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21Yearwood, John
10Taheri, Sona
8Bai, Fusheng
8Tilakaratne, Chandima
8Wu, Zhiyou
6Rubinov, Alex
6Saunders, Gary
5Hajilarov, Eldar
5Kuznetsov, Alexey
5Morris, Sidney
5Sultan, Ibrahim
4Bagirov, Adil
4Kasimbeyli, Refail
4Yang, Y. J.
4Zhao, Lei
3Banerjee, Arunava
3Ivanov, Anatoli
3Kouhbor, Shahnaz
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130102 Applied Mathematics
100103 Numerical and Computational Mathematics
80101 Pure Mathematics
8Classification
8Global optimization
8Optimization
7Data mining
7Optimisation
60802 Computation Theory and Mathematics
4Drug reaction
4Multi-label classification
4Newton's method
4Turnpike property
30801 Artificial Intelligence and Image Processing
3Adverse drug reaction
3Algorithm
3Asymptotical stability
3Australia
3Bayesian networks
3Data classification

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Vibration analysis : Optimization of parameters of the two mass model based on Kelvin elements

- Kuznetsov, Alexey, Mammadov, Musa, Sultan, Ibrahim, Hajilarov, Eldar

**Authors:**Kuznetsov, Alexey , Mammadov, Musa , Sultan, Ibrahim , Hajilarov, Eldar**Date:**2010**Type:**Text , Conference paper**Relation:**Paper presented at 8th IEEE International Conference on Control and Automation, ICCA 2010, Asia Gulf Hotel, Xiamen, China : 9th-11th June 2010 p. 1326-1332**Full Text:****Description:**In this paper we consider the problem of finding optimal parameters of the two mass model that represents vehicle suspension systems. The analysis of the problem is based on finding analytical solution of the system of coupled Ordinary Differential Equations (ODE). Such a technique allows us to generate optimization problem, where an objective function should be minimized, in accordance with ISO 2631 standard formula of admissible acceleration levels. That ensures maximum comfort for a driver and passenger in a moving vehicle on the considered highways.**Description:**2003008232

**Authors:**Kuznetsov, Alexey , Mammadov, Musa , Sultan, Ibrahim , Hajilarov, Eldar**Date:**2010**Type:**Text , Conference paper**Relation:**Paper presented at 8th IEEE International Conference on Control and Automation, ICCA 2010, Asia Gulf Hotel, Xiamen, China : 9th-11th June 2010 p. 1326-1332**Full Text:****Description:**In this paper we consider the problem of finding optimal parameters of the two mass model that represents vehicle suspension systems. The analysis of the problem is based on finding analytical solution of the system of coupled Ordinary Differential Equations (ODE). Such a technique allows us to generate optimization problem, where an objective function should be minimized, in accordance with ISO 2631 standard formula of admissible acceleration levels. That ensures maximum comfort for a driver and passenger in a moving vehicle on the considered highways.**Description:**2003008232

Using links to aid web classification

- Xie, Wei, Mammadov, Musa, Yearwood, John

**Authors:**Xie, Wei , Mammadov, Musa , Yearwood, John**Date:**2007**Type:**Text , Conference paper**Relation:**Paper presented at 6th IEEE/ACIS International Conference on Computer and Information Science, ICIS 2007, Melbourne, Victoria : 11th-13th July 2007 p. 981-986**Full Text:****Description:**In this paper, we will present a new approach of using link information to improve the accuracy and efficiency of web classification. However, different from others, we only use the mappings between linked documents and their own class or classes. In this case, we only need to add a few features called linked-class features into the datasets. We apply SVM and BoosTexter for classification. We show that the classification accuracy can be improved based on mixtures of ordinary word features and out-linked-class features. We analyze and discuss the reason of this improvement.**Description:**2003005438

**Authors:**Xie, Wei , Mammadov, Musa , Yearwood, John**Date:**2007**Type:**Text , Conference paper**Relation:**Paper presented at 6th IEEE/ACIS International Conference on Computer and Information Science, ICIS 2007, Melbourne, Victoria : 11th-13th July 2007 p. 981-986**Full Text:****Description:**In this paper, we will present a new approach of using link information to improve the accuracy and efficiency of web classification. However, different from others, we only use the mappings between linked documents and their own class or classes. In this case, we only need to add a few features called linked-class features into the datasets. We apply SVM and BoosTexter for classification. We show that the classification accuracy can be improved based on mixtures of ordinary word features and out-linked-class features. We analyze and discuss the reason of this improvement.**Description:**2003005438

