Metric projection onto a closed set : Necessary and sufficient conditions for the global minimum
- Authors: Mohebi, Hossein , Rubinov, Alex
- Date: 2006
- Type: Text , Journal article
- Relation: Mathematics of Operations Research Vol. 31, no. 1 (2006), p. 124-132
- Full Text: false
- Reviewed:
- Description: Necessary and sufficient conditions for a local minimum form a well-developed chapter of optimization theory. Determination of such conditions for the global minimum is a challenging problem. Useful conditions are currently known only for a few classes of nonconvex optimization problems. It is important to find different classes of problems for which the required conditions can be obtained. In this paper we examine one of these classes: the minimization of the distance to an arbitrary closed set in a class of ordered normed spaces. We use the structure of the objective function in order to present necessary and sufficient conditions that give a clear understanding of the structure of a global minimizer and can be easily verified for some problems under consideration. © 2006 INFORMS.
- Description: C1
- Description: 2003001835
Complete solutions and triality theory to a nonconvex optimization problem with double-well potential in Rn
- Authors: Morales-Silva, Daniel , Gao, David
- Date: 2013
- Type: Text , Journal article
- Relation: Numerical Algebra, Control and Optimization Vol. 3, no. 2 (2013), p. 271-282
- Full Text:
- Reviewed:
- Description: The main purpose of this research note is to show that the triality theory can always be used to identify both global minimizer and the biggest local maximizer in global optimization. An open problem left on the double- min duality is solved for a nonconvex optimization problem with double-well potential in ℝn, which leads to a complete set of analytical solutions. Also a convergency theorem is proved for linear perturbation canonical dual method, which can be used for solving global optimization problems with multiple so- lutions. The methods and results presented in this note pave the way towards the proof of the triality theory in general cases.
Canonical duality theory and triality for solving general global optimization problems in complex systems
- Authors: Morales-Silva, Daniel , Gao, David
- Date: 2015
- Type: Text , Journal article
- Relation: Mathematics and Mechanics of Complex Systems Vol. 3, no. 2 (2015), p. 139-161
- Full Text:
- Reviewed:
- Description: General nonconvex optimization problems are studied by using the canonical duality-triality theory. The triality theory is proved for sums of exponentials and quartic polynomials, which solved an open problem left in 2003. This theory can be used to find the global minimum and local extrema, which bridges a gap between global optimization and nonconvex mechanics. Detailed applications are illustrated by several examples. © 2015 Mathematical Sciences Publishers.
Second-order variational analysis in conic programming with applications to optimality and stability
- Authors: Mordukhovich, Boris , Outrata, Jiri , Ramírez, Hector
- Date: 2015
- Type: Text , Journal article
- Relation: SIAM Journal on Optimization Vol. 25, no. 1 (2015), p. 76-101
- Relation: http://purl.org/au-research/grants/arc/DP110102011
- Full Text:
- Reviewed:
- Description: This paper is devoted to the study of a broad class of problems in conic programming modeled via parameter-dependent generalized equations. In this framework we develop a second-order generalized differential approach of variational analysis to calculate appropriate derivatives and coderivatives of the corresponding solution maps. These developments allow us to resolve some important issues related to conic programming. They include verifiable conditions for isolated calmness of the considered solution maps, sharp necessary optimality conditions for a class of mathematical programs with equilibrium constraints, and characterizations of tilt-stable local minimizers for cone-constrained problems. The main results obtained in the general conic programming setting are specified for and illustrated by the second-order cone programming. © 2015 Society for Industrial and Applied Mathematics.
Full stability of locally optimal solutions in second-order cone programs
- Authors: Mordukhovich, Boris , Outrata, Jiri , Sarabi, Ebrahim
- Date: 2014
- Type: Text , Journal article
- Relation: SIAM Journal on Optimization Vol. 24, no. 4 (2014), p. 1581-1613
- Full Text:
- Reviewed:
- Description: The paper presents complete characterizations of Lipschitzian full stability of locally optimal solutions to second-order cone programs (SOCPs) expressed entirely in terms of their initial data. These characterizations are obtained via appropriate versions of the quadratic growth and strong second-order sufficient conditions under the corresponding constraint qualifications. We also establish close relationships between full stability of local minimizers for SOCPs and strong regularity of the associated generalized equations at nondegenerate points. Our approach is mainly based on advanced tools of second-order variational analysis and generalized differentiation.
