This paper proposed a new libration decoupling analytical speed function(LD-ASF)in lieu of the classic analytical speed function to control the climber's speed along a partial space elevator to improve libration s...This paper proposed a new libration decoupling analytical speed function(LD-ASF)in lieu of the classic analytical speed function to control the climber's speed along a partial space elevator to improve libration stability in cargo transportation.The LD-ASF is further optimized for payload transportation efficiency by a novel coordinate game theory to balance competing control objectives among payload transport speed,stable end body's libration,and overall control input via model predictive control.The transfer period is divided into several sections to reduce computational burden.The validity and efficacy of the proposed LD-ASF and coordinate game-based model predictive control are demonstrated by computer simulation.Numerical results reveal that the optimized LD-ASF results in higher transportation speed,stable end body's libration,lower thrust fuel consumption,and more flexible optimization space than the classic analytical speed function.展开更多
Improving the efficiency of ship optimization is crucial for modem ship design. Compared with traditional methods, multidisciplinary design optimization （MDO） is a more promising approach. For this reason, Collabora...Improving the efficiency of ship optimization is crucial for modem ship design. Compared with traditional methods, multidisciplinary design optimization （MDO） is a more promising approach. For this reason, Collaborative Optimization （CO） is discussed and analyzed in this paper. As one of the most frequently applied MDO methods, CO promotes autonomy of disciplines while providing a coordinating mechanism guaranteeing progress toward an optimum and maintaining interdisciplinary compatibility. However, there are some difficulties in applying the conventional CO method, such as difficulties in choosing an initial point and tremendous computational requirements. For the purpose of overcoming these problems, optimal Latin hypercube design and Radial basis function network were applied to CO. Optimal Latin hypercube design is a modified Latin Hypercube design. Radial basis function network approximates the optimization model, and is updated during the optimization process to improve accuracy. It is shown by examples that the computing efficiency and robustness of this CO method are higher than with the conventional CO method.展开更多
In this study,2~5 （five factors at two-level factorial design） design of experiment was applied to investigate a set of optimal machining parameters to achieve a minimum surface roughness value for Abies nordmannian...In this study,2~5 （five factors at two-level factorial design） design of experiment was applied to investigate a set of optimal machining parameters to achieve a minimum surface roughness value for Abies nordmanniana.Wood specimens were prepared using different values of spindle speed,feed rate,depth of cut,tool radius,and cutting directions.Average surface roughness （R_z） values were applied using a stylus.The objectives were to：（1）obtain the effective variables of wood surface roughness;（2） analyze which of these factors had an impact on variability in the CNC machining process;（3） evaluate the optimal cutting values within the range of different cutting levels of machining parameters.The results indicate that the design of experiment（DOE） based on the desirability function approach determined the optimal machining parameters successfully,leading to minimum R_a compared to the observed value.Minimum surface roughness values of tangential and radial cutting directions were 3.58 and 3.21 μm,respectively.展开更多
In this paper, both Fritz John and Karush-Kuhn-Tucker necessary optimality conditions are established for a （weakly） LU-efficient solution in the considered nonsmooth multiobjective programming problem with the mult...In this paper, both Fritz John and Karush-Kuhn-Tucker necessary optimality conditions are established for a （weakly） LU-efficient solution in the considered nonsmooth multiobjective programming problem with the multiple interval-objective function. Further, the sufficient optimality conditions for a （weakly） LU-efficient solution and several duality results in Mond-Weir sense are proved under assumptions that the functions constituting the considered nondifferentiable multiobjective programming problem with the multiple interval- objective function are convex.展开更多
Efficiently solving partial differential equations(PDEs)is a long-standing challenge in mathematics and physics research.In recent years,the rapid development of artificial intelligence technology has brought deep lea...Efficiently solving partial differential equations(PDEs)is a long-standing challenge in mathematics and physics research.In recent years,the rapid development of artificial intelligence technology has brought deep learning-based methods to the forefront of research on numerical methods for partial differential equations.Among them,physics-informed neural networks(PINNs)are a new class of deep learning methods that show great potential in solving PDEs and predicting complex physical phenomena.In the field of nonlinear science,solitary waves and rogue waves have been important research topics.In this paper,we propose an improved PINN that enhances the physical constraints of the neural network model by adding gradient information constraints.In addition,we employ meta-learning optimization to speed up the training process.We apply the improved PINNs to the numerical simulation and prediction of solitary and rogue waves.We evaluate the accuracy of the prediction results by error analysis.The experimental results show that the improved PINNs can make more accurate predictions in less time than that of the original PINNs.展开更多
