On computation of generalized derivatives of the normal-cone mapping and their applications
- Authors: Gfrerer, Helmut , Outrata, Jiri
- Date: 2016
- Type: Text , Journal article
- Relation: Mathematics of Operations Research Vol. 41, no. 4 (2016), p. 1535-1556
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- Description: The paper concerns the computation of the graphical derivative and the regular (Fréchet) coderivative of the normal-cone mapping related to C2 inequality constraints under very weak qualification conditions. This enables us to provide the graphical derivative and the regular coderivative of the solution map to a class of parameterized generalized equations with the constraint set of the investigated type. On the basis of these results, we finally obtain a characterization of the isolated calmness property of the mentioned solution map and derive strong stationarity conditions for an MPEC with control constraints. © 2016 INFORMS.
On computation of limiting coderivatives of the normal-cone mapping to inequality systems and their applications
- Authors: Gfrerer, Helmut , Outrata, Jiri
- Date: 2016
- Type: Text , Journal article
- Relation: Optimization Vol. 65, no. 4 (2016), p. 671-700
- Relation: http://purl.org/au-research/grants/arc/DP110102011
- Full Text: false
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- Description: The paper concerns the computation of the limiting coderivative of the normal-cone mapping related to inequality constraints under weak qualification conditions. The obtained results are applied to verify the Aubin property of solution maps to a class of parameterized generalized equations.
On lipschitzian properties of implicit multifunctions
- Authors: Gfrerer, Helmut , Outrata, Jiri
- Date: 2016
- Type: Text , Journal article
- Relation: SIAM Journal on Optimization Vol. 26, no. 4 (2016), p. 2160-2189
- Relation: http://purl.org/au-research/grants/arc/DP160100854
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- Description: This paper is devoted to the development of new sufficient conditions for the calmness and the Aubin property of implicit multifunctions. As the basic tool we employ the directional limiting coderivative which, together with the graphical derivative, enables a fine analysis of the local behavior of the investigated multifunction along relevant directions. For verification of the calmness property, in addition, a new condition has been discovered which parallels the missing implicit function paradigm and permits us to replace the original multifunction by a substantially simpler one. Moreover, as an auxiliary tool, a handy formula for the computation of the directional limiting coderivative of the normal-cone map with a polyhedral set has been derived which perfectly matches the framework of [A. L. Dontchev and R. T. Rockafellar, SIAM J. Optim., 6 (1996), pp. 1087{1105]. All important statements are illustrated by examples. © 2016 Society for Industrial and Applied Mathematics.
On the extrema of a nonconvex functional with double-well potential in 1D
- Authors: Gao, David , Lu, Xioajun
- Date: 2016
- Type: Text , Journal article
- Relation: Zeitschrift fur Angewandte Mathematik und Physik Vol. 67, no. 3 (2016), p. 1-7
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- Description: This paper mainly investigates the extrema of a nonconvex functional with double-well potential in 1D through the approach of nonlinear differential equations. Based on the canonical duality method, the corresponding Euler–Lagrange equation with Neumann boundary condition can be converted into a cubic dual algebraic equation, which will help find the local extrema for the primal problem. © 2016, Springer International Publishing.
Quasistatic thermoviscoelastic problem with normal compliance, multivalued friction and wear diffusion
- Authors: Gasi , Ochal, Anna , Shillor, Meir
- Date: 2016
- Type: Text , Journal article
- Relation: Nonlinear Analysis: Real World Applications Vol. 27, no. (2016), p. 183-202
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- Description: This paper presents and analyzes a model for quasistatic frictional contact between a thermoviscoelastic body and a moving foundation that involves wear of the contacting surface and the diffusion of the wear debris. The constitutive law includes temperature effects and the evolution of the temperature is described by a parabolic equation with a subdifferential heat exchange boundary condition. Contact is modeled with normal compliance together with a subdifferential frictional law. The rate of wear of the contact surface is described by the differential form of the Archard condition. The effects of the diffusion of the wear particles on the contact surface are taken into account. Such situations arise in mechanical joints and in orthopedic biomechanics where the wear debris is trapped, diffuses and influences the properties of joint prosthesis and implants. The variational formulation of the problem leads to a system with a time-dependent hemivariational inequality for the displacement, a parabolic hemivariational inequality for the temperature and a parabolic equation on the contact boundary for the wear diffusion. The existence of a unique weak solution is proved by using recent results from the theory of hemivariational inequalities, variational diffusion equation, and a fixed point argument. © 2015 Elsevier Ltd.
Towards supremum-sum subdifferential calculus free of qualification conditions
- Authors: Correa, Rafael , Hantoute, Abderrahim , López, Marco
- Date: 2016
- Type: Text , Journal article
- Relation: Siam Journal on Optimization Vol. 26, no. 4 (2016), p. 2219-2234
- Relation: http://purl.org/au-research/grants/arc/DP160100854
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- Description: We give a formula for the subdifferential of the sum of two convex functions where one of them is the supremum of an arbitrary family of convex functions. This is carried out under a weak assumption expressing a natural relationship between the lower semicontinuous envelopes of the data functions in the domain of the sum function. We also provide a new rule for the subdifferential of the sum of two convex functions, which uses a strategy of augmenting the involved functions. The main feature of our analysis is that no continuity-type condition is required. Our approach allows us to unify, recover, and extend different results in the recent literature.