- Saunders, Gary, Mammadov, Musa, Ivkovic, Sasha

**Authors:**Saunders, Gary , Mammadov, Musa , Ivkovic, Sasha**Date:**2005**Type:**Text , Conference paper**Relation:**Paper presented at the Sixteenth Australasian Workshop on Combinatorial Algorithms, Ballarat, Victoria : 18th - 21st September, 2005**Full Text:**false**Reviewed:****Description:**E1**Description:**2003001449

Two level clustering using SOM and dynamical systems

- Ghosh, Ranadhir, Mammadov, Musa, Ghosh, Moumita, Yearwood, John

**Authors:**Ghosh, Ranadhir , Mammadov, Musa , Ghosh, Moumita , Yearwood, John**Date:**2004**Type:**Text , Conference paper**Relation:**Paper presented at ICOTA6: 6th International Conference on Optimization - Techniques and Applications, Ballarat, Victoria : 9th December, 2004**Full Text:**false**Reviewed:****Description:**E1**Description:**2003000871

Turnpike theory : Stability of optimal trajectories

**Authors:**Mammadov, Musa**Date:**2009**Type:**Text , Book chapter**Relation:**Encyclopedia of Optimization Chapter p. 3948-3955**Full Text:**false

Turnpike theorem for an infinite horizon optimal control problem with time delay

**Authors:**Mammadov, Musa**Date:**2014**Type:**Text , Journal article**Relation:**SIAM Journal on Control and Optimization Vol. 52, no. 1 (2014), p. 420-438**Full Text:****Reviewed:****Description:**An optimal control problem for systems described by a special class of nonlinear differential equations with time delay is considered. The cost functional adopted could be considered as an analogue of the terminal functional defined over an infinite time horizon. The existence of optimal solutions as well as the asymptotic stability of optimal trajectories (that is, the turnpike property) are established under some quite mild restrictions on the nonlinearities of the functions involved in the description of the problem. Such mild restrictions on the nonlinearities allowed us to apply these results to a blood cell production model. Â© 2014 Society for Industrial and Applied Mathematics.

**Authors:**Mammadov, Musa**Date:**2014**Type:**Text , Journal article**Relation:**SIAM Journal on Control and Optimization Vol. 52, no. 1 (2014), p. 420-438**Full Text:****Reviewed:****Description:**An optimal control problem for systems described by a special class of nonlinear differential equations with time delay is considered. The cost functional adopted could be considered as an analogue of the terminal functional defined over an infinite time horizon. The existence of optimal solutions as well as the asymptotic stability of optimal trajectories (that is, the turnpike property) are established under some quite mild restrictions on the nonlinearities of the functions involved in the description of the problem. Such mild restrictions on the nonlinearities allowed us to apply these results to a blood cell production model. Â© 2014 Society for Industrial and Applied Mathematics.

Tree augmented naive bayes based on optimization

- Taheri, Sona, Mammadov, Musa

**Authors:**Taheri, Sona , Mammadov, Musa**Date:**2011**Type:**Text , Conference paper**Relation:**42 Annual Iranian Mathematics Conference Vali-a-Asr University of Rasanjan 5th-8th September, 2011 p. 594-598**Full Text:**false**Reviewed:****Description:**Tree augmented naive Bayes is a semi-naive Bayesian Learning method. It relaxes the naive Bayes attribute independence assumption by employing a tree structure, in which each attribute only depends on the class and one other attribute. A maximum weighted spanning tree that maximizes the likelihood of the training data is used to perform classification.**Description:**2003009354

To be fair or efficient or a bit of both

- Zukerman, Moshe, Mammadov, Musa, Tan, Liansheng, Ouveysi, Iradj, Andrew, Lachlan