Well production optimization is a complex and time-consuming task in the oilfield development.The combination of reservoir numerical simulator with optimization algorithms is usually used to optimize well production.T...Well production optimization is a complex and time-consuming task in the oilfield development.The combination of reservoir numerical simulator with optimization algorithms is usually used to optimize well production.This method spends most of computing time in objective function evaluation by reservoir numerical simulator which limits its optimization efficiency.To improve optimization efficiency,a well production optimization method using streamline features-based objective function and Bayesian adaptive direct search optimization(BADS)algorithm is established.This new objective function,which represents the water flooding potential,is extracted from streamline features.It only needs to call the streamline simulator to run one time step,instead of calling the simulator to calculate the target value at the end of development,which greatly reduces the running time of the simulator.Then the well production optimization model is established and solved by the BADS algorithm.The feasibility of the new objective function and the efficiency of this optimization method are verified by three examples.Results demonstrate that the new objective function is positively correlated with the cumulative oil production.And the BADS algorithm is superior to other common algorithms in convergence speed,solution stability and optimization accuracy.Besides,this method can significantly accelerate the speed of well production optimization process compared with the objective function calculated by other conventional methods.It can provide a more effective basis for determining the optimal well production for actual oilfield development.展开更多
To solve the global optimization problems which have several local minimizers,a new F-C function is proposes by combining a lled function and a cross function.The properties of the F-C function are discussed and the c...To solve the global optimization problems which have several local minimizers,a new F-C function is proposes by combining a lled function and a cross function.The properties of the F-C function are discussed and the corresponding algorithm is given in this paper.F-C function has the same local minimizers with the objective function.Therefore,the F-C function method only needs to minimize the objective function once in the rst iteration.Numerical experiments are performed and the results show that the proposed method is very effective.展开更多
In this note,new classes of generalized type-I functions are introduced for functions between Banach spaces.These generalized type-I functions are then utilized to establish sufficient optimality conditions and dualit...In this note,new classes of generalized type-I functions are introduced for functions between Banach spaces.These generalized type-I functions are then utilized to establish sufficient optimality conditions and duality results for a vector optimization problem with functions defined on a Banach space.展开更多
In this paper, auxiliary function method for global optimization with box constraints is considered. First, a new non-parameter filled function which has the same local minimizers of the objective function is proposed...In this paper, auxiliary function method for global optimization with box constraints is considered. First, a new non-parameter filled function which has the same local minimizers of the objective function is proposed. By the character that having same local minimizers, and these minimizers are all better than the current minimizer of the objective function, it does not need to minimize the objective function except for thefirst iteration in the filled function method. It changes the frame of conventional filled function methods that objective function and filled function are minimized alternately,and can effectively reduce the iterations of the algorithm and accelerate the speed of global optimization. And then the theoretical properties of the filled function are discussed and the corresponding algorithm is established. Finally, numerical experiments are made and comparisons on several test problems are shown which exhibit the feasibility and effectiveness of the algorithm.展开更多
A filled function with adjustable parameters is suggested in this paper for finding a global minimum point of a general class of nonlinear programming problems with a bounded and closed domain. This function has two a...A filled function with adjustable parameters is suggested in this paper for finding a global minimum point of a general class of nonlinear programming problems with a bounded and closed domain. This function has two adjustable parameters. We will discuss the properties of the proposed filled function. Conditions on this function and on the values of parameters are given so that the constructed function has the desired properties of traditional filled function.展开更多
