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Optimization Methods and Software (10294937)
We develop an algorithm based on the idea of the bundle trust-region method to solve nonsmooth nonconvex constrained optimization problems. The resulting algorithm inherits some attractive features from both bundle and trust-region methods. Moreover, it allows effective control of the size of trust-region subproblems via the compression and aggregation techniques of bundle methods. On the other hand, the trust-region strategy is used to manage the search region and accept a candidate point as a new successful iterate. Global convergence of the developed algorithm is studied under some mild assumptions and its encouraging preliminary computational results are reported. © 2025 Informa UK Limited, trading as Taylor & Francis Group.
Dehghan manshadi, Z. ,
Sarafraz M.R. ,
Bahrami, F. ,
Nobakhtian, S. ,
Raisi, S.A.R. ,
Marandi, S.M. BMC Psychology (20507283) (1)pp. 175-184
Childhood maltreatment has profound and long-lasting effects, not only on the victims but also on their offspring when they become parents later in life. This study aimed to investigate the role of two key mediating factors—parental reflective functioning and perceived social support—in the relationship between maternal childhood maltreatment and children's emotional and behavioral problems. We conducted a cross-sectional study in Iran from March to June 2024. Mothers of preschool children (4–6 years old) with emotional or behavioral problems (N = 222; Mean age = 34.06 ± 4.2 years) completed measures of Childhood maltreatment Questionnaire (CTQ), Parental Reflective Functioning Questionnaire (PRFQ), Perceived Social Support (PSS), and children's emotional and behavioral outcomes, as assessed using the Strengths and Difficulties Questionnaire (SDQ). The SDQ includes subscales for emotional symptoms, conduct problems, hyperactivity/inattention, peer relationship problems, and prosocial behavior. Structural equation modeling (SEM) was used to assess the conceptual model. The results revealed that childhood maltreatment had no direct association with children’s emotional and behavioral problems. However, childhood maltreatment was positively and indirectly related to children’s emotional and behavioral problems (including emotional symptoms, conduct problems, hyperactivity/inattention, and peer relationship problems) and negatively and indirectly related to children’s prosocial behavior. These associations were mediated through pre-mentalizing modes of parental reflective functioning. Additionally, the analysis revealed no significant mediating role of perceived social support in this relationship. The findings highlight the significant indirect association between childhood maltreatment and various aspects of children’s emotional and behavioral problems through pre-mentalizing modes of parental reflective functioning. This underscores the critical role of enhancing parental reflective abilities to mitigate the adverse outcomes of childhood maltreatment on children’s emotional regulation and behavioral adjustment. © The Author(s) 2025.; • This study addresses a significant gap in understanding the nuanced relationship between maternal childhood maltreatment and children's emotional and behavioral problems, focusing on mediating factors such as parental reflective functioning and perceived social support. • Maternal childhood maltreatment is found to have an indirect but significant association with various aspects of children's emotional and behavioral problems, including emotional symptoms, conduct problems, hyperactivity/inattention, peer relationship difficulties, and prosocial behavior, through pre-mentalizing modes of parental reflective functioning. This indirect pathway underscores the complexity of familial influences on child development. • The indirect association of maternal childhood maltreatment with children's emotional and behavioral problems through the sequential pathway of perceived social support and parental reflective functioning was not significant. • The findings emphasize the critical importance of interventions aimed at enhancing parental reflective abilities. These interventions have the potential to mitigate the adverse associations of maternal childhood maltreatment with children's emotional regulation and behavioral adjustment, offering valuable insights for both clinical practices and policy initiatives. © The Author(s) 2025.
