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Journal of Optimization Theory and Applications (00223239) 107(1)pp. 89-122
For a general fixed-duration optimal control problem, the proximal aiming technique of nonsmooth analysis is employed in order to construct a discontinuous feedback law, whose Euler solutions are all optimal to within a prescribed tolerance, universally for all initial data in a prescribed bounded set. The technique is adapted in order to construct universal near-saddle points for two-player fixed-duration differential games of the Krasovskii-Subbotin type.
International Journal Of Mathematics And Mathematical Sciences (16870425) 2006
We are concerned with a nonsmooth multiobjective optimization problem with inequality constraints. In order to obtain our main results, we give the definitions of the generalized convex functions based on the generalized directional derivative. Under the above generalized convexity assumptions, sufficient and necessary conditions for optimality are given without the need of a constraint qualification. Then we formulate the dual problem corresponding to the primal problem, and some duality results are obtained without a constraint qualification. Copyright © 2006 Hindawi Publishing Corporation. All rights reserved.
Computers and Mathematics with Applications (08981221) 51(9-10)pp. 1385-1394
In this paper, we consider notion of infine functions and we establish necessary and sufficient optimality conditions for a feasible solution of a multiobjective optimization problem involving mixed constraints (equality and inequality) to be an efficient or properly efficient solution. We also obtain duality theorems for Wolf type and Mond-Weir type duals under the generalized invexity assumptions. © 2006 Elsevier Ltd.
Journal of Global Optimization (09255001) 35(4)pp. 593-606
In this paper a generalization of invexity is considered in a general form, by means of the concept of K-directional derivative. Then in the case of nonlinear multiobjective programming problems where the functions involved are nondifferentiable, we established sufficient optimality conditions without any convexity assumption of the K-directional derivative. Then we obtained some duality results. © Springer 2006.
Journal of Optimization Theory and Applications (00223239) 130(2)pp. 359-365
In this paper, we consider a generalization of convexity for nonsmooth multiobjective programming problems. We obtain sufficient optimality conditions under generalized (F, ρ)-convexity. © 2006 Springer Science+Business Media, Inc.
Numerical Functional Analysis and Optimization (15322467) 28(11-12)pp. 1355-1367
A class of nonsmooth multiobjective fractional programming is formulated. We establish the necessary and sufficient optimality conditions without the need of a constraint qualification. Then a mixed dual is introduced for a class of nonsmooth fractional programming problems, and various duality theorems are established without a constraint qualification.
Security quantification is a topic that has gained a lot of interest in the research community during the recent years. In this paper, a new method is proposed for modeling and quantifying attack effects on a computer system. In this work, intrusion process is considered as atomic sequential steps. Each atomic step changes the current system state. On the other hand, system tries to prevent and detect the attacker activity and therefore can transfer the current system state to a secure state. Intrusion process modeling is done by a semi-Markov chain (SMC). Distribution functions assigned to SMC transitions are uniform distributions. Uniform distributions represent the sojourn time of the attacker or the system in the transient states. Then the SMC is converted into a discrete-time Markov chain (DTMC). The DTMC is analyzed and then the probability of attacker success is computed based on mathematical theorems. The SMC has two absorbing for representing success and failure states of intrusion process.©2008 IEEE.
Journal of Optimization Theory and Applications (00223239) 136(1)pp. 77-85
We consider a nonsmooth vector optimization continuous-time problem. We establish weak and strong duality theorems under generalized convexity assumptions. © 2007 Springer Science+Business Media, LLC.
Journal of Optimization Theory and Applications (00223239) 136(1)pp. 61-68
A mixed-type dual for a nonsmooth multiobjective optimization problem with inequality and equality constraints is formulated. We obtain weak and strong duality theorems for a mixed-type dual without requiring the regularity assumptions and the nonnegativeness of the Lagrange multipliers associated to the equality constraints. We apply also a nonsmooth constraint qualification for multiobjective programming to establish strong duality results. In this case, our constraint qualification assures the existence of positive Lagrange multipliers associated with the vector-valued objective function. © 2007 Springer Science+Business Media, LLC.
Journal of Global Optimization (09255001) 41(1)pp. 103-115
We consider nonsmooth multiobjective fractional programming problems with inequality and equality constraints. We establish the necessary and sufficient optimality conditions under various generalized invexity assumptions. In addition, we formulate a mixed dual problem corresponding to primal problem, and discuss weak, strong and strict converse duality theorems. © 2007 Springer Science+Business Media LLC.
Journal of Optimization Theory and Applications (00223239) 136(1)pp. 69-76
A nonsmooth multiobjective continuous-time problem is introduced. We establish the necessary and sufficient optimality conditions under generalized convexity assumptions on the functions involved. © 2007 Springer Science+Business Media, LLC.