In the paper,to solve the global optimization problems,we propose a novel parameter-free filled function.Based on the non-parameter filled function,a new filled function algorithm is designed.In the algorithm,the sele...In the paper,to solve the global optimization problems,we propose a novel parameter-free filled function.Based on the non-parameter filled function,a new filled function algorithm is designed.In the algorithm,the selection and adjustment of parameters can be ignored by the characteristic that the filled function is parameter-free.In addition,in the region lower than the current local minimizer of the objective function,the filled function is continuously differentiable which enables any gradient descent method to be used as a local search method in the algorithm.Through numerical experiments by solving two test problems,the effectiveness of the algorithm is verified.展开更多
Statistical Energy Analysis(SEA) is one of the conventional tools for predicting vehicle high-frequency acoustic responses.This study proposes a new method that can provide customized optimization solutions to meet NV...Statistical Energy Analysis(SEA) is one of the conventional tools for predicting vehicle high-frequency acoustic responses.This study proposes a new method that can provide customized optimization solutions to meet NVH targets based on the specific needs of different project teams during the initial project stages.This approach innovatively integrates dynamic optimization,Radial Basis Function(RBF),and Fuzzy Design Variables Genetic Algorithm(FDVGA) into the optimization process of Statistical Energy Analysis(SEA),and also takes vehicle sheet metal into account in the optimization of sound packages.In the implementation process,a correlation model is established through Python scripts to link material density with acoustic parameters,weight,and cost.By combining Optimus and VaOne software,an optimization design workflow is constructed and the optimization design process is successfully executed.Under various constraints related to acoustic performance,weight and cost,a globally optimal design is achieved.This technology has been effectively applied in the field of Battery Electric Vehicle(BEV).展开更多
Based on the ordering of fuzzy numbers proposed by Goetschel and Voxman,the representations and some properties of strongly preinvex fuzzy-valued function are defined and obtained, several new concepts of strongly mon...Based on the ordering of fuzzy numbers proposed by Goetschel and Voxman,the representations and some properties of strongly preinvex fuzzy-valued function are defined and obtained, several new concepts of strongly monotonicities fuzzy functions are introduced, the relationship among the strongly preinvex, strongly invex and monotonicities under some suitable and appropriate conditions is established and a necessary condition for strongly pseudoinvex functions is given. As an application, the conditions of local optimal solution and global optimal solution in the mathematical programming problem are discussed.展开更多
This article introduces a singleton type-1 fuzzy logic system(T1-SFLS) controller and Fuzzy-WDO hybrid for the autonomous mobile robot navigation and collision avoidance in an unknown static and dynamic environment. T...This article introduces a singleton type-1 fuzzy logic system(T1-SFLS) controller and Fuzzy-WDO hybrid for the autonomous mobile robot navigation and collision avoidance in an unknown static and dynamic environment. The WDO(Wind Driven Optimization) algorithm is used to optimize and tune the input/output membership function parameters of the fuzzy controller. The WDO algorithm is working based on the atmospheric motion of infinitesimal small air parcels navigates over an N-dimensional search domain. The performance of this proposed technique has compared through many computer simulations and real-time experiments by using Khepera-Ⅲ mobile robot. As compared to the T1-SFLS controller the Fuzzy-WDO algorithm is found good agreement for mobile robot navigation.展开更多
New classes of functions namely (V, ρ)_(h,φ)-type I, quasi (V, ρ)_(h,φ)-type I and pseudo (V, ρ)_(h,φ)-type I functions are defined for multiobjective programming problem by using BenTal's generalized algebr...New classes of functions namely (V, ρ)_(h,φ)-type I, quasi (V, ρ)_(h,φ)-type I and pseudo (V, ρ)_(h,φ)-type I functions are defined for multiobjective programming problem by using BenTal's generalized algebraic operation. The examples of (V, ρ)_(h,φ)-type I functions are given. The sufficient optimality conditions are obtained for multi-objective programming problem involving above new generalized convexity.展开更多
The question of optimal portfolio is that finds the trading strategy satisfying the maximal expected utility function subject to some constraints. There is the optimal trading strategy under the risk neutral probabili...The question of optimal portfolio is that finds the trading strategy satisfying the maximal expected utility function subject to some constraints. There is the optimal trading strategy under the risk neutral probability measure （martingale measure） if and only if there is no-arbitrage opportunity in the market. This paper argues the optimal wealth and the optimal value of expected utility with different utility function.展开更多