Real-time localisation system for GPS-denied open areas using smart street furniture
- Authors: Nassar, Mohamed , Luxford, Len , Cole, Peter , Oatley, Giles , Koutsakis, Polychronis
- Date: 2021
- Type: Text , Journal article
- Relation: Simulation Modelling Practice and Theory Vol. 112, no. (2021), p.
- Full Text: false
- Reviewed:
- Description: Wifi-based localisation systems have gained significant interest with many researchers proposing different localisation techniques using publicly available datasets. However, these datasets are limited because they only contain Wifi fingerprints collected and labelled by users, and they are restricted to indoor locations. We have generated the first Wifi-based localisation datasets for a GPS-denied open area. We selected a busy open area at Murdoch University to generate the datasets using so-called “smart bins”, which are rubbish bins that we enabled to work as access points. The data gathered consists of two different datasets. In the first, four users generated labelled WiFi fingerprints for all available Reference Points using four different smartphones. The second dataset includes 2450865 auto-generated rows received from more than 1000 devices. We have developed a light-weight algorithm to label the second dataset from the first and we proposed a localisation approach that converts the second dataset from asynchronous format to synchronous, applies feature engineering and a deep learning classifier. Finally, we have demonstrated via simulations that by using this approach we achieve higher prediction accuracy, with up to 19% average improvement, compared with using only the fingerprint dataset. © 2021 Elsevier B.V.
A scaled boundary finite element formulation over arbitrary faceted star convex polyhedra
- Authors: Natarajan, Sundararajan , Ooi, Ean Tat , Saputra, Albert , Song, Chongmin
- Date: 2017
- Type: Text , Journal article
- Relation: Engineering Analysis with Boundary Elements Vol. 80, no. (2017), p. 218-229
- Full Text: false
- Reviewed:
- Description: In this paper, a displacement based finite element framework for general three-dimensional convex polyhedra is presented. The method is based on a semi-analytical framework, the scaled boundary finite element method. The method relies on the definition of a scaling center from which the entire boundary is visible. The salient feature of the method is that the discretizations are restricted to the surfaces of the polyhedron, thus reducing the dimensionality of the problem by one. Hence, an explicit form of the shape functions inside the polyhedron is not required. Conforming shape functions defined over arbitrary polygon, such as the Wachpress interpolants are used over each surface of the polyhedron. Analytical integration is employed within the polyhedron. The proposed method passes patch test to machine precision. The convergence and the accuracy properties of the method is discussed by solving few benchmark problems in linear elasticity. © 2017 Elsevier Ltd
Finite element computations over quadtree meshes : Strain smoothing and semi-analytical formulation
- Authors: Natarajan, Sundararajan , Ooi, Ean Tat , Song, Chongmin
- Date: 2013
- Type: Text , Journal article
- Relation: International Journal of Advances in Engineering Sciences and Applied Mathematics Vol. 7, no. 3 (2013), p. 124-133
- Full Text:
- Reviewed:
- Description: In this paper, we discuss two alternate techniques to treat hanging nodes in a quadtree mesh. Both the techniques share similarities, in that, they require only boundary information. Moreover, they do not require an explicit form of the shape functions, unlike the conventional approaches, for example, as in the work of Gupta (Int J Numer Methods Eng 12:35, 1978) or Tabarraei and Sukumar (Finite Elem Anal Des 41:686, 2005). Hence, no special numerical integration technique is required. One of the techniques relies on the strain projection procedure, whilst the other is based on the scaled boundary finite element method. Numerical examples are presented to demonstrate the accuracy and the convergence properties of the two techniques.
Directional Holder metric regularity
- Authors: Ngai, Huynh Van , Tron, Nguyen Huu , Thera, Michel
- Date: 2016
- Type: Text , Journal article
- Relation: Journal of Optimization Theory and Applications Vol. 171, no. 3 (2016), p. 785-819
- Full Text:
- Reviewed:
- Description: This paper sheds new light on regularity of multifunctions through various characterizations of directional Holder/Lipschitz metric regularity, which are based on the concepts of slope and coderivative. By using these characterizations, we show that directional Holder/Lipschitz metric regularity is stable, when the multifunction under consideration is perturbed suitably. Applications of directional Holder/Lipschitz metric regularity to investigate the stability and the sensitivity analysis of parameterized optimization problems are also discussed.
