matrix uniformly–instead they form micro-scale colonies. To which extent such colony formation limits the bioavailability and biodegradation of a substrate is poorly understood. We used a high-resolution numerical model of a single pore channel inhabited by bacterial colonies to simulate the transport and biodegradation of organic substrates. These high-resolution 2D simulation results were compared to 1D simulations that were based on effective rate laws for bioavailability-limited biodegradation. We (i) quantified the observed bioavailability limitations and (ii) evaluated the applicability of previously established effective rate concepts if microorganisms are heterogeneously distributed. Effective bioavailability reductions of up to more than one order of magnitude were observed, showing that the micro-scale aggregation of bacterial cells into colonies can severely restrict the bioavailability of a substrate and reduce in situ degradation rates. Effective rate laws proved applicable for upscaling when using the introduced effective colony sizes.

- Zhang, Jinjia, Cliff, David, Xu, Kaili, You, Greg

**Authors:**Zhang, Jinjia , Cliff, David , Xu, Kaili , You, Greg**Date:**2018**Type:**Text , Journal article**Relation:**Process Safety and Environmental Protection Vol. 117, no. (2018), p. 390-398**Full Text:**false**Reviewed:****Description:**Extraordinarily severe gas explosion accidents (ESGEAs) (thirty fatalities or more in one accident) have a high occurrence frequency in Chinese coal mines. There are 126 ESGEAs that occurred in China from 1950 to 2015, and they were investigated through statistical methods in this study to review the overall circumstances and to provide quantitative information on ESGEAs. Statistical characteristics about accident-related factors, such as gas accumulation, ignition sources, operating locations, accident time, coal mine regions and coal mine ownership, were assessed in this paper. The statistical analysis shows that disorganized ventilation fan management was the most frequent cause of gas accumulation in ESGEAs, while illegal blasting was the most prominent cause of the ignition source in ESGEAs. Furthermore, ESGEAs were found to occur frequently in certain provinces (e.g., Shanxi, Henan and Heilongjiang) and during November and December of the year. Moreover, most accidents and the largest death tolls generally occur in state-owned coal mines. Based on the results of statistical studies, some countermeasures were proposed in this study.

- Zhong, Hong, Li, Hongjun, Ooi, Ean Tat, Song, Chongmin

**Authors:**Zhong, Hong , Li, Hongjun , Ooi, Ean Tat , Song, Chongmin**Date:**2018**Type:**Text , Journal article**Relation:**Engineering Analysis with Boundary Elements Vol. 88, no. (2018), p. 41-53**Full Text:**false**Reviewed:****Description:**The scaled boundary finite element method coupled with the cohesive crack model is extended to investigate the hydraulic fracture at the dam-foundation interface. The concrete and rock bulk are modeled by the scaled boundary polygons. Cohesive interface elements model the fracture process zone between the crack faces. The cohesive tractions are modeled as side-face tractions in the scaled boundary polygons. The solution of the stress field around the crack tip is expressed semi-analytically as a power series. Accurate displacement field, stress field and stress intensity factors can be obtained without asymptotic enrichment or local mesh refinement. The proposed procedure is verified by the hydraulic fracture of a rectangular embankment on rigid foundation and applied to the modeling of hydraulic fracture on the dam-foundation interface of a benchmark dam. Different distributions of water pressure inside the crack are investigated. It is found that the water pressure inside the crack decreases the peak overflow to less than 20% of the case without water in the crack. Considering the water lag or not is significant to the response, while different distribution of pressure following the water lag region in the fracture process zone has negligible influence.

- Khandelwal, Manoj, Marto, Aminaton, Fatemi, Seyed, Ghoroqi, Mahyar, Armaghani, Danial, Singh, Trilok, Tabrizi, Omid

**Authors:**Khandelwal, Manoj , Marto, Aminaton , Fatemi, Seyed , Ghoroqi, Mahyar , Armaghani, Danial , Singh, Trilok , Tabrizi, Omid**Date:**2018**Type:**Text , Journal article**Relation:**Engineering with Computers Vol. 34, no. 2 (2018), p. 307-317**Full Text:**false**Reviewed:****Description:**Shear strength parameters such as cohesion are the most significant rock parameters which can be utilized for initial design of some geotechnical engineering applications. In this study, evaluation and prediction of rock material cohesion is presented using different approaches i.e., simple and multiple regression, artificial neural network (ANN) and genetic algorithm (GA)-ANN. For this purpose, a database including three model inputs i.e., p-wave velocity, uniaxial compressive strength and Brazilian tensile strength and one output which is cohesion of limestone samples was prepared. A meaningful relationship was found for all of the model inputs with suitable performance capacity for prediction of rock cohesion. Additionally, a high level of accuracy (coefficient of determination, R2 of 0.925) was observed developing multiple regression equation. To obtain higher performance capacity, a series of ANN and GA-ANN models were built. As a result, hybrid GA-ANN network provides higher performance for prediction of rock cohesion compared to ANN technique. GA-ANN model results (R2 = 0.976 and 0.967 for train and test) were better compared to ANN model results (R2 = 0.949 and 0.948 for train and test). Therefore, this technique is introduced as a new one in estimating cohesion of limestone samples. © 2017, Springer-Verlag London Ltd., part of Springer Nature.