A new class of hybrid particle swarm optimization （PSO） algorithm is developed for solving the premature convergence caused by some particles in standard PSO fall into stagnation. In this algorithm, the linearly dec...A new class of hybrid particle swarm optimization （PSO） algorithm is developed for solving the premature convergence caused by some particles in standard PSO fall into stagnation. In this algorithm, the linearly decreasing inertia weight technique （LDIW） and the mutative scale chaos optimization algorithm （MSCOA） are combined with standard PSO, which are used to balance the global and local exploration abilities and enhance the local searching abilities, respectively. In order to evaluate the performance of the new method, three benchmark functions are used. The simulation results confirm the proposed algorithm can greatly enhance the searching ability and effectively improve the premature convergence.展开更多
In this article, we focus to study about modified objective function approach for multiobjective optimization problem with vanishing constraints. An equivalent η-approximated multiobjective optimization problem is co...In this article, we focus to study about modified objective function approach for multiobjective optimization problem with vanishing constraints. An equivalent η-approximated multiobjective optimization problem is constructed by a modification of the objective function in the original considered optimization problem. Furthermore, we discuss saddle point criteria for the aforesaid problem. Moreover, we present some examples to verify the established results.展开更多
基金funded by the National Natural Science Foundation of China(12102487)Basic and Applied Basic Research Foundation of Guangdong Province,China(2023A1515012339)+1 种基金Shenzhen Science and Technology Program(ZDSYS20210623091808026)the Discovery Grant(RGPIN-2024-06290)of the Natural Sciences and Engineering Research Council of Canada。
文摘This paper proposed a new libration decoupling analytical speed function(LD-ASF)in lieu of the classic analytical speed function to control the climber's speed along a partial space elevator to improve libration stability in cargo transportation.The LD-ASF is further optimized for payload transportation efficiency by a novel coordinate game theory to balance competing control objectives among payload transport speed,stable end body's libration,and overall control input via model predictive control.The transfer period is divided into several sections to reduce computational burden.The validity and efficacy of the proposed LD-ASF and coordinate game-based model predictive control are demonstrated by computer simulation.Numerical results reveal that the optimized LD-ASF results in higher transportation speed,stable end body's libration,lower thrust fuel consumption,and more flexible optimization space than the classic analytical speed function.
文摘Improving the efficiency of ship optimization is crucial for modem ship design. Compared with traditional methods, multidisciplinary design optimization （MDO） is a more promising approach. For this reason, Collaborative Optimization （CO） is discussed and analyzed in this paper. As one of the most frequently applied MDO methods, CO promotes autonomy of disciplines while providing a coordinating mechanism guaranteeing progress toward an optimum and maintaining interdisciplinary compatibility. However, there are some difficulties in applying the conventional CO method, such as difficulties in choosing an initial point and tremendous computational requirements. For the purpose of overcoming these problems, optimal Latin hypercube design and Radial basis function network were applied to CO. Optimal Latin hypercube design is a modified Latin Hypercube design. Radial basis function network approximates the optimization model, and is updated during the optimization process to improve accuracy. It is shown by examples that the computing efficiency and robustness of this CO method are higher than with the conventional CO method.
基金performed in the laboratuary of Istanbul University,Faculty of Forestry where the authors are employed at present
文摘In this study,2~5 （five factors at two-level factorial design） design of experiment was applied to investigate a set of optimal machining parameters to achieve a minimum surface roughness value for Abies nordmanniana.Wood specimens were prepared using different values of spindle speed,feed rate,depth of cut,tool radius,and cutting directions.Average surface roughness （R_z） values were applied using a stylus.The objectives were to：（1）obtain the effective variables of wood surface roughness;（2） analyze which of these factors had an impact on variability in the CNC machining process;（3） evaluate the optimal cutting values within the range of different cutting levels of machining parameters.The results indicate that the design of experiment（DOE） based on the desirability function approach determined the optimal machining parameters successfully,leading to minimum R_a compared to the observed value.Minimum surface roughness values of tangential and radial cutting directions were 3.58 and 3.21 μm,respectively.
文摘In this paper, both Fritz John and Karush-Kuhn-Tucker necessary optimality conditions are established for a （weakly） LU-efficient solution in the considered nonsmooth multiobjective programming problem with the multiple interval-objective function. Further, the sufficient optimality conditions for a （weakly） LU-efficient solution and several duality results in Mond-Weir sense are proved under assumptions that the functions constituting the considered nondifferentiable multiobjective programming problem with the multiple interval- objective function are convex.
基金Project supported by the National Natural Science Foundation of China(Grant Nos.42005003 and 41475094).