Vibrations of a mass-spring system using a granular-material damper
- Authors: Zalewski, Robert , Chodkiewicz, Pawel , Shillor, Meir
- Date: 2016
- Type: Text , Journal article
- Relation: Applied Mathematical Modelling Vol. 40, no. 17-18 (2016), p. 8033-8047
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- Description: The control of vibrations of a mass-spring system with a damper made of granular material is modeled, numerically simulated and experimentally verified. The damper consists of a hermetically closed flexible sleeve filled with granular material. Pumping air out of or into the sleeve increases or decreases the under-pressure, which increases or decreases the compression of the granules, causing the system to become more or less rigid. This, in turn, increases or decreases the energy dissipation, which provides the damping control mechanism of the system's vibrations. The spring is assumed to be nonlinear and once the coils are fully compressed it becomes essentially rigid. The changes to the damping characteristics of the damper caused by the rearrangement and compacting of the granules are described by a 'damage-like' variable- the granules rearrangement function. The model consists of a nonlinear ordinary differential equation for the mass-spring-damper system and a differential inclusion for the granules rearrangement function. A numerical algorithm for the problem is presented and simulations of the system behavior depicted. In particular, the changes in the oscillations of the system as the grain rearrangement progresses are shown. Then, the predictions of a version of the model are compared to experimental results that are presented briefly. The numerical results are fully supported by the experiments. It is concluded that a granular material damper may be an easy to implement and cost effective way to dampen vibrations of a mechanical system. (C) 2016 Elsevier Inc. All rights reserved.
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
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- 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.
A BMI approach to guaranteed cost control of discrete-time uncertain system with both state and input delays
- Authors: Zhou, Xiaojun , Dong, Tianxue , Tang, Xiaolin , Yang, Chunhua , Gui, Weihua
- Date: 2015
- Type: Text , Journal article
- Relation: Optimal Control Applications and Methods Vol. 36, no. 6 (2015), p. 844-852
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- Description: In this study, the guaranteed cost control of discrete time uncertain system with both state and input delays is considered. Sufficient conditions for the existence of a memoryless state feedback guaranteed cost control law are given in the bilinear matrix inequality form, which needs much less auxiliary matrix variables and storage space. Furthermore, the design of guaranteed cost controller is reformulated as an optimization problem with a linear objective function, bilinear, and linear matrix inequalities constraints. A nonlinear semi-definite optimization solver - PENLAB is used as a solution technique. A numerical example is given to demonstrate the effectiveness of the proposed method. Copyright © 2014 John Wiley & Sons, Ltd.
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
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- 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.
A reliability-based design optimization model for electricity power networks
- Authors: Ezzati, Ghasem
- Date: 2015
- Type: Text , Journal article
- Relation: Dynamics of Continuous, Discrete and Impulsive Systems Series B: Applications and Algorithms Vol. 22, no. 5 (2015), p. 339-357
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- Description: Significant attentions have recently been attracted by electricity power net- works where many optimization models are applied to optimize distributed power. Many optimization models are available for electricity networks that mainly take into accoun- t total cost. Reliability related issues of electricity networks are also considered in the literature. However, there is a lack to formulate a reliability-based design optimization (RBDO) model of these networks. An RBDO model is introduced in this paper to deal with probabilistic constraints in an optimization model for electricity networks. In our suggested approach, an optimization problem is firstly solved to find optimal parameters of the network. Then, the optimal solution is adjusted using an RBDO problem. Our main aim is to minimize an extra cost that is experienced by considering reliability. It is expected to have a higher extra cost for a lower failure probability. © 2015 Watam Press.
An incremental clustering algorithm based on hyperbolic smoothing
- Authors: Bagirov, Adil , Ordin, Burak , Ozturk, Gurkan , Xavier, Adilson
- Date: 2015
- Type: Text , Journal article
- Relation: Computational Optimization and Applications Vol. 61, no. 1 (2015), p. 219-241
- Relation: http://purl.org/au-research/grants/arc/DP140103213
- Full Text: false
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- Description: Clustering is an important problem in data mining. It can be formulated as a nonsmooth, nonconvex optimization problem. For the most global optimization techniques this problem is challenging even in medium size data sets. In this paper, we propose an approach that allows one to apply local methods of smooth optimization to solve the clustering problems. We apply an incremental approach to generate starting points for cluster centers which enables us to deal with nonconvexity of the problem. The hyperbolic smoothing technique is applied to handle nonsmoothness of the clustering problems and to make it possible application of smooth optimization algorithms to solve them. Results of numerical experiments with eleven real-world data sets and the comparison with state-of-the-art incremental clustering algorithms demonstrate that the smooth optimization algorithms in combination with the incremental approach are powerful alternative to existing clustering algorithms.
