In this paper,we study the minimax linear fractional programming problem on a non-empty bounded set,called problem(MLFP),and we design a branch and bound algorithm to find a globally optimal solution of(MLFP).Firstly,...In this paper,we study the minimax linear fractional programming problem on a non-empty bounded set,called problem(MLFP),and we design a branch and bound algorithm to find a globally optimal solution of(MLFP).Firstly,we convert the problem(MLFP)to a problem(EP2)that is equivalent to it.Secondly,by applying the convex relaxation technique to problem(EP2),a convex quadratic relaxation problem(CQRP)is obtained.Then,the overall framework of the algorithm is given and its convergence is proved,the worst-case iteration number is also estimated.Finally,experimental data are listed to illustrate the effectiveness of the algorithm.展开更多
First, a class of higher order exponential type hybrid （α,β, γ, η, p, h（.,.）, κ（., .）, w（.,., .）, ω（.,.,.）, θ）-invexities is introduced, second, some parametrically sufficient efficiency conditions ba...First, a class of higher order exponential type hybrid （α,β, γ, η, p, h（.,.）, κ（., .）, w（.,., .）, ω（.,.,.）, θ）-invexities is introduced, second, some parametrically sufficient efficiency conditions based on the higher order exponential type hybrid invexities are established, and finally some parametrically sufficient efficiency results under the higher order exponential type hybrid （a,β, γ, ρ, h（.,.）, k（.,-）, w（-,., .）, w（.,., .）, 0）-invexities are investigated to the context of solving semiinfinite multiobjective fractional programming problems. The notions of the higher order exponential type hybrid （a, β, γ η, p, h（., .）, n（., .）, w（-,.,-）, ω（.,.,.）, 0）-invexities encompass most of the generalized invexities in the literature. To the best of our knowledge, the results on semiinfinite multiobjective fractional programming problems established in this communication are new and application-oriented toward multitime multi- objectve problems as well as multiobiective control problems.展开更多
A new concept of(Φ,ρ,α)-V-invexity for differentiable vector-valued functions is introduced,which is a generalization of differentiable scalar-valued(Φ,ρ)-invexity.Based upon the(Φ,ρ,α)-V-invex functions,suffi...A new concept of(Φ,ρ,α)-V-invexity for differentiable vector-valued functions is introduced,which is a generalization of differentiable scalar-valued(Φ,ρ)-invexity.Based upon the(Φ,ρ,α)-V-invex functions,sufficient optimality conditions and MondWeir type dual theorems are derived for a class of nondifferentiable multiobjective fractional programming problems in which every component of the objective function and each constraint function contain a term involving the support function of a compact convex set.展开更多
This paper studies a class of multiobjective generalized fractional programming problems, where the numerators of objective functions are the sum of differentiable function and convex function, while the denominators ...This paper studies a class of multiobjective generalized fractional programming problems, where the numerators of objective functions are the sum of differentiable function and convex function, while the denominators are the difference of differentiable function and convex function. Under the assumption of Calmness Constraint Qualification the Kuhn-Tucker type necessary conditions for efficient solution are given, and the Kuhn-Tucker type sufficient conditions for efficient solution are presented under the assumptions of （F, α, ρ, d）-V-convexity. Subsequently, the optimality conditions for two kinds of duality models are formulated and duality theorems are proved.展开更多
In this article,the authors discuss the optimal conditions of the linear fractionalprogramming problem and prove that a locally optional solution is a globally optional solution and the locally optimal solution can be...In this article,the authors discuss the optimal conditions of the linear fractionalprogramming problem and prove that a locally optional solution is a globally optional solution and the locally optimal solution can be attained at a basic feasible solution withconstraint condition.展开更多
In this paper, two duality results are established under generalized ρ-convexity conditions for a class of multiobjective fractional programmign involvign differentiable n-sten functions.