文摘Efficiently solving partial differential equations(PDEs)is a long-standing challenge in mathematics and physics research.In recent years,the rapid development of artificial intelligence technology has brought deep learning-based methods to the forefront of research on numerical methods for partial differential equations.Among them,physics-informed neural networks(PINNs)are a new class of deep learning methods that show great potential in solving PDEs and predicting complex physical phenomena.In the field of nonlinear science,solitary waves and rogue waves have been important research topics.In this paper,we propose an improved PINN that enhances the physical constraints of the neural network model by adding gradient information constraints.In addition,we employ meta-learning optimization to speed up the training process.We apply the improved PINNs to the numerical simulation and prediction of solitary and rogue waves.We evaluate the accuracy of the prediction results by error analysis.The experimental results show that the improved PINNs can make more accurate predictions in less time than that of the original PINNs.
基金supported partly by the National Science and Technology Major Project of China(Grant No.2016ZX05025-001006)Major Science and Technology Project of CNPC(Grant No.ZD2019-183-007)
文摘Well production optimization is a complex and time-consuming task in the oilfield development.The combination of reservoir numerical simulator with optimization algorithms is usually used to optimize well production.This method spends most of computing time in objective function evaluation by reservoir numerical simulator which limits its optimization efficiency.To improve optimization efficiency,a well production optimization method using streamline features-based objective function and Bayesian adaptive direct search optimization(BADS)algorithm is established.This new objective function,which represents the water flooding potential,is extracted from streamline features.It only needs to call the streamline simulator to run one time step,instead of calling the simulator to calculate the target value at the end of development,which greatly reduces the running time of the simulator.Then the well production optimization model is established and solved by the BADS algorithm.The feasibility of the new objective function and the efficiency of this optimization method are verified by three examples.Results demonstrate that the new objective function is positively correlated with the cumulative oil production.And the BADS algorithm is superior to other common algorithms in convergence speed,solution stability and optimization accuracy.Besides,this method can significantly accelerate the speed of well production optimization process compared with the objective function calculated by other conventional methods.It can provide a more effective basis for determining the optimal well production for actual oilfield development.
基金Supported by National Natural Science Foundation of China(No.11471102)Basic research projects for key scientific research projects in Henan Province(No.20ZX001)。
文摘To solve the global optimization problems which have several local minimizers,a new F-C function is proposes by combining a lled function and a cross function.The properties of the F-C function are discussed and the corresponding algorithm is given in this paper.F-C function has the same local minimizers with the objective function.Therefore,the F-C function method only needs to minimize the objective function once in the rst iteration.Numerical experiments are performed and the results show that the proposed method is very effective.
基金Foundation item： Supported by the National Natural Science Foundation of China（60574075） University, engaged in optimization theory and application.
文摘In this note,new classes of generalized type-I functions are introduced for functions between Banach spaces.These generalized type-I functions are then utilized to establish sufficient optimality conditions and duality results for a vector optimization problem with functions defined on a Banach space.
基金Supported by National Natural Science Foundation of China (Grant No. 11471102, 11701150,12071112)Basic research projects for key scientific research projects in Henan Province (Grant No. 20ZX001)。
文摘In this paper, auxiliary function method for global optimization with box constraints is considered. First, a new non-parameter filled function which has the same local minimizers of the objective function is proposed. By the character that having same local minimizers, and these minimizers are all better than the current minimizer of the objective function, it does not need to minimize the objective function except for thefirst iteration in the filled function method. It changes the frame of conventional filled function methods that objective function and filled function are minimized alternately,and can effectively reduce the iterations of the algorithm and accelerate the speed of global optimization. And then the theoretical properties of the filled function are discussed and the corresponding algorithm is established. Finally, numerical experiments are made and comparisons on several test problems are shown which exhibit the feasibility and effectiveness of the algorithm.
基金Supported by the National Science Foundation of China(10171118)Supported by the Science Foundation of University of Science and Technology of Henan(2003ZY06)
文摘A filled function with adjustable parameters is suggested in this paper for finding a global minimum point of a general class of nonlinear programming problems with a bounded and closed domain. This function has two adjustable parameters. We will discuss the properties of the proposed filled function. Conditions on this function and on the values of parameters are given so that the constructed function has the desired properties of traditional filled function.
文摘In the paper,to solve the global optimization problems,we propose a novel parameter-free filled function.Based on the non-parameter filled function,a new filled function algorithm is designed.In the algorithm,the selection and adjustment of parameters can be ignored by the characteristic that the filled function is parameter-free.In addition,in the region lower than the current local minimizer of the objective function,the filled function is continuously differentiable which enables any gradient descent method to be used as a local search method in the algorithm.Through numerical experiments by solving two test problems,the effectiveness of the algorithm is verified.