**Authors:**Zukerman, Moshe , Mammadov, Musa , Tan, Liansheng , Ouveysi, Iradj , Andrew, Lachlan**Date:**2008**Type:**Text , Journal article**Relation:**Computers and Operations Research Vol. 35, no. 12 (2008), p. 3787-3806**Full Text:****Reviewed:****Description:**IIntroducing a new concept of (®, ¯)-fairness, which allows for a bounded fairness compromise, so that a source is allocated a rate neither less than 0 · ® · 1, nor more than ¯ ¸ 1, times its fair share, this paper provides a framework to optimize efficiency (utilization, throughput or revenue) subject to fairness constraints in a general telecommunications network for an arbitrary fairness criterion and cost functions. We formulate a non-linear program (NLP) that finds the optimal bandwidth allocation by maximizing efficiency subject to (®, ¯)-fairness constraints. This leads to what we call an efficiency-fairness function, which shows the benefit in efficiency as a function of the extent to which fairness is compromised. To solve the NLP we use two algorithms. The first is a well known branch-and-bound-based algorithm called Lipschitz Global Optimization and the second is a recently developed algorithm called Algorithm for Global Optimization Problems (AGOP). We demonstrate the applicability of the framework to a range of example from sharing a single link to efficiency fairness issues associated with serving customers in remote communities.**Description:**C1

**Authors:**Zukerman, Moshe , Mammadov, Musa , Tan, Liansheng , Ouveysi, Iradj , Andrew, Lachlan**Date:**2008**Type:**Text , Journal article**Relation:**Computers and Operations Research Vol. 35, no. 12 (2008), p. 3787-3806**Full Text:****Reviewed:****Description:**IIntroducing a new concept of (®, ¯)-fairness, which allows for a bounded fairness compromise, so that a source is allocated a rate neither less than 0 · ® · 1, nor more than ¯ ¸ 1, times its fair share, this paper provides a framework to optimize efficiency (utilization, throughput or revenue) subject to fairness constraints in a general telecommunications network for an arbitrary fairness criterion and cost functions. We formulate a non-linear program (NLP) that finds the optimal bandwidth allocation by maximizing efficiency subject to (®, ¯)-fairness constraints. This leads to what we call an efficiency-fairness function, which shows the benefit in efficiency as a function of the extent to which fairness is compromised. To solve the NLP we use two algorithms. The first is a well known branch-and-bound-based algorithm called Lipschitz Global Optimization and the second is a recently developed algorithm called Algorithm for Global Optimization Problems (AGOP). We demonstrate the applicability of the framework to a range of example from sharing a single link to efficiency fairness issues associated with serving customers in remote communities.**Description:**C1

The study of drug-reaction relationships using global optimization techniques

- Mammadov, Musa, Rubinov, Alex, Yearwood, John

**Authors:**Mammadov, Musa , Rubinov, Alex , Yearwood, John**Date:**2007**Type:**Text , Journal article**Relation:**Optimization Methods and Software Vol. 22, no. 1 (2007), p. 99-126**Full Text:**false**Reviewed:****Description:**In this paper we develop an optimization approach for the study of adverse drug reaction (ADR) problems. 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 it can be used for solving many ADR problems, we concentrate on two of them here: the accurate identification of drugs that are responsible for reactions that have occurred, and drug-drug interactions. Based on drug-reaction relationships, we formulate these problems as an optimization problem. The approach is applied to cardiovascularn-type reactions from the Australian Adverse Drug Reaction Advisory Committee (ADRAC) database. Software based on this approach has been developed and could have beneficial use in prescribing.**Description:**C1**Description:**2003002217

The effect of regularization on drug-reaction relationships

- Mammadov, Musa, Zhao, L., Zhang, Jianjun

**Authors:**Mammadov, Musa , Zhao, L. , Zhang, Jianjun**Date:**2012**Type:**Text , Journal article**Relation:**Optimization Vol. 61, no. 4 (2012), p. 405-422**Full Text:****Reviewed:****Description:**The least-squares method is a standard approach used in data fitting that has important applications in many areas in science and engineering including many finance problems. In the case when the problem under consideration involves large-scale sparse matrices regularization methods are used to obtain more stable solutions by relaxing the data fitting. In this article, a new regularization algorithm is introduced based on the Karush-Kuhn-Tucker conditions and the Fisher-Burmeister function. The Newton method is used for solving corresponding systems of equations. The advantages of the proposed method has been demonstrated in the establishment of drug-reaction relationships based on the Australian Adverse Drug Reaction Advisory Committee database. © 2012 Copyright Taylor and Francis Group, LLC.