Journal of Pure and Applied Algebra (00224049) 212(4)pp. 727-734
In this paper we study the probability that the commutator of two randomly chosen elements in a finite group is equal to a given element of that group. Explicit computations are obtained for groups G which | G′ | is prime and G′ ≤ Z (G) as well as for groups G which | G′ | is prime and G′ ∩ Z (G) = 1. This paper extends results of Rusin [see D.J. Rusin, What is the probability that two elements of a finite group commute? Pacific J. Math. 82 (1) (1979) 237-247]. © 2007 Elsevier Ltd. All rights reserved.
Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) (03029743) 5430pp. 200-214
The aim is to propose a new approach for stochastic modeling of an intrusion process and quantitative evaluation of the probability of the attacker success. In many situations of security analysis, it is necessary to obtain the probabilities of success for attackers in an intrusion process. In the proposed method, the intrusion process is considered as elementary attack phases. In each atomic phase the attacker and the system interact and this interaction can transfer the current system state to a secure or failure state. Intrusion process modeling is done by a semi-Markov chain (SMC). The distribution functions assigned to the SMC transitions are a linear combination of some uniform distributions. These mixture distributions represent the time distribution of the attacker or the system in the transient states. In order to evaluate the security measure, the SMC is converted into a discrete-time Markov chain (DTMC) and then the resulting DTMC is analyzed and the probability of the attacker success is com uted based on mathematical theorems. The desired security measure is evaluated with respect to the temporal aspects of the attacker behavior. ©Springer-Verlag Berlin Heidelberg 2009.
Finite Fields and their Applications (10715797) 15(3)pp. 387-391
Linear cyclic codes of length pk over the Galois ring GR (p2, m), that is ideals of the ring GR (p2, m) [u] / 〈 upk - 1 〉, are studied. The form of the dual codes is analyzed and self-dual codes are identified. © 2009 Elsevier Inc. All rights reserved.
Set-Valued and Variational Analysis (09276947) 17(1)pp. 63-95
We study nonsmooth mathematical programs with equilibrium constraints. First we consider a general disjunctive program which embeds a large class of problems with equilibrium constraints. Then, we establish several constraint qualifications for these optimization problems. In particular, we generalize the Abadie and Guignard-type constraint qualifications. Subsequently, we specialize these results to mathematical program with equilibrium constraints. In our investigation, we show that a local minimum results in a so-called M-stationary point under a very weak constraint qualification. © 2009 Springer Science+Business Media B.V.
Discrete Applied Mathematics (0166218X) 157(13)pp. 2892-2903
This paper deals with cyclic codes over the Galois ring GR (p2, m). A unique set of generators for these codes and an algorithm for finding these generators are presented. The form of dual codes is studied. The obtained results on cyclic codes are extended to the class of negacyclic codes. © 2009 Elsevier B.V. All rights reserved.
The aim is to develop a suitable method for quantifying security. We use stochastic modeling techniques for this purpose. An intrusion process is considered as a series of elementary attack phases and at each phase the interactions between the attacker and the system are analyzed rigorously. It is assumed that a typical attacker needs some time to perform an elementary attack phase. On the other hand, it is assumed that the attacker may be detected by the system and thus the overall intrusion process is interrupted. The attacker skill level and the system's abilities are characterized by the uniform distribution functions assigned to the transitions of the model. The underlying stochastic model is recognized as a semi- Markov chain. For security analysis, some valid assumptions about intrusion process are considered. Also, two quantitative security measures are defined and evaluated based on the model. The proposed method is demonstrated by modeling a complicated attack process and evaluating the desired security measures © 2009 IEEE.
Numerical Functional Analysis and Optimization (15322467) 30(3-4)pp. 337-351
We study a multiobjective problem with a feasible set defined by equality and inequality constraints. Then, by using the concept of K-directional derivative, we prove general optimality conditions as well as results concerning duality theorems.
Journal of Convex Analysis (09446532) 16(1)pp. 187-210
We consider a mathematical program with equilibrium constraints (MPBC). First we obtain a Lagrange multiplier rule based on the linear sub differential involving equality, inequality and set constraints. Then we propose new constraint qualifications for M-stationary condition to hold. Finally we establish the Fritz John and Karush-Kuhn Tucker M-stationary necessary conditions for a nonsmooth (MPBC) based on the Michel-Penot subdifferential. © Heldermann Verlag.
Journal of Global Optimization (09255001) 43(4)pp. 593-606
A nonsmooth multiobjective continuous-time problem is considered. The definition of invexity and its generalizations for continuous-time functions are extended. Then, optimality conditions under generalized invexity assumptions are established. Subsequently, these optimality conditions are utilized as a basis for formulating dual problems. Duality results are also obtained for Wolfe as well as Mond-Weir type dual, using generalized invexity on the functions involved. © 2008 Springer Science+Business Media, LLC.