文摘Statistical Energy Analysis(SEA) is one of the conventional tools for predicting vehicle high-frequency acoustic responses.This study proposes a new method that can provide customized optimization solutions to meet NVH targets based on the specific needs of different project teams during the initial project stages.This approach innovatively integrates dynamic optimization,Radial Basis Function(RBF),and Fuzzy Design Variables Genetic Algorithm(FDVGA) into the optimization process of Statistical Energy Analysis(SEA),and also takes vehicle sheet metal into account in the optimization of sound packages.In the implementation process,a correlation model is established through Python scripts to link material density with acoustic parameters,weight,and cost.By combining Optimus and VaOne software,an optimization design workflow is constructed and the optimization design process is successfully executed.Under various constraints related to acoustic performance,weight and cost,a globally optimal design is achieved.This technology has been effectively applied in the field of Battery Electric Vehicle(BEV).
基金Supported by Natural Science Foundation of Gansu Province of China (Grant No.18JR3RM238)Research Foundation of Higher Education of Gansu Province of China (Grant No. 2018A-101)Innovation Ability promotion Project of Higher Education of Gansu Province of China (Grant No. 2019A-117)。
文摘Based on the ordering of fuzzy numbers proposed by Goetschel and Voxman,the representations and some properties of strongly preinvex fuzzy-valued function are defined and obtained, several new concepts of strongly monotonicities fuzzy functions are introduced, the relationship among the strongly preinvex, strongly invex and monotonicities under some suitable and appropriate conditions is established and a necessary condition for strongly pseudoinvex functions is given. As an application, the conditions of local optimal solution and global optimal solution in the mathematical programming problem are discussed.
文摘This article introduces a singleton type-1 fuzzy logic system(T1-SFLS) controller and Fuzzy-WDO hybrid for the autonomous mobile robot navigation and collision avoidance in an unknown static and dynamic environment. The WDO(Wind Driven Optimization) algorithm is used to optimize and tune the input/output membership function parameters of the fuzzy controller. The WDO algorithm is working based on the atmospheric motion of infinitesimal small air parcels navigates over an N-dimensional search domain. The performance of this proposed technique has compared through many computer simulations and real-time experiments by using Khepera-Ⅲ mobile robot. As compared to the T1-SFLS controller the Fuzzy-WDO algorithm is found good agreement for mobile robot navigation.
基金Supported by the NSF of Shaanxi Provincial Educational Department(06JK152)
文摘New classes of functions namely (V, ρ)_(h,φ)-type I, quasi (V, ρ)_(h,φ)-type I and pseudo (V, ρ)_(h,φ)-type I functions are defined for multiobjective programming problem by using BenTal's generalized algebraic operation. The examples of (V, ρ)_(h,φ)-type I functions are given. The sufficient optimality conditions are obtained for multi-objective programming problem involving above new generalized convexity.
文摘The question of optimal portfolio is that finds the trading strategy satisfying the maximal expected utility function subject to some constraints. There is the optimal trading strategy under the risk neutral probability measure （martingale measure） if and only if there is no-arbitrage opportunity in the market. This paper argues the optimal wealth and the optimal value of expected utility with different utility function.
文摘A new class of hybrid particle swarm optimization （PSO） algorithm is developed for solving the premature convergence caused by some particles in standard PSO fall into stagnation. In this algorithm, the linearly decreasing inertia weight technique （LDIW） and the mutative scale chaos optimization algorithm （MSCOA） are combined with standard PSO, which are used to balance the global and local exploration abilities and enhance the local searching abilities, respectively. In order to evaluate the performance of the new method, three benchmark functions are used. The simulation results confirm the proposed algorithm can greatly enhance the searching ability and effectively improve the premature convergence.
基金financially supported by the CSIR,New Delhi,India through Grant no.:25(0266)/17/EMR-II
文摘In this article, we focus to study about modified objective function approach for multiobjective optimization problem with vanishing constraints. An equivalent η-approximated multiobjective optimization problem is constructed by a modification of the objective function in the original considered optimization problem. Furthermore, we discuss saddle point criteria for the aforesaid problem. Moreover, we present some examples to verify the established results.