In this paper we define higher order (F,α,β,ρ,d,E)-convex function with respect to E-differentiable function K and obtain optimality conditions for nonlinear programming problem (NP) from the concept of higher orde...In this paper we define higher order (F,α,β,ρ,d,E)-convex function with respect to E-differentiable function K and obtain optimality conditions for nonlinear programming problem (NP) from the concept of higher order (F,α,β,ρ,d)-convexity. Here, we establish Mond-Weir and Wolfe duality for (NP) and utilize these duality in nonlinear fractional programming problem.展开更多
In this paper, we introduce a class of generalized second order （F,α,ρ , d,p）-univex functions. Two types of second order dual models are considered for a minimax fractional programming problem and the duality res...In this paper, we introduce a class of generalized second order （F,α,ρ , d,p）-univex functions. Two types of second order dual models are considered for a minimax fractional programming problem and the duality results are established by using the assumptions on the functions involved.展开更多
The mixed-integer quadratically constrained quadratic fractional programming(MIQCQFP)problem often appears in various fields such as engineering practice,management science and network communication.However,most of th...The mixed-integer quadratically constrained quadratic fractional programming(MIQCQFP)problem often appears in various fields such as engineering practice,management science and network communication.However,most of the solutions to such problems are often designed for their unique circumstances.This paper puts forward a new global optimization algorithm for solving the problem MIQCQFP.We first convert the MIQCQFP into an equivalent generalized bilinear fractional programming(EIGBFP)problem with integer variables.Secondly,we linearly underestimate and linearly overestimate the quadratic functions in the numerator and the denominator respectively,and then give a linear fractional relaxation technique for EIGBFP on the basis of non-negative numerator.After that,combining rectangular adjustment-segmentation technique and midpointsampling strategy with the branch-and-bound procedure,an efficient algorithm for solving MIQCQFP globally is proposed.Finally,a series of test problems are given to illustrate the effectiveness,feasibility and other performance of this algorithm.展开更多
A semi-infinite programming problem is a mathematical programming problem with a finite number of variables and infinitely many constraints. Duality theories and generalized convexity concepts are important research t...A semi-infinite programming problem is a mathematical programming problem with a finite number of variables and infinitely many constraints. Duality theories and generalized convexity concepts are important research topics in mathematical programming. In this paper, we discuss a fairly large number of paramet- ric duality results under various generalized （η,ρ）-invexity assumptions for a semi-infinite minmax fractional programming problem.展开更多
This paper presents an efficient algorithm for globally solving a generalized linear fractional programming problem.For establishing this algorithm,we firstly construct a two-level linear relaxation method,and by util...This paper presents an efficient algorithm for globally solving a generalized linear fractional programming problem.For establishing this algorithm,we firstly construct a two-level linear relaxation method,and by utilizing the method,we can convert the initial generalized linear fractional programming problem and its subproblems into a series of linear programming relaxation problems.Based on the branch-and-bound framework and linear programming relaxation problems,a branch-and-bound algorithm is presented for globally solving the generalized linear fractional programming problem,and the computational complexity of the algorithm is given.Finally,numerical experimental results demonstrate the feasibility and efficiency of the proposed algorithm.展开更多
In this paper, we discuss a large number of sets of global parametric sufficient optimality conditions under various generalized （η,ρ）-invexity assumptions for a semi-infinite minmax fractional programming problem.