We propose a method for nonsmooth nonconvex multiobjective optimization problems that is based on a bundle-type approach. The proposed method employs an extension of the redistributed proximal bundle algorithm and uses an augmented improvement function to handle different objectives. In which, at each iteration, a common piecewise linear model is used to approximate the augmented improvement function. Contrary to many existing multiobjective optimization methods, this algorithm works directly with objective functions, without using any kind of a priori chosen parameters or employing any scalarization. Under appropriate assumptions, we discuss convergence to points which satisfy a necessary condition for Pareto optimality. We provide numerical results for a set of nonsmooth convex and nonconvex multiobjective optimization problems in the form of tables, figures and performance profiles. The numerical results confirm the superiority of the proposed algorithm in the considered test problems compared to other multiobjective solvers, in the computational effort necessary to compute weakly Pareto stationary points. © 2024 Informa UK Limited, trading as Taylor & Francis Group.
Computational Optimization and Applications (15732894) 88(3)pp. 871-902
This paper develops an iterative algorithm to solve nonsmooth nonconvex optimization problems on complete Riemannian manifolds. The algorithm is based on the combination of the well known trust region and bundle methods. According to the process of the most bundle methods, the objective function is approximated by a piecewise linear working model which is updated by adding cutting planes at unsuccessful trial steps. Then at each iteration, by solving a subproblem that employs the working model in the objective function subject to the trust region, a candidate descent direction is obtained. We study the algorithm from both theoretical and practical points of view and its global convergence is verified to stationary points for locally Lipschitz functions. Moreover, in order to demonstrate the reliability and efficiency, a MATLAB implementation of the proposed algorithm is prepared and results of numerical experiments are reported. © The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature 2024.
Optimization (10294945) 72(12)pp. 2893-2923
This paper focuses on the constrained minimax location problem with the closest distance. Some properties concerning the existence and uniqueness of the optimal solution are provided. To achieve these results, we apply non-smooth approach that allow us to give the explicit solution structure of the constrained problem. Moreover, we develop an effective algorithm for solving this class of problems and we provide its convergence under some mild assumptions. At the end, some computational test problems are provided to illustrate the effectiveness of the method and certify the theoretical results. © 2022 Informa UK Limited, trading as Taylor & Francis Group.
Numerical Algorithms (10171398) 94(2)pp. 765-787
In this paper, we focus on a descent algorithm for solving nonsmooth nonconvex optimization problems. The proposed method is based on the proximal bundle algorithm and the gradient sampling method and uses the advantages of both. In addition, this algorithm has the ability to handle inexact information, which creates additional challenges. The global convergence is proved with probability one. More precisely, every accumulation point of the sequence of serious iterates is either a stationary point if exact values of gradient are provided or an approximate stationary point if only inexact information of the function and gradient values is available. The performance of the proposed algorithm is demonstrated using some academic test problems. We further compare the new method with a general nonlinear solver and two other methods specifically designed for nonconvex nonsmooth optimization problems. © 2023, The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature.
Reliability Engineering and System Safety (18790836) 230
Integrity data (DI) attacks are considered malicious cyber threats to the economic performance of power markets in current power systems. A cyber attacker could mislead the system operator by implementing a DI attack, through the deviation of measured information, and causes non-optimal power distribution and erroneous participation in the electricity market (EM). This paper proposes a placement scheme of phasor measurement units (PMUs) to defend against these attacks, so that network observability is guaranteed; the possibility of detecting DI attacks by the operator is increased; and the effect of electricity price fluctuations caused by these attacks is prevented. For this purpose, we introduce two possible indices to determine the degree of attack detectability and the magnitude of system congestion variation. Accordingly, the two-objective placement model of PMUs is upgraded, in which the minimum number of PMUs and their placement must be specified to improve the proposed indices so as to minimize the possibility of financial misconduct taking place in the real time market. Using IEEE standard systems, the effectiveness of this PMU placement-based defense scheme has been confirmed. © 2022 Elsevier Ltd
Numerical Algorithms (10171398) 89(2)pp. 637-674
In this paper, a proximal bundle-based method for solving nonsmooth nonconvex constrained multiobjective optimization problems with inexact information is proposed and analyzed. In this method, each objective function is treated individually without employing any scalarization. Using the improvement function, we transform the problem into an unconstrained one. At each iteration, by the proximal bundle method, a piecewise linear model is built and by solving a convex subproblem, a new candidate iterate is obtained. For locally Lipschitz objective and constraint functions, we study the problem of computing an approximate substationary point (a substationary point), when only inexact (exact) information about the functions and subgradient values are accessible. At the end, some numerical experiments are provided to illustrate the effectiveness of the method and certify the theoretical results. © 2021, The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature.