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.

**Authors:**Bagirov, Adil , Ugon, Julien**Date:**2018**Type:**Text , Journal article**Relation:**Optimization Methods and Software Vol. 33, no. 1 (2018), p. 194-219**Relation:**http://purl.org/au-research/grants/arc/DP140103213**Full Text:**false**Reviewed:****Description:**The clusterwise linear regression problem is formulated as a nonsmooth nonconvex optimization problem using the squared regression error function. The objective function in this problem is represented as a difference of convex functions. Optimality conditions are derived, and an algorithm is designed based on such a representation. An incremental approach is proposed to generate starting solutions. The algorithm is tested on small to large data sets. © 2017 Informa UK Limited, trading as Taylor & Francis Group.

**Authors:**Bagirov, Adil , Ugon, Julien**Date:**2018**Type:**Text , Journal article**Relation:**Optimization Methods and Software Vol. 33, no. 1 (2018), p. 194-219**Full Text:**false**Reviewed:****Description:**The clusterwise linear regression problem is formulated as a nonsmooth nonconvex optimization problem using the squared regression error function. The objective function in this problem is represented as a difference of convex functions. Optimality conditions are derived, and an algorithm is designed based on such a representation. An incremental approach is proposed to generate starting solutions. The algorithm is tested on small to large data sets.

On modeling and complete solutions to general fixpoint problems in multi-scale systems with applications

**Authors:**Ruan, Ning , Gao, David**Date:**2018**Type:**Text , Journal article**Relation:**Fixed Point Theory and Applications Vol. 2018, no. 1 (2018), p. 1-19**Full Text:****Reviewed:****Description:**This paper revisits the well-studied fixed point problem from a unified viewpoint of mathematical modeling and canonical duality theory, i.e., the general fixed point problem is first reformulated as a nonconvex optimization problem, its well-posedness is discussed based on the objectivity principle in continuum physics; then the canonical duality theory is applied for solving this challenging problem to obtain not only all fixed points, but also their stability properties. Applications are illustrated by problems governed by nonconvex polynomial, exponential, and logarithmic operators. This paper shows that within the framework of the canonical duality theory, there is no difference between the fixed point problems and nonconvex analysis/optimization in multidisciplinary studies.

**Authors:**Ruan, Ning , Gao, David**Date:**2018**Type:**Text , Journal article**Relation:**Fixed Point Theory and Applications Vol. 2018, no. 1 (2018), p. 1-19**Full Text:****Reviewed:****Description:**This paper revisits the well-studied fixed point problem from a unified viewpoint of mathematical modeling and canonical duality theory, i.e., the general fixed point problem is first reformulated as a nonconvex optimization problem, its well-posedness is discussed based on the objectivity principle in continuum physics; then the canonical duality theory is applied for solving this challenging problem to obtain not only all fixed points, but also their stability properties. Applications are illustrated by problems governed by nonconvex polynomial, exponential, and logarithmic operators. This paper shows that within the framework of the canonical duality theory, there is no difference between the fixed point problems and nonconvex analysis/optimization in multidisciplinary studies.

On SPD method for solving canonical dual problem in post buckling of large deformed elastic beam

**Authors:**Ali, Elaf , Gao, David**Date:**2018**Type:**Text , Journal article**Relation:**Communications in Mathematical Sciences Vol. 16, no. 5 (2018), p. 1225-1240**Full Text:****Reviewed:****Description:**This paper presents a new methodology and algorithm for solving post buckling problems of a large deformed elastic beam. The total potential energy of this beam is a nonconvex functional, which can be used to model both pre- and post-buckling phenomena. By using a canonical dual finite element method, a new primal-dual semi-definite programming (PD-SDP) algorithm is presented, which can be used to obtain all possible post-buckled solutions. Applications are illustrated by several numerical examples with different boundary conditions. We find that the global minimum solution of the nonconvex potential leads to a stable configuration of the buckled beam, the local maximum solution leads to the unbuckled state, and both of these two solutions are numerically stable. However, the local minimum solution leads to an unstable buckled state, which is very sensitive to axial compressive forces, thickness of beam, numerical precision, and the size of finite elements. The method and algorithm proposed in this paper can be used for solving general nonconvex variational problems in engineering and sciences.