This paper considers a nonsmooth semi-infinite minimax fractional programming problem（SIMFP） involving locally Lipschitz invex functions. The authors establish necessary optimality conditions for SIMFP. The authors ...This paper considers a nonsmooth semi-infinite minimax fractional programming problem（SIMFP） involving locally Lipschitz invex functions. The authors establish necessary optimality conditions for SIMFP. The authors establish the relationship between an optimal solution of SIMFP and saddle point of scalar Lagrange function for SIMFP. Further, the authors study saddle point criteria of a vector Lagrange function defined for SIMFP.展开更多
For optimizing the water-use structure and increasing plantation benefit of unit water consumption,a multi-objective model for water resources utilization was established based on fractional programming(FP).Meanwhile,...For optimizing the water-use structure and increasing plantation benefit of unit water consumption,a multi-objective model for water resources utilization was established based on fractional programming(FP).Meanwhile,considering the stochasticity of water availability in the study area,the impact of the risk factor(λ)from a quantitative and qualitative perspective was analyzed.The chance-constrained programming(CCP)and conditional value-at-risk(CVaR)models were introduced into five important major grain production areas in Sanjiang Plain,and the crop planting structure under this condition was optimized.The results showed that,after optimization,overall benefit of cultivation increased from 42.07 billion Yuan to 42.47 billion Yuan,water consumption decreased from 15.90 billion m3 to 11.95 billion m3,the plantation benefit of unit water consumption increased from 2.65 Yuan/m3 to 3.55 Yuan/m3.Furthermore,the index of water consumption,benefit of cultivation and plantation benefit of unit water consumption showed an increasing trend with the increase of violation likelihood.However,through the quantification ofλfrom an economic perspective,the increasing ofλcould not enhance plantation benefit of unit water consumption significantly.展开更多
In this paper, we present several parametric duality results under various generalized （a,v,p）-V- invexity assumptions for a semiinfinite multiobjective fractional programming problem.
This paper aims at studying optimality conditions of robust weak efficient solutions for a nonsmooth uncertain multi-objective fractional programming problem(NUMFP).The concepts of two types of generalized convex func...This paper aims at studying optimality conditions of robust weak efficient solutions for a nonsmooth uncertain multi-objective fractional programming problem(NUMFP).The concepts of two types of generalized convex function pairs,called type-I functions and pseudo-quasi-type-I functions,are introduced in this paper for(NUMFP).Under the assumption that(NUMFP)satisfies the robust constraint qualification with respect to Clarke subdifferential,necessary optimality conditions of the robust weak efficient solution are given.Sufficient optimality conditions are obtained under pseudo-quasi-type-I generalized convexity assumption.Furthermore,we introduce the concept of robust weak saddle points to(NUMFP),and prove two theorems about robust weak saddle points.The main results in the present paper are verified by concrete examples.展开更多
Abstract In this paper, we discuss numerous sets of global parametric sufficient efficiency conditions under various generalized （a,n, p）-V-invexity assumptions for a semiinfinite multiobjective fractional programmi...Abstract In this paper, we discuss numerous sets of global parametric sufficient efficiency conditions under various generalized （a,n, p）-V-invexity assumptions for a semiinfinite multiobjective fractional programming problem.展开更多
In this paper, some necessary and sufficient optimality conditions are obtained for a fractional multiple objective programming involving semilocal E-convex and related functions. Also, some dual results are establish...In this paper, some necessary and sufficient optimality conditions are obtained for a fractional multiple objective programming involving semilocal E-convex and related functions. Also, some dual results are established under this kind of generalized convex functions. Our results generalize the ones obtained by Preda[J Math Anal Appl, 288（2003） 365-382].展开更多
Interference significantly impacts the performance of the Global Navigation Satellite Systems(GNSS),highlighting the need for advanced interference localization technology to bolster anti-interference and defense capa...Interference significantly impacts the performance of the Global Navigation Satellite Systems(GNSS),highlighting the need for advanced interference localization technology to bolster anti-interference and defense capabilities.The Uniform Circular Array(UCA)enables concurrent estimation of the Direction of Arrival(DOA)in both azimuth and elevation.Given the paramount importance of stability and real-time performance in interference localization,this work proposes an innovative approach to reduce the complexity and increase the robustness of the DOA estimation.The proposed method reduces computational complexity by selecting a reduced number of array elements to reconstruct a non-uniform sparse array from a UCA.To ensure DOA estimation accuracy,minimizing the Cramér-Rao Bound(CRB)is the objective,and the Spatial Correlation Coefficient(SCC)is incorporated as a constraint to mitigate side-lobe.The optimization model is a quadratic fractional model,which is solved by Semi-Definite Relaxation(SDR).When the array has perturbations,the mathematical expressions for CRB and SCC are re-derived to enhance the robustness of the reconstructed array.Simulation and hardware experiments validate the effectiveness of the proposed method in estimating interference DOA,showing high robustness and reductions in hardware and computational costs associated with DOA estimation.展开更多