An induction theorem and nonlinear regularity models
- Authors: Khanh, Phan , Kruger, Alexander , Thao, Nguyen
- Date: 2015
- Type: Text , Journal article
- Relation: Siam Journal on Optimization Vol. 25, no. 4 (2015), p. 2561-2588
- Relation: http://purl.org/au-research/grants/arc/DP110102011
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- Description: A general nonlinear regularity model for a set-valued mapping F : X x R+ paired right arrows Y, where X and Y are metric spaces, is studied using special iteration procedures, going back to Banach, Schauder, Lyusternik, and Graves. Namely, we revise the induction theorem from Khanh [J. Math. Anal. Appl., 118 (1986), pp. 519-534] and employ it to obtain basic estimates for exploring regularity/openness properties. We also show that it can serve as a substitution for the Ekeland variational principle when establishing other regularity criteria. Then, we apply the induction theorem and the mentioned estimates to establish criteria for both global and local versions of regularity/openness properties for our model and demonstrate how the definitions and criteria translate into the conventional setting of a set-valued mapping F : X paired right arrows Y. An application to second-order necessary optimality conditions for a nonsmooth set-valued optimization problem with mixed constraints is provided.
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
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- 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.
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
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- 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.
Error bounds and metric subregularity
- Authors: Kruger, Alexander
- Date: 2015
- Type: Text , Journal article
- Relation: Optimization Vol. 64, no. 1 (2015), p. 49-79
- Relation: http://purl.org/au-research/grants/arc/DP110102011
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- Description: Necessary and sufficient criteria for metric subregularity (or calmness) of set-valued mappings between general metric or Banach spaces are treated in the framework of the theory of error bounds for a special family of extended real-valued functions of two variables. A classification scheme for the general error bound and metric subregularity criteria is presented. The criteria are formulated in terms of several kinds of primal and subdifferential slopes.
Global optimality conditions and optimization methods for constrained polynomial programming problems
- Authors: Wu, Zhiyou , Tian, Jing , Ugon, Julien , Zhang, Liang
- Date: 2015
- Type: Text , Journal article
- Relation: Applied Mathematics and Computation Vol. 262, no. (2015), p. 312-325
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- Description: The general constrained polynomial programming problem (GPP) is considered in this paper. Problem (GPP) has a broad range of applications and is proved to be NP-hard. Necessary global optimality conditions for problem (GPP) are established. Then, a new local optimization method for this problem is proposed by exploiting these necessary global optimality conditions. A global optimization method is proposed for this problem by combining this local optimization method together with an auxiliary function. Some numerical examples are also given to illustrate that these approaches are very efficient. (C) 2015 Elsevier Inc. All rights reserved.
Global optimality conditions and optimization methods for polynomial programming problems
- Authors: Wu, Zhiyou , Tian, Jing , Ugon, Julien
- Date: 2015
- Type: Text , Journal article
- Relation: Journal of Global Optimization Vol. 62, no. 4 (2015), p. 617-641
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- Description: This paper is concerned with the general polynomial programming problem with box constraints, including global optimality conditions and optimization methods. First, a necessary global optimality condition for a general polynomial programming problem with box constraints is given. Then we design a local optimization method by using the necessary global optimality condition to obtain some strongly or -strongly local minimizers which substantially improve some KKT points. Finally, a global optimization method, by combining the new local optimization method and an auxiliary function, is designed. Numerical examples show that our methods are efficient and stable.
Global solutions to fractional programming problem with ratio of nonconvex functions
- Authors: Ruan, Ning , Gao, David
- Date: 2015
- Type: Text , Journal article
- Relation: Applied Mathematics and Computation Vol. 255, no. (2015), p. 66-72
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- Description: This paper presents a canonical dual approach for minimizing a sum of quadratic function and a ratio of nonconvex functions in Rn. By introducing a parameter, the problem is first equivalently reformed as a nonconvex polynomial minimization with elliptic constraint. It is proved that under certain conditions, the canonical dual is a concave maximization problem in R2 that exhibits no duality gap. Therefore, the global optimal solution of the primal problem can be obtained by solving the canonical dual problem. © 2014 Elsevier Inc. All rights reserved.
Mixed finite element solutions to contact problems of nonlinear Gao beam on elastic foundation
- Authors: Gao, David , Machalova, Jitka , Netuka, Horymir
- Date: 2015
- Type: Text , Journal article
- Relation: Nonlinear Analysis: Real World Applications Vol. 22, no. (2015), p. 537-550
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- Description: This paper analyzes nonlinear contact problems of a large deformed beam on an elastic foundation. The beam model is governed by a nonlinear fourth-order differential equation developed by Gao (1996); while the elastic foundation model is assumed as Winkler's type. Based on a decomposition method, the nonlinear variational inequality problem is able to be reformed as a min-max problem of a saddle Lagrangian. Therefore, by using mixed finite element method with independent discretization-interpolations for foundation and beam elements, the nonlinear contact problem in continuous space is eventually converted as a nonlinear mixed complementarity problem, which can be solved by combination of interior-point and Newton methods. Applications are illustrated by different boundary conditions. Results show that the nonlinear Gao beam is more stiffer than the Euler-Bernoulli beam.