Annals of Operations Research (02545330) 311(2)pp. 1123-1154
For a class of nonsmooth nonconvex multiobjective problems, we develop an inexact multiple proximal bundle method. In our approach instead of scalarization, we find descent direction for every objective function separately by utilizing the inexact proximal bundle method. Then we attempt to find a common descent direction for all objective functions. We study the effect of the inexactness of the objective and subgradient values on the new proposed method and obtain the reasonable convergence properties. We further consider a class of difficult nonsmooth nonconvex problems, made even more difficult by inserting the inexactness in the available information. At the end, to demonstrate the efficiency of the proposed algorithm, some encouraging numerical experiments are provided. © 2020, Springer Science+Business Media, LLC, part of Springer Nature.
Optimization Letters (18624472) 16(5)pp. 1495-1511
A proximal bundle algorithm is proposed for solving unconstrained nonsmooth nonconvex optimization problems. At each iteration, using already generated information, the algorithm defines a convex model of the augmented objective function. Then by solving a quadratic subproblem a new candidate iterate is obtained and the algorithm is repeated. The novelty in our approach is that the objective function can be any arbitrary locally Lipschitz function without any additional assumptions. The global convergence, starting from any point, is also studied. At the end, some encouraging numerical results with a MATLAB implementation are reported. © 2021, The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature.
International Journal of Electrical Power and Energy Systems (01420615) 138
False data injection (FDI) attacks can significantly impact on economic performance of electricity markets in modern power systems. These attacks can be stealthily accomplished by cyber-attackers for the purpose of profitability through financial arbitrage in electricity markets. In this paper, a new strategy of FDI attack based on Monte Carlo is proposed for an attacker participating in an electricity market, who has overmuch imperfect level of the network information. This piece of information, including both the connection /disconnection situation and admittance values of the transmission lines is denominated as topology and parametric uncertainties, respectively. Herein, a probable model is offered for analyzing the uncertainties by the Monte Carlo simulation (MCS). Afterwards, considering the probable errors of uncertainties, the attack strategy is designed in such a manner that the attacker obtains the most profit based on the contribution of each transmission line. The numerical results on two PJM 5-bus and IEEE 30-bus test networks could obviously demonstrate the success of such limited attackers in current electricity markets. © 2022 Elsevier Ltd
Optimization (10294945) 71(10)pp. 2979-3005
In this paper, we consider the problem of minimizing a continuously differentiable function subject to sparsity constraints. We formulate this problem as an equivalent disjunctive constrained optimization program. Then, we extend some of the well-known constraint qualifications by using the contingent and normal cones of the sparsity set and show that these constraint qualifications can be applied to obtain the first-order optimality conditions. In addition, we give the first-order sufficient optimality conditions by defining a new generalized convexity notion. Furthermore, we present the second-order necessary and sufficient optimality conditions for sparsity constrained optimization problems. Finally, we provide some examples and special cases to illustrate the obtained results. © 2021 Informa UK Limited, trading as Taylor & Francis Group.