Integrated sensing and communication(ISAC) is considered an effective technique to solve spectrum congestion in the future. In this paper, we consider a hybrid reconfigurable intelligent surface(RIS)-assisted downlink...Integrated sensing and communication(ISAC) is considered an effective technique to solve spectrum congestion in the future. In this paper, we consider a hybrid reconfigurable intelligent surface(RIS)-assisted downlink ISAC system that simultaneously serves multiple single-antenna communication users and senses multiple targets. Hybrid RIS differs from fully passive RIS in that it is composed of both active and passive elements, with the active elements having the effect of amplifying the signal in addition to phase-shifting. We maximize the achievable sum rate of communication users by collaboratively improving the beamforming matrix at the dual function base station(DFBS) and the phase-shifting matrix of the hybrid RIS, subject to the transmit power constraint at the DFBS, the signal-to-interference-plus-noise-ratio(SINR) constraint of the radar echo signal and the RIS constraint are satisfied at the same time. The builtin RIS-assisted ISAC design problem model is significantly non-convex due to the fractional objective function of this optimization problem and the coupling of the optimization variables in the objective function and constraints. As a result, we provide an effective alternating optimization approach based on fractional programming(FP) with block coordinate descent(BCD)to solve the optimization variables. Results from simulations show that the hybrid RIS-assisted ISAC system outperforms the other benchmark solutions.展开更多
基金Supported by the National Natural Science Foundation of China(Grant Nos.12071133 and 11871196).
文摘In this paper,we study the minimax linear fractional programming problem on a non-empty bounded set,called problem(MLFP),and we design a branch and bound algorithm to find a globally optimal solution of(MLFP).Firstly,we convert the problem(MLFP)to a problem(EP2)that is equivalent to it.Secondly,by applying the convex relaxation technique to problem(EP2),a convex quadratic relaxation problem(CQRP)is obtained.Then,the overall framework of the algorithm is given and its convergence is proved,the worst-case iteration number is also estimated.Finally,experimental data are listed to illustrate the effectiveness of the algorithm.
文摘First, a class of higher order exponential type hybrid （α,β, γ, η, p, h（.,.）, κ（., .）, w（.,., .）, ω（.,.,.）, θ）-invexities is introduced, second, some parametrically sufficient efficiency conditions based on the higher order exponential type hybrid invexities are established, and finally some parametrically sufficient efficiency results under the higher order exponential type hybrid （a,β, γ, ρ, h（.,.）, k（.,-）, w（-,., .）, w（.,., .）, 0）-invexities are investigated to the context of solving semiinfinite multiobjective fractional programming problems. The notions of the higher order exponential type hybrid （a, β, γ η, p, h（., .）, n（., .）, w（-,.,-）, ω（.,.,.）, 0）-invexities encompass most of the generalized invexities in the literature. To the best of our knowledge, the results on semiinfinite multiobjective fractional programming problems established in this communication are new and application-oriented toward multitime multi- objectve problems as well as multiobiective control problems.
基金National Natural Science Foundation of China(No.11071110)
文摘A new concept of(Φ,ρ,α)-V-invexity for differentiable vector-valued functions is introduced,which is a generalization of differentiable scalar-valued(Φ,ρ)-invexity.Based upon the(Φ,ρ,α)-V-invex functions,sufficient optimality conditions and MondWeir type dual theorems are derived for a class of nondifferentiable multiobjective fractional programming problems in which every component of the objective function and each constraint function contain a term involving the support function of a compact convex set.
基金Supported by Chongqing Key Lab. of Operations Research and System Engineering
文摘This paper studies a class of multiobjective generalized fractional programming problems, where the numerators of objective functions are the sum of differentiable function and convex function, while the denominators are the difference of differentiable function and convex function. Under the assumption of Calmness Constraint Qualification the Kuhn-Tucker type necessary conditions for efficient solution are given, and the Kuhn-Tucker type sufficient conditions for efficient solution are presented under the assumptions of （F, α, ρ, d）-V-convexity. Subsequently, the optimality conditions for two kinds of duality models are formulated and duality theorems are proved.