**Authors:**Mammadov, Musa , Zhao, L. , Zhang, Jianjun**Date:**2012**Type:**Text , Journal article**Relation:**Optimization Vol. 61, no. 4 (2012), p. 405-422**Full Text:****Reviewed:****Description:**The least-squares method is a standard approach used in data fitting that has important applications in many areas in science and engineering including many finance problems. In the case when the problem under consideration involves large-scale sparse matrices regularization methods are used to obtain more stable solutions by relaxing the data fitting. In this article, a new regularization algorithm is introduced based on the Karush-Kuhn-Tucker conditions and the Fisher-Burmeister function. The Newton method is used for solving corresponding systems of equations. The advantages of the proposed method has been demonstrated in the establishment of drug-reaction relationships based on the Australian Adverse Drug Reaction Advisory Committee database. © 2012 Copyright Taylor and Francis Group, LLC.

The core of a sequence of fuzzy numbers

- Aytar, Salih, Pehlivan, Serpil, Mammadov, Musa

**Authors:**Aytar, Salih , Pehlivan, Serpil , Mammadov, Musa**Date:**2008**Type:**Text , Journal article**Relation:**Fuzzy Sets and Systems Vol. 159, no. 24 (2008), p. 3369-3379**Full Text:**false**Reviewed:****Description:**In this paper, based on level sets we define the limit inferior and limit superior of a bounded sequence of fuzzy numbers and prove some properties. We extend the concept of the core of a sequence of complex numbers, first introduced by Knopp in 1930, to a bounded sequence of fuzzy numbers and prove that the core of a sequence of fuzzy numbers is the interval [ν, μ] where ν and μ are extreme limit points of the sequence. © 2008 Elsevier B.V. All rights reserved.

Target learning : A novel framework to mine significant dependencies for unlabeled data

- Wang, Limin, Chen, Shenglei, Mammadov, Musa

**Authors:**Wang, Limin , Chen, Shenglei , Mammadov, Musa**Date:**2018**Type:**Text , Conference proceedings**Relation:**22nd Pacific-Asia Conference on Advances in Knowledge Discovery and Data Mining, PAKDD 2018; Melbourne, Australia; 3rd-6th June 2018; published in Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) Vol. 10937 LNAI, p. 106-117**Full Text:**false**Reviewed:****Description:**To mine significant dependencies among predictiveattributes, much work has been carried out to learn Bayesian netwrok classifiers (BNC T s) from labeled training data set T. However, if BNC T does not capture the “right” dependencies that would be most relevant to unlabeled testing instance, that will result in performance degradation. To address this issue we propose a novel framework, called target learning, that takes each unlabeled testing instance as a target and builds an “unstable” Bayesian model BNC P for it. To make BNC P and BNC T complementary to each other and work efficiently in combination, the same learning strategy is applied to build them. Experimental comparison on 32 large data sets from UCI machine learning repository shows that, for BNCs with different degrees of dependence target learning always helps improve the generalization performance with minimal additional computation.**Description:**Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

Structure learning of Bayesian Networks using global optimization with applications in data classification

- Taheri, Sona, Mammadov, Musa

**Authors:**Taheri, Sona , Mammadov, Musa**Date:**2014**Type:**Text , Journal article**Relation:**Optimization Letters Vol. 9, no. 5 (2014), p. 931-948**Full Text:****Reviewed:****Description:**Bayesian Networks are increasingly popular methods of modeling uncertainty in artificial intelligence and machine learning. A Bayesian Network consists of a directed acyclic graph in which each node represents a variable and each arc represents probabilistic dependency between two variables. Constructing a Bayesian Network from data is a learning process that consists of two steps: learning structure and learning parameter. Learning a network structure from data is the most difficult task in this process. This paper presents a new algorithm for constructing an optimal structure for Bayesian Networks based on optimization. The algorithm has two major parts. First, we define an optimization model to find the better network graphs. Then, we apply an optimization approach for removing possible cycles from the directed graphs obtained in the first part which is the first of its kind in the literature. The main advantage of the proposed method is that the maximal number of parents for variables is not fixed a priory and it is defined during the optimization procedure. It also considers all networks including cyclic ones and then choose a best structure by applying a global optimization method. To show the efficiency of the algorithm, several closely related algorithms including unrestricted dependency Bayesian Network algorithm, as well as, benchmarks algorithms SVM and C4.5 are employed for comparison. We apply these algorithms on data classification; data sets are taken from the UCI machine learning repository and the LIBSVM. © 2014, Springer-Verlag Berlin Heidelberg.