Electric Power Systems Research (03787796) 205
The economic operations of real time (RT) electricity markets are vulnerable to false data injection (FDI) attacks, designed by cyber-attackers. Strategically, the RT locational marginal prices (LMPs) are stealthily altered by manipulating some of measurement data and it provides conditions for profitable financial misconduct in the electricity market. This paper proposes a new Monte Carlo-based FDI attack strategy for a cyber-attacker, who has very limited knowledge about the topology and parametric information of targeted network, which called an attacker with model topology-parametric uncertainties (TPUs). The main feature of the proposed attack is that despite the model errors, the attacker can guarantee the stealthy and profitable attack in advance, since the attack is designed based on an optimization problem of worst-case robust against uncertainties. Two 5-bus PJM and 30 IEEE bus systems are used to demonstrate the success of such cyber-attacks in real-time electricity markets. © 2021 Elsevier B.V.
TOP (11345764) 30(2)pp. 270-295
We focus on optimality conditions for an important class of nonconvex and nonsmooth optimization problems, where the objective and constraint functions are presented as a difference of two tangentially convex functions. The main contribution of this paper is to clarify several kinds of stationary solutions and their relations, and establish local optimality conditions with a nonconvex feasible set. Finally, several examples are given to illustrate the effectiveness of the obtained results. © 2021, Sociedad de Estadística e Investigación Operativa.
Journal of Global Optimization (09255001) 79(1)pp. 1-37
A filter proximal bundle algorithm is presented for nonsmooth nonconvex constrained optimization problems. The new algorithm is based on the proximal bundle method and utilizes the improvement function to regularize the constraint. At every iteration by solving a convex piecewise-linear subproblem a trial point is obtained. The process of the filter technique is employed either to accept the trial point as a serious iterate or to reject it as a null iterate. Under some mild and standard assumptions and for every possible choice of a starting point, it is shown that every accumulation point of the sequence of serious iterates is feasible. In addition, there exists at least one accumulation point which is stationary for the improvement function. Finally, some encouraging numerical results show that the proposed algorithm is effective. © 2020, Springer Science+Business Media, LLC, part of Springer Nature.
Positivity (13851292) 25(2)pp. 701-729
In this paper, the robust approach (the worst case approach) for nonsmooth nonconvex optimization problems with uncertainty data is studied. First various robust constraint qualifications are introduced based on the concept of tangential subdifferential. Further, robust necessary and sufficient optimality conditions are derived in the absence of the convexity of the uncertain sets and the concavity of the related functions with respect to the uncertain parameters. Finally, the results are applied to obtain the necessary and sufficient optimality conditions for robust weakly efficient solutions in multiobjective programming problems. In addition, several examples are provided to illustrate the advantages of the obtained outcomes. © 2020, Springer Nature Switzerland AG.
Computational Optimization and Applications (15732894) 74(2)pp. 443-480
Proximal bundle method has usually been presented for unconstrained convex optimization problems. In this paper, we develop an infeasible proximal bundle method for nonsmooth nonconvex constrained optimization problems. Using the improvement function we transform the problem into an unconstrained one and then we build a cutting plane model. The resulting algorithm allows effective control of the size of quadratic programming subproblems via the aggregation techniques. The novelty in our approach is that the objective and constraint functions can be any arbitrary (regular) locally Lipschitz functions. In addition the global convergence, starting from any point, is proved in the sense that every accumulation point of the iterative sequence is stationary for the improvement function. At the end, some encouraging numerical results with a MATLAB implementation are also reported. © 2019, Springer Science+Business Media, LLC, part of Springer Nature.
Optimization Letters (18624472) 13(5)pp. 1027-1038
This paper concerns a nonsmooth sparsity constrained optimization problem. We present first and second-order necessary and sufficient optimality conditions by using the concept of normal and tangent cones to the sparsity constraint set. Moreover, second-order tangent set to the sparsity constraint is described and then a new second-order necessary optimality condition is established. The results are illustrated by several examples. © 2018, Springer-Verlag GmbH Germany, part of Springer Nature.
Numerical Functional Analysis and Optimization (15322467) 40(16)pp. 1918-1938
We present optimality conditions for a class of nonsmooth and nonconvex constrained optimization problems. To achieve this aim, various well-known constraint qualifications are extended based on the concept of tangential subdifferential and the relations between them are investigated. Moreover, local and global necessary and sufficient optimality conditions are derived in the absence of convexity of the feasible set. In addition to the theoretical results, several examples are provided to illustrate the advantage of our outcomes. © 2019, © 2019 Taylor & Francis Group, LLC.