Directional metric regularity of multifunctions
- Authors: Ngai, Huynh Van , Thera, Michel
- Date: 2015
- Type: Text , Journal article
- Relation: Mathematics of Operations Research Vol. 40, no. 4 (2015), p. 969-991
- Relation: http://purl.org/au-research/grants/arc/DP110102011
- Full Text:
- Reviewed:
- Description: In this paper, we study relative metric regularity of set-valued mappings with emphasis on directional metric regularity. We establish characterizations of relative metric regularity without assuming the completeness of the image spaces, by using the relative lower semicontinuous envelopes of the distance functions to set-valued mappings. We then apply these characterizations to establish a coderivative type criterion for directional metric regularity as well as for the robustness of metric regularity.
- Description: In this paper, we study relative metric regularity of set-valued mappings with emphasis on directional metric regularity. We establish characterizations of relative metric regularity without assuming the completeness of the image spaces, by using the relative lower semicontinuous envelopes of the distance functions to set-valued mappings. We then apply these characterizations to establish a coderivative type criterion for directional metric regularity as well as for the robustness of metric regularity. © 2015 INFORMS.
Numerical investigation of the meshless radial basis integral equation method for solving 2D anisotropic potential problems
- Authors: Ooi, Ean Hin , Ooi, Ean Tat , Ang, Whye Teong
- Date: 2015
- Type: Text , Journal article
- Relation: Engineering Analysis with Boundary Elements Vol. 53, no. (2015), p. 27-39
- Full Text: false
- Reviewed:
- Description: The radial basis integral equation (RBIE) method is derived for the first time to solve potential problems involving material anisotropy. The coefficients of the anisotropic conductivity require the gradient term to be modified accordingly when deriving the boundary integral equation so that the flux expression can be properly accounted. Analyses of the behavior of the anisotropic fundamental solution and its spatial gradients showed that their variations along the subdomain boundaries may be large and they increase as the diagonal components of the material anisotropy become larger. The accuracy of the anisotropic RBIE was found to depend primarily on the accuracy of the influence coefficients evaluations and this precedes the number of nodes used. Root mean squared errors of less than 10(-4)% can be obtained if evaluations of the influence coefficients are sufficiently accurate. An alternative formulation of the anisotropic RBIE was derived. The levels of accuracy obtained were not significantly different from the standard formulation. (C) 2014 Elsevier Ltd. All rights reserved.
A heuristic algorithm for solving the minimum sum-of-squares clustering problems
- Authors: Ordin, Burak , Bagirov, Adil
- Date: 2015
- Type: Text , Journal article
- Relation: Journal of Global Optimization Vol. 61, no. 2 (2015), p. 341-361
- Relation: http://purl.org/au-research/grants/arc/DP140103213
- Full Text: false
- Reviewed:
- Description: Clustering is an important task in data mining. It can be formulated as a global optimization problem which is challenging for existing global optimization techniques even in medium size data sets. Various heuristics were developed to solve the clustering problem. The global k-means and modified global k-means are among most efficient heuristics for solving the minimum sum-of-squares clustering problem. However, these algorithms are not always accurate in finding global or near global solutions to the clustering problem. In this paper, we introduce a new algorithm to improve the accuracy of the modified global k-means algorithm in finding global solutions. We use an auxiliary cluster problem to generate a set of initial points and apply the k-means algorithm starting from these points to find the global solution to the clustering problems. Numerical results on 16 real-world data sets clearly demonstrate the superiority of the proposed algorithm over the global and modified global k-means algorithms in finding global solutions to clustering problems.
An incremental nonsmooth optimization algorithm for clustering using L1 and L∞ norms
- Authors: Ordin, Burak , Bagirov, Adil , Mohebi, Ehsam
- Date: 2020
- Type: Text , Journal article
- Relation: Journal of Industrial and Management Optimization Vol. 16, no. 6 (2020), p. 2757-2779
- Relation: http://purl.org/au-research/grants/arc/DP190100580
- Full Text: false
- Reviewed:
- Description: An algorithm is developed for solving clustering problems with the similarity measure defined using the L1and L∞ norms. It is based on an incremental approach and applies nonsmooth optimization methods to find cluster centers. Computational results on 12 data sets are reported and the proposed algorithm is compared with the X-means algorithm. ©
On computation of optimal strategies in oligopolistic markets respecting the cost of change
- Authors: Outrata, Jiri , Valdman, Jan
- Date: 2020
- Type: Text , Journal article
- Relation: Mathematical Methods of Operations Research Vol. 92, no. 3 (2020), p. 489-509
- Relation: http://purl.org/au-research/grants/arc/DP160100854
- Full Text:
- Reviewed:
- Description: The paper deals with a class of parameterized equilibrium problems, where the objectives of the players do possess nonsmooth terms. The respective Nash equilibria can be characterized via a parameter-dependent variational inequality of the second kind, whose Lipschitzian stability, under appropriate conditions, is established. This theory is then applied to evolution of an oligopolistic market in which the firms adapt their production strategies to changing input costs, while each change of the production is associated with some “costs of change”. We examine both the Cournot-Nash equilibria as well as the two-level case, when one firm decides to take over the role of the Leader (Stackelberg equilibrium). The impact of costs of change is illustrated by academic examples. © 2020, Springer-Verlag GmbH Germany, part of Springer Nature.