**Authors:**Taheri, Sona , Mammadov, Musa**Date:**2014**Type:**Text , Journal article**Relation:**Optimization Letters Vol. 9, no. 5 (2014), p. 931-948**Full Text:****Reviewed:****Description:**Bayesian Networks are increasingly popular methods of modeling uncertainty in artificial intelligence and machine learning. A Bayesian Network consists of a directed acyclic graph in which each node represents a variable and each arc represents probabilistic dependency between two variables. Constructing a Bayesian Network from data is a learning process that consists of two steps: learning structure and learning parameter. Learning a network structure from data is the most difficult task in this process. This paper presents a new algorithm for constructing an optimal structure for Bayesian Networks based on optimization. The algorithm has two major parts. First, we define an optimization model to find the better network graphs. Then, we apply an optimization approach for removing possible cycles from the directed graphs obtained in the first part which is the first of its kind in the literature. The main advantage of the proposed method is that the maximal number of parents for variables is not fixed a priory and it is defined during the optimization procedure. It also considers all networks including cyclic ones and then choose a best structure by applying a global optimization method. To show the efficiency of the algorithm, several closely related algorithms including unrestricted dependency Bayesian Network algorithm, as well as, benchmarks algorithms SVM and C4.5 are employed for comparison. We apply these algorithms on data classification; data sets are taken from the UCI machine learning repository and the LIBSVM. © 2014, Springer-Verlag Berlin Heidelberg.

Structure learning of Bayesian networks using a new unrestricted dependency algorithm

- Taheri, Sona, Mammadov, Musa

**Authors:**Taheri, Sona , Mammadov, Musa**Date:**2012**Type:**Text , Conference proceedings**Full Text:****Description:**Bayesian Networks have deserved extensive attentions in data mining due to their efficiencies, and reasonable predictive accuracy. A Bayesian Network is a directed acyclic graph in which each node represents a variable and each arc a probabilistic dependency between two variables. Constructing a Bayesian Network from data is the learning process that is divided in two steps: learning structure and learning parameter. In many domains, the structure is not known a priori and must be inferred from data. This paper presents an iterative unrestricted dependency algorithm for learning structure of Bayesian Networks for binary classification problems. Numerical experiments are conducted on several real world data sets, where continuous features are discretized by applying two different methods. The performance of the proposed algorithm is compared with the Naive Bayes, the Tree Augmented Naive Bayes, and the k

**Authors:**Taheri, Sona , Mammadov, Musa**Date:**2012**Type:**Text , Conference proceedings**Full Text:****Description:**Bayesian Networks have deserved extensive attentions in data mining due to their efficiencies, and reasonable predictive accuracy. A Bayesian Network is a directed acyclic graph in which each node represents a variable and each arc a probabilistic dependency between two variables. Constructing a Bayesian Network from data is the learning process that is divided in two steps: learning structure and learning parameter. In many domains, the structure is not known a priori and must be inferred from data. This paper presents an iterative unrestricted dependency algorithm for learning structure of Bayesian Networks for binary classification problems. Numerical experiments are conducted on several real world data sets, where continuous features are discretized by applying two different methods. The performance of the proposed algorithm is compared with the Naive Bayes, the Tree Augmented Naive Bayes, and the k

Statistical limit inferior and limit superior for sequences of fuzzy numbers

- Aytar, Salih, Mammadov, Musa, Pehlivan, Serpil

**Authors:**Aytar, Salih , Mammadov, Musa , Pehlivan, Serpil**Date:**2006**Type:**Text , Journal article**Relation:**Fuzzy Sets and Systems Vol. 157, no. 7 (2006), p. 976-985**Full Text:**false**Reviewed:****Description:**In this paper, we extend the concepts of statistical limit superior and limit inferior (as introduced by Fridy and Orhan [Statistical limit superior and limit inferior, Proc. Amer. Math. Soc. 125 (12) (1997) 3625-3631. [12]]) to statistically bounded sequences of fuzzy numbers and give some fuzzy-analogues of properties of statistical limit superior and limit inferior for sequences of real numbers. © 2005 Elsevier B.V. All rights reserved.**Description:**C1**Description:**2003001832