Mathematical Methods of Operations Research (14325217) 88(1)pp. 81-98
In this paper, we consider the rectilinear distance location problem with box constraints (RDLPBC) and we show that RDLPBC can be reduced to the rectilinear distance location problem (RDLP). A necessary and sufficient condition of optimality is provided for RDLP. A fast algorithm is presented for finding the optimal solution set of RDLP. Global convergence of the method is proved by a necessary and sufficient condition. The new proposed method is provably more efficient in finding the optimal solution set. Computational experiments highlight the magnitude of the theoretical efficiency. © 2018, Springer-Verlag GmbH Germany, part of Springer Nature.
Optimization (10294945) 67(8)pp. 1265-1286
In this paper, we propose a nonsmooth trust region algorithm for nonconvex optimization problems. The algorithm is based on notion of the Goldstein ϵ-subdifferential, which are subgradients computed in some neighbourhoods of a point. The proposed method contains a new quadratic model of the classical trust region method, in which the gradient vector is replaced by a quasisecant. Then we apply a combined approach based on the Cauchy point and the dog-leg methods in order to solve the obtained model. The global convergence is established under some suitable assumptions. Finally, the algorithm is implemented in the MATLAB environment and applied on some nonsmooth test problems. Numerical results on some small-scale and large-scale nonsmooth optimization test problems illustrate the efficiency of the proposed algorithm in the practical computation. © 2018, © 2018 Informa UK Limited, trading as Taylor & Francis Group.
4OR (16142411) 16(4)pp. 343-377
In this paper, we consider a multi-source Weber problem of m new facilities with respect to n demand regions in order to minimize the sum of the transportation costs between these facilities and the demand regions. We find a point on the border of each demand region from which the facilities serve the demand regions at these points. We present an algorithm including a location phase and an allocation phase in each iteration for solving this problem. An algorithm is also proposed for carrying out the location phase. Moreover, global convergence of the new algorithm is proved under mild assumptions, and some numerical results are presented. © 2017, Springer-Verlag GmbH Germany, part of Springer Nature.
Numerical Functional Analysis and Optimization (15322467) 39(1)pp. 11-37
In this paper, we study necessary optimality conditions for local Pareto and weak Pareto solutions of multiobjective problems involving inequality and equality constraints in terms of convexificators. We develop the enhanced Karush–Kuhn–Tucker conditions and introduce the associated pseudonormality and quasinormality conditions. We also introduce several other new constraint qualifications which entirely depend on the feasible set. Then a connecting link between these constraint qualifications is presented. Moreover, we provide several examples that clarify the interrelations between the different results that we have established. © 2018 Taylor & Francis.
Rendiconti del Circolo Matematico di Palermo (0009725X) 67(3)pp. 453-464
We state the Abadie constraint qualification for multiobjective optimization problems involving inequality, equality constraints and a closed abstract set constraint and derive the necessary optimality conditions at weak Pareto optimal solutions based on the limiting subdifferential. In contrast to the most developed works on multiobjective problems, this constraint qualification entirely depends on the feasible set. Moreover, the relationship between the Abadie constraint qualification and the local error bound condition is studied. Finally some examples are provided to clarify our results. © 2017, Springer-Verlag Italia S.r.l., part of Springer Nature.
Journal of Optimization Theory and Applications (00223239) 179(3)pp. 778-799
A nonsmooth and nonconvex general optimization problem is considered. Using the idea of pseudo-Jacobians, nonsmooth versions of the Robinson and Mangasarian–Fromovitz constraint qualifications are defined and relationships between them and the local error bound property are investigated. A new necessary optimality condition, based on the pseudo-Jacobians, is derived under the local error bound constraint qualification. These results are applied for computing and estimating the Fréchet and limiting subdifferentials of value functions. Moreover, several examples are provided to clarify the obtained results. © 2018, Springer Science+Business Media, LLC, part of Springer Nature.