**Authors:**Ali, Elaf , Gao, David**Date:**2018**Type:**Text , Journal article**Relation:**Communications in Mathematical Sciences Vol. 16, no. 5 (2018), p. 1225-1240**Full Text:****Reviewed:****Description:**This paper presents a new methodology and algorithm for solving post buckling problems of a large deformed elastic beam. The total potential energy of this beam is a nonconvex functional, which can be used to model both pre- and post-buckling phenomena. By using a canonical dual finite element method, a new primal-dual semi-definite programming (PD-SDP) algorithm is presented, which can be used to obtain all possible post-buckled solutions. Applications are illustrated by several numerical examples with different boundary conditions. We find that the global minimum solution of the nonconvex potential leads to a stable configuration of the buckled beam, the local maximum solution leads to the unbuckled state, and both of these two solutions are numerically stable. However, the local minimum solution leads to an unstable buckled state, which is very sensitive to axial compressive forces, thickness of beam, numerical precision, and the size of finite elements. The method and algorithm proposed in this paper can be used for solving general nonconvex variational problems in engineering and sciences.

Perturbation of error bounds

- Kruger, Alexander, López, Marco, Théra, Michel

**Authors:**Kruger, Alexander , López, Marco , Théra, Michel**Date:**2018**Type:**Text , Journal article**Relation:**Mathematical Programming Vol. 168, no. 1-2 (2018), p. 533-554**Relation:**http://purl.org/au-research/grants/arc/DP160100854**Full Text:****Reviewed:****Description:**Our aim in the current article is to extend the developments in Kruger et al. (SIAM J Optim 20(6):3280–3296, 2010. doi:10.1137/100782206) and, more precisely, to characterize, in the Banach space setting, the stability of the local and global error bound property of inequalities determined by lower semicontinuous functions under data perturbations. We propose new concepts of (arbitrary, convex and linear) perturbations of the given function defining the system under consideration, which turn out to be a useful tool in our analysis. The characterizations of error bounds for families of perturbations can be interpreted as estimates of the ‘radius of error bounds’. The definitions and characterizations are illustrated by examples. © 2017, Springer-Verlag Berlin Heidelberg and Mathematical Optimization Society.

**Authors:**Kruger, Alexander , López, Marco , Théra, Michel**Date:**2018**Type:**Text , Journal article**Relation:**Mathematical Programming Vol. 168, no. 1-2 (2018), p. 533-554**Relation:**http://purl.org/au-research/grants/arc/DP160100854**Full Text:****Reviewed:****Description:**Our aim in the current article is to extend the developments in Kruger et al. (SIAM J Optim 20(6):3280–3296, 2010. doi:10.1137/100782206) and, more precisely, to characterize, in the Banach space setting, the stability of the local and global error bound property of inequalities determined by lower semicontinuous functions under data perturbations. We propose new concepts of (arbitrary, convex and linear) perturbations of the given function defining the system under consideration, which turn out to be a useful tool in our analysis. The characterizations of error bounds for families of perturbations can be interpreted as estimates of the ‘radius of error bounds’. The definitions and characterizations are illustrated by examples. © 2017, Springer-Verlag Berlin Heidelberg and Mathematical Optimization Society.

Set regularities and feasibility problems

- Kruger, Alexander, Luke, Russell, Thao, Nguyen

**Authors:**Kruger, Alexander , Luke, Russell , Thao, Nguyen**Date:**2018**Type:**Text , Journal article**Relation:**Mathematical Programming Vol. 168, no. 1-2 (2018), p. 279-311**Relation:**http://purl.org/au-research/grants/arc/DP160100854**Full Text:****Reviewed:****Description:**We synthesize and unify notions of regularity, both of individual sets and of collections of sets, as they appear in the convergence theory of projection methods for consistent feasibility problems. Several new characterizations of regularities are presented which shed light on the relations between seemingly different ideas and point to possible necessary conditions for local linear convergence of fundamental algorithms

**Authors:**Kruger, Alexander , Luke, Russell , Thao, Nguyen**Date:**2018**Type:**Text , Journal article**Relation:**Mathematical Programming Vol. 168, no. 1-2 (2018), p. 279-311**Relation:**http://purl.org/au-research/grants/arc/DP160100854**Full Text:****Reviewed:****Description:**We synthesize and unify notions of regularity, both of individual sets and of collections of sets, as they appear in the convergence theory of projection methods for consistent feasibility problems. Several new characterizations of regularities are presented which shed light on the relations between seemingly different ideas and point to possible necessary conditions for local linear convergence of fundamental algorithms