Spline regression models for complex multi-modal regulatory networks
- Authors: Ozmen, Ayse , Kropat, Erik , Weber, Gerhard-Wilhelm
- Date: 2014
- Type: Text , Journal article
- Relation: Optimization Methods and Software Vol. 29, no. 3 (2014), p. 515-534
- Full Text: false
- Reviewed:
- Description: Complex regulatory networks often have to be further expanded and improved with regard to the unknown effects of additional parameters and factors that can emit a disturbing influence on the key variables under consideration. The concept of target-environment (TE) networks provides a holistic framework for the analysis of such parameter-dependent multi-modal systems. In this study, we consider time-discrete TE regulatory systems with spline entries. We introduce a new regression model for these particular two-modal systems that allows us to determine the unknown system parameters by applying the multivariate adaptive regression spline (MARS) technique and the newly developed conic multivariate adaptive regression spline (CMARS) method. We obtain a relaxation by means of continuous optimization, especially, conic quadratic programming (CQP) that could be conducted by interior point methods. Finally, a numerical example demonstrates the efficiency of the spline-based approach.
ZERO++ : Harnessing the power of zero appearances to detect anomalies in large-scale data sets
- Authors: Pang, Guansong , Ting, Kaiming , Albrecht, David , Jin, Huidong
- Date: 2016
- Type: Text , Journal article
- Relation: Journal of Artificial Intelligence Research Vol. 57, no. (2016), p. 593-620
- Full Text: false
- Reviewed:
- Description: This paper introduces a new unsupervised anomaly detector called ZERO++ which employs the number of zero appearances in subspaces to detect anomalies in categorical data. It is unique in that it works in regions of subspaces that are not occupied by data; whereas existing methods work in regions occupied by data. ZERO++ examines only a small number of low dimensional subspaces to successfully identify anomalies. Unlike existing frequency-based algorithms, ZERO++ does not involve subspace pattern searching. We show that ZERO++ is better than or comparable with the state-of-the-art anomaly detection methods over a wide range of real-world categorical and numeric data sets; and it is efficient with linear time complexity and constant space complexity which make it a suitable candidate for large-scale data sets.
Generalised rational approximation and its application to improve deep learning classifiers
- 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: https://purl.org/au-research/grants/arc/DP180100602
- Full Text: false
- Reviewed:
- 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).
A complementarity partition theorem for multifold conic systems
- Authors: Peña, Javier , Roshchina, Vera
- Date: 2012
- Type: Text , Journal article
- Relation: Mathematical Programming Vol.142 , no.1-2 (2012), p.579-589
- Full Text: false
- Reviewed:
- Description: Consider a homogeneous multifold convex conic system {Mathematical expression}and its alternative system {Mathematical expression}, where K 1,..., K r are regular closed convex cones. We show that there is a canonical partition of the index set {1,..., r} determined by certain complementarity sets associated to the most interior solutions to the two systems. Our results are inspired by and extend the Goldman-Tucker Theorem for linear programming. © 2012 Springer and Mathematical Optimization Society.
Some preconditioners for systems of linear inequalities
- Authors: Peña, Javier , Roshchina, Vera , Soheili, Negar
- Date: 2014
- Type: Text , Journal article
- Relation: Optimization Letters Vol. 8, no. 7 (2014), p. 2145-2152
- Full Text: false
- Reviewed:
- Description: We show that a combination of two simple preprocessing steps would generally improve the conditioning of a homogeneous system of linear inequalities. Our approach is based on a comparison among three different but related notions of conditioning for linear inequalities. © 2014, Springer-Verlag Berlin Heidelberg.
New largest known graphs of diameter 6
- Authors: Pineda-Villavicencio, Guillermo , Gómez, José , Miller, Mirka , Pérez-Rosés, Hebert
- Date: 2009
- Type: Text , Journal article
- Relation: Networks Vol. 53, no. 4 (2009), p. 315-328
- Full Text:
- Reviewed:
- Description: In the pursuit of obtaining largest graphs of given maximum degree
- Description: 2003007890