TOP (11345764) 25(2)pp. 271-287
In this paper, an approximation algorithm for solving nonconvex multiobjective programming problems (NCMOPs) is presented. We modify Benson’s method using cones instead of hyperplanes. This algorithm uses an inner approximation and an outer approximation to generate (weakly) efficient solutions and (weakly ε-) nondominated points of NCMOPs. Some numerical examples are presented to clarify the proposed algorithm. © 2016, Sociedad de Estadística e Investigación Operativa.
Optimization (10294945) 66(9)pp. 1445-1463
In this paper we consider a nonsmooth optimization problem with equality, inequality and set constraints. We propose new constraint qualifications and Kuhn–Tucker type necessary optimality conditions for this problem involving locally Lipschitz functions. The main tool of our approach is the notion of convexificators. We introduce a nonsmooth version of the Mangasarian–Fromovitz constraint qualification and show that this constraint qualification is necessary and sufficient for the Kuhn–Tucker multipliers set to be nonempty and bounded. © 2017 Informa UK Limited, trading as Taylor & Francis Group.
Journal of Industrial and Management Optimization (15475816) 13(5)pp. 623-631
In this paper, the idea of convexificators is used to derive the Karush-Kuhn-Tucker necessary optimality conditions for local weak efficient solutions of multiobjective fractional problems involving inequality and equality constraints. In this regard, several well known constraint qualifications are generalized and relationships between them are investigated. Moreover, some examples are provided to clarify our results. © 2018, Journal of Industrial and Management Optimization.
Operations Research Letters (01676377) 45(4)pp. 348-352
In this paper we present sensitivity analysis for a nonsmooth optimization problem with equality and inequality constraints. A necessary optimality condition, based on the convexificators, under the local error bound constraint qualification is derived. Then, we employ them to establish upper estimates for Fréchet and limiting subdifferentials of the value function. Furthermore, we present sufficient conditions for Lipschitzness of the value function at the point of interest. Also, some examples are provided to clarify our results. © 2017 Elsevier B.V.
Mathematical Reports (15823067) 18(2)pp. 279-297
In this paper, using the idea of upper semi-regular convexificators, we propose constraint qualification and study existence and boundedness of the strong Karush-Kuhn-Tucker multipliers for proper and isolated efficiencies in nonsmooth multiobjective optimization problems with inequality, equality constraints and an arbitrary set constraint. Moreover, sufficient optimality conditions are studied for a (local) properly efficient solution. © 2016, Editura Academiei Romane. All rights reserved.
Set-Valued and Variational Analysis (18770541) 24(3)pp. 483-497
In this paper, we consider a mathematical program with equilibrium constraints (MPEC) formulated as a mathematical program with complementarity constraints. We obtain necessary conditions of Fritz John (FJ) and Karush-Kuhn-Tucker (KKT) types for a nonsmooth (MPEC) problem in terms of the lower Hadamard directional derivative. In particular sufficient conditions for MPECs are given where the involved functions have pseudoconvex sublevel sets. The functions with pseudoconvex sublevel sets is a class of generalized convex functions that include quasiconvex functions. © 2015, Springer Science+Business Media Dordrecht.
Optimization (10294945) 65(1)pp. 67-85
In this paper, we deal with constraint qualifications, stationary concepts and optimality conditions for a nonsmooth mathematical program with equilibrium constraints (MPEC). The main tool in our study is the notion of convexificator. Using this notion, standard and MPEC Abadie and several other constraint qualifications are proposed and a comparison between them is presented. We also define nonsmooth stationary conditions based on the convexificators. In particular,we show that GS-stationary is the first-order optimality condition under generalized standardAbadie constraint qualification. Finally, sufficient conditions for global or local optimality are derived under some MPEC generalized convexity assumptions. © 2014 Taylor & Francis.