Statistical convergence and turnpike theory

**Authors:**Mammadov, Musa**Date:**2009**Type:**Text , Book chapter**Relation:**Encyclopedia of Optimization Chapter p. 3713-3718**Full Text:**false**Description:**This article considers the application of the notion of statistical convergence in turnpike theory. The first results have been obtained recently [, , ]. We briefly discuss the importance of this conjunction, present some results obtained and, finally, we formulate a challenge problem for future investigations.**Description:**2003007533

Statistical cluster points of sequences in finite dimensional spaces

- Pehlivan, Serpil, Guncan, A., Mammadov, Musa

**Authors:**Pehlivan, Serpil , Guncan, A. , Mammadov, Musa**Date:**2004**Type:**Text , Journal article**Relation:**Czechoslovak Mathematical Journal Vol. 54, no. 1 (2004), p. 95-102**Full Text:**false**Reviewed:****Description:**In this paper we study the set of statistical cluster points of sequences in m-dimensional spaces. We show that some properties of the set of statistical cluster points of the real number sequences remain in force for the sequences in m-dimensional spaces too. We also define a notion of T-statistical convergence. A sequence x is**Description:**C1**Description:**2003000896

Solving systems of nonlinear equations using a globally convergent optimization algorithm

- Taheri, Sona, Mammadov, Musa

**Authors:**Taheri, Sona , Mammadov, Musa**Date:**2012**Type:**Text , Journal article**Relation:**Global Journal of Technology & Optimization Vol. 3, no. (2012), p. 132-138**Full Text:****Reviewed:****Description:**Solving systems of nonlinear equations is a relatively complicated problem for which a number of different approaches have been presented. In this paper, a new algorithm is proposed for the solutions of systems of nonlinear equations. This algorithm uses a combination of the gradient and the Newton’s methods. A novel dynamic combinatory is developed to determine the contribution of the methods in the combination. Also, by using some parameters in the proposed algorithm, this contribution is adjusted. We use the gradient method due to its global convergence property, and the Newton’s method to speed up the convergence rate. We consider two different combinations. In the first one, a step length is determined only along the gradient direction. The second one is finding a step length along both the gradient and the Newton’s directions. The performance of the proposed algorithm in comparison to the Newton’s method, the gradient method and an existing combination method is explored on several well known test problems in solving systems of nonlinear equations. The numerical results provide evidence that the proposed combination algorithm is generally more robust and efficient than other mentioned methods on someimportant and difficult problems.

Solving a system of nonlinear integral equations by an RBF network

- Golbabai, A., Mammadov, Musa, Seifollahi, Sattar

**Authors:**Golbabai, A. , Mammadov, Musa , Seifollahi, Sattar**Date:**2009**Type:**Text , Journal article**Relation:**Computers & Mathematics with Applications Vol. 57, no. 10 (2009), p. 1651-1658**Full Text:**false**Reviewed:****Description:**In this paper, a novel learning strategy for radial basis function networks (RBFN) is proposed. By adjusting the parameters of the hidden layer, including the RBF centers and widths, the weights of the output layer are adapted by local optimization methods. A new local optimization algorithm based on a combination of the gradient and Newton methods is introduced. The efficiency of some local optimization methods to Update the weights of RBFN is Studied in solving systems of nonlinear integral equations. (C) 2009 Elsevier Ltd. All rights reserved.

Sigma supporting cone and optimality conditions in non-convex problems

- Hassani, Sara, Mammadov, Musa

**Authors:**Hassani, Sara , Mammadov, Musa**Date:**2014**Type:**Text , Journal article**Relation:**Far East Journal of Mathematical Sciences Vol. 91, no. 2 (2014), p. 169-190**Full Text:**false**Reviewed:****Description:**In this paper, a new supporting function for characterizing non-convex sets is introduced. The notions of Ïƒ-supporting cone and maximal conic gap are proposed and some properties are investigated. By applying these new notions, we establish the optimality conditions considered in [7] for a broader class of finite dimensional normed spaces in terms of weak subdifferentials.

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