Journal of Mathematical Analysis and Applications (10960813) 351(1)pp. 170-181
We consider a nonsmooth semi-infinite programming problem with a feasible set defined by inequality and equality constraints and a set constraint. First, we study some alternative theorems which involve linear and sublinear functions and a convex set and we propose several generalizations of them. Then, alternative theorems are applied to obtain, under different constraint qualifications, several necessary optimality conditions in the type of Fritz-John and Karush-Kuhn-Tucker. © 2008 Elsevier Inc. All rights reserved.
IEEE Transactions on Communications (00906778) 58(5)pp. 1333-1337
This letter proposes a new and comprehensive set of convergence conditions for a distributed optimal traffic engineering method for connectionless networks. The method was originally proposed in [2], but we show in this letter that the convergence conditions accompanying the method in [2] are not comprehensive and work only in some conditions. More precisely, it is shown that the adaptation laws in [2] will misbehave in some situations. Consequently, they can not achieve optimal resource allocation. This letter presents the correct form of the convergence conditions under which the adaptation laws can effectively lead to optimal network utilization. © 2006 IEEE.
A group G is an A-group if x(alpha)x = xx(alpha) for all x is an element of G and all automorphisms alpha of G. Such groups have nilpotency class at most 3; we construct the first example having class precisely 3.
Nonlinear Analysis, Theory, Methods and Applications (0362546X) 72(5)pp. 2694-2705
In this paper we consider a mathematical program with equilibrium constraints (MPEC) formulated as a mathematical program with complementarity constraints. Then, we derive a necessary optimality result for nonsmooth MPEC on any Asplund space. Also, under generalized convexity assumptions, we establish sufficient optimality conditions for this program in Banach spaces. © 2009 Elsevier Ltd. All rights reserved.
European Journal of Operational Research (03772217) 205(2)pp. 253-261
This paper is devoted to the study of nonsmooth generalized semi-infinite programming problems in which the index set of the inequality constraints depends on the decision vector and all emerging functions are assumed to be locally Lipschitz. We introduce a constraint qualification which is based on the Mordukhovich subdifferential. Then, we derive a Fritz-John type necessary optimality condition. Finally, interrelations between the new and the existing constraint qualifications such as the Mangasarian-Fromovitz, linear independent, and the Slater are investigated. © 2010 Elsevier B.V. All rights reserved.
In this chapter we present some classes of nonsmooth continuous-time problems. Optimality conditions under certain structure of generalized convexity are derived for these classes. Subsequently, two dual models are formulated and weak and strong duality theorems are established.All rights reserved - © 2010 Bentham Science Publishers Ltd. All rights reserved.
Optimization (10294945) 59(5)pp. 717-727
This article deals with a class of non-smooth semi-infinite programming (SIP) problems in which the index set of the inequality constraints is an arbitrary set not necessarily finite. We introduce several kinds of constraint qualifications for these non-smooth SIP problems and we study the relationships between them. Finally, necessary and sufficient optimality conditions are investigated. © 2010 Taylor & Francis.
IEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences (17451337) 93(4)pp. 808-813
A generalized Gray map for codes over the ring Fq〈u〉/ 〈ut+1〉 is introduced, where q = pm is a prime power. It is shown that the generalized Gray image of a linear length-N (1 - ut)-cyclic code over Fq[u]/〈ut+1〉 is a distance-invariant linear length-qtN quasi-cyclic code of index qt/p over Fq. It turns out that if (N, p) = 1 then every linear code over Fq that is the generalized Gray image of a length-N cyclic code over Fq[u]/〈ut+1〉, is also equivalent to a linear length-qtN quasi-cyclic code of index q t/p over Fq. The relationship between linear length-p N cyclic codes with (N, p) = 1 over Fp and linear length-N cyclic codes over Fp + uFp is explicitly determined. Copyright © 2010 The Institute of Electronics, Information and Communication Engineers.
Numerical Functional Analysis and Optimization (15322467) 32(11)pp. 1175-1189
We introduce a concept of generalized invexity for the nonsmooth continuous time optimization problems, namely, the concept of Karush-Kuhn-Tucker (KKT) invexity. Then, we prove that this notion is necessary and sufficient for global optimality of a KKT point. We also extend the notion of weak-invexity for nonsmooth continuous time optimization problems. Further, we show that weak-invexity is a necessary and sufficient condition for weak duality. Copyright © Taylor & Francis Group, LLC.
International Journal of Geometric Methods in Modern Physics (02198878) 8(8)pp. 1747-1762
A gauge invariant partition function is defined for gauge theories which leads to the standard quantization. It is shown that the descent equations and consequently the consistent anomalies and Schwinger terms can be extracted from this modified partition function naturally. © 2011 World Scientific Publishing Company.
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