Optimization (10294945) 64(8)pp. 1669-1681
We show how to use intensively local cone approximations to obtain results in some fields of optimization theory as optimality conditions, constraint qualifications, mean value theorems and error bound. © 2014 Taylor & Francis.
Numerical Functional Analysis and Optimization (15322467) 36(9)pp. 1087-1106
This article is devoted to the study of Fritz John and strong Kuhn-Tucker necessary conditions for properly efficient solutions, efficient solutions and isolated efficient solutions of a nonsmooth multiobjective optimization problem involving inequality and equality constraints and a set constraints in terms of the lower Hadamard directional derivative. Sufficient conditions for the existence of such solutions are also provided where the involved functions have pseudoconvex sublevel sets. Our results are based on the concept of pseudoconvex sublevel sets. The functions with pseudoconvex sublevel sets are a class of generalized convex functions that include quasiconvex functions. © 2015 Taylor & Francis Group, LLC.
Positivity (13851292) 19(2)pp. 221-236
This study is devoted to the semidefinite optimization problems with inequality constraints. We use the notion of convexificators to establish optimality conditions for nonsmooth semidefinite optimization problems. Moreover, we introduce appropriate constraint qualifications to present the Karush–Kuhn–Tucker multipliers. © 2014, Springer Basel.
Optimization Letters (18624472) 8(4)pp. 1517-1528
This paper is devoted to the study of nonsmooth multiobjective semi-infinite programming 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 problems, and then necessary optimality conditions for weakly efficient solutions are investigated. Finally by imposing assumptions of generalized convexity we give sufficient conditions for efficient solutions. © 2013 Springer-Verlag Berlin Heidelberg.
Optimization (10294945) 62(6)pp. 783-795
We consider a nonsmooth multiobjective programming problem with inequality and set constraints. By using the notion of convexificator, we extend the Abadie constraint qualification, and derive the strong Kuhn-Tucker necessary optimality conditions. Some other constraint qualifications have been generalized and their interrelations are investigated. © 2013 Copyright Taylor and Francis Group, LLC.
Positivity (13851292) 17(3)pp. 711-732
We consider a multiobjective optimization problem with a feasible set defined by inequality and equality constraints and a set constraint, where the objective and constraint functions are locally Lipschitz. Several constraint qualifications are given in such a way that they generalize the classical ones, when the functions are differentiable. The relationships between them are analyzed. Then, we establish strong Kuhn-Tucker necessary optimality conditions in terms of the Clarke subdifferentials such that the multipliers of the objective function are all positive. Furthermore, sufficient optimality conditions under generalized convexity assumptions are derived. Moreover, the concept of efficiency is used to formulate duality for nonsmooth multiobjective problems. Wolf and Mond-Weir type dual problems are formulated. We also establish the weak and strong duality theorems. © 2012 Springer Basel AG.
Computers and Mathematics with Applications (08981221) 64(4)pp. 550-557
This study is devoted to constraint qualifications and strong Kuhn-Tucker necessary optimality conditions for nonsmooth multiobjective optimization problems. The main tool of the study is the concept of convexificators. Mangasarian-Fromovitz type constraint qualification and several other qualifications are proposed and their relationships are investigated. In addition, sufficient optimality conditions are studied. © 2011 Elsevier Ltd. All rights reserved.
Journal of Optimization Theory and Applications (00223239) 152(1)pp. 245-255
In this paper, we consider a class of nonsmooth fractional continuous-time problems. Optimality conditions under certain structure of generalized invexity are derived for this class. Subsequently, two parameter-free dual models are formulated. Finally weak, strong, and strict converse duality theorems are proved in the framework of generalized invexity. © 2010 Springer Science+Business Media, LLC.
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.
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.
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.
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.
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.
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.
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.
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.
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.