Strong metric subregularity of mappings in variational analysis and optimization

- Cibulka, Radek, Dontchev, Asen, Kruger, Alexander

**Authors:**Cibulka, Radek , Dontchev, Asen , Kruger, Alexander**Date:**2018**Type:**Text , Journal article**Relation:**Journal of Mathematical Analysis and Applications Vol. 457, no. 2 (2018), p. 1247-1287**Relation:**http://purl.org/au-research/grants/arc/DP160100854**Full Text:**false**Reviewed:****Description:**Although the property of strong metric subregularity of set-valued mappings has been present in the literature under various names and with various (equivalent) definitions for more than two decades, it has attracted much less attention than its older “siblings”, the metric regularity and the strong (metric) regularity. The purpose of this paper is to show that the strong metric subregularity shares the main features of these two most popular regularity properties and is not less instrumental in applications. We show that the strong metric subregularity of a mapping F acting between metric spaces is stable under perturbations of the form f+F, where f is a function with a small calmness constant. This result is parallel to the Lyusternik–Graves theorem for metric regularity and to the Robinson theorem for strong regularity, where the perturbations are represented by a function f with a small Lipschitz constant. Then we study perturbation stability of the same kind for mappings acting between Banach spaces, where f is not necessarily differentiable but admits a set-valued derivative-like approximation. Strong metric q-subregularity is also considered, where q is a positive real constant appearing as exponent in the definition. Rockafellar's criterion for strong metric subregularity involving injectivity of the graphical derivative is extended to mappings acting in infinite-dimensional spaces. A sufficient condition for strong metric subregularity is established in terms of surjectivity of the Fréchet coderivative, and it is shown by a counterexample that surjectivity of the limiting coderivative is not a sufficient condition for this property, in general. Then various versions of Newton's method for solving generalized equations are considered including inexact and semismooth methods, for which superlinear convergence is shown under strong metric subregularity. As applications to optimization, a characterization of the strong metric subregularity of the KKT mapping is obtained, as well as a radius theorem for the optimality mapping of a nonlinear programming problem. Finally, an error estimate is derived for a discrete approximation in optimal control under strong metric subregularity of the mapping involved in the Pontryagin principle.

A comparison of bidding strategies for online auctions using fuzzy reasoning and negotiation decision functions

- Kaur, Preetinder, Goyal, Madhu, Lu, Jie

**Authors:**Kaur, Preetinder , Goyal, Madhu , Lu, Jie**Date:**2017**Type:**Text , Journal article**Relation:**IEEE Transactions on Fuzzy Systems Vol. 25, no. 2 (2017), p. 425-438**Full Text:****Reviewed:****Description:**Bidders often feel challenged when looking for the best bidding strategies to excel in the competitive environment of multiple and simultaneous online auctions for same or similar items. Bidders face complicated issues for deciding which auction to participate in, whether to bid early or late, and how much to bid. In this paper, we present the design of bidding strategies, which aim to forecast the bid amounts for buyers at a particular moment in time based on their bidding behavior and their valuation of an auctioned item. The agent develops a comprehensive methodology for final price estimation, which designs bidding strategies to address buyers' different bidding behaviors using two approaches: Mamdani method with regression analysis and negotiation decision functions. The experimental results show that the agents who follow fuzzy reasoning with a regression approach outperform other existing agents in most settings in terms of their success rate and expected utility.

**Authors:**Kaur, Preetinder , Goyal, Madhu , Lu, Jie**Date:**2017**Type:**Text , Journal article**Relation:**IEEE Transactions on Fuzzy Systems Vol. 25, no. 2 (2017), p. 425-438**Full Text:****Reviewed:****Description:**Bidders often feel challenged when looking for the best bidding strategies to excel in the competitive environment of multiple and simultaneous online auctions for same or similar items. Bidders face complicated issues for deciding which auction to participate in, whether to bid early or late, and how much to bid. In this paper, we present the design of bidding strategies, which aim to forecast the bid amounts for buyers at a particular moment in time based on their bidding behavior and their valuation of an auctioned item. The agent develops a comprehensive methodology for final price estimation, which designs bidding strategies to address buyers' different bidding behaviors using two approaches: Mamdani method with regression analysis and negotiation decision functions. The experimental results show that the agents who follow fuzzy reasoning with a regression approach outperform other existing agents in most settings in terms of their success rate and expected utility.

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