基金Supported by the Natural Science Foundation of Henan Province（0511012000 0511013600） Supported by the Science Foundation for Pure Research of Natural Science of the Education Department of Henan Province（200512950001）
文摘In this article,the authors discuss the optimal conditions of the linear fractionalprogramming problem and prove that a locally optional solution is a globally optional solution and the locally optimal solution can be attained at a basic feasible solution withconstraint condition.
文摘In this paper, two duality results are established under generalized ρ-convexity conditions for a class of multiobjective fractional programmign involvign differentiable n-sten functions.
文摘In this paper we define higher order (F,α,β,ρ,d,E)-convex function with respect to E-differentiable function K and obtain optimality conditions for nonlinear programming problem (NP) from the concept of higher order (F,α,β,ρ,d)-convexity. Here, we establish Mond-Weir and Wolfe duality for (NP) and utilize these duality in nonlinear fractional programming problem.
基金Supported by the National Natural Science Foundation of China(Grant No.11101016)
文摘In this paper, we introduce a class of generalized second order （F,α,ρ , d,p）-univex functions. Two types of second order dual models are considered for a minimax fractional programming problem and the duality results are established by using the assumptions on the functions involved.
基金supported by the National Natural Science Foundation of China(Grant 11961001)the construction project of first-class subjects in Ningxia Higher Education(Grant NXYLXK2017B09)by the major proprietary funded project of North Minzu University(Grant ZDZX201901).
文摘The mixed-integer quadratically constrained quadratic fractional programming(MIQCQFP)problem often appears in various fields such as engineering practice,management science and network communication.However,most of the solutions to such problems are often designed for their unique circumstances.This paper puts forward a new global optimization algorithm for solving the problem MIQCQFP.We first convert the MIQCQFP into an equivalent generalized bilinear fractional programming(EIGBFP)problem with integer variables.Secondly,we linearly underestimate and linearly overestimate the quadratic functions in the numerator and the denominator respectively,and then give a linear fractional relaxation technique for EIGBFP on the basis of non-negative numerator.After that,combining rectangular adjustment-segmentation technique and midpointsampling strategy with the branch-and-bound procedure,an efficient algorithm for solving MIQCQFP globally is proposed.Finally,a series of test problems are given to illustrate the effectiveness,feasibility and other performance of this algorithm.
文摘A semi-infinite programming problem is a mathematical programming problem with a finite number of variables and infinitely many constraints. Duality theories and generalized convexity concepts are important research topics in mathematical programming. In this paper, we discuss a fairly large number of paramet- ric duality results under various generalized （η,ρ）-invexity assumptions for a semi-infinite minmax fractional programming problem.
基金the National Natural Science Foundation of China(Nos.11871196,12071133 and 12071112)the China Postdoctoral Science Foundation(No.2017M622340)+1 种基金the Key Scientific and Technological Research Projects of Henan Province(Nos.202102210147 and 192102210114)the Science and Technology Climbing Program of Henan Institute of Science and Technology(No.2018JY01).
文摘This paper presents an efficient algorithm for globally solving a generalized linear fractional programming problem.For establishing this algorithm,we firstly construct a two-level linear relaxation method,and by utilizing the method,we can convert the initial generalized linear fractional programming problem and its subproblems into a series of linear programming relaxation problems.Based on the branch-and-bound framework and linear programming relaxation problems,a branch-and-bound algorithm is presented for globally solving the generalized linear fractional programming problem,and the computational complexity of the algorithm is given.Finally,numerical experimental results demonstrate the feasibility and efficiency of the proposed algorithm.
文摘In this paper, we discuss a large number of sets of global parametric sufficient optimality conditions under various generalized （η,ρ）-invexity assumptions for a semi-infinite minmax fractional programming problem.
基金supported by the Council of Scientific and Industrial Research(CSIR),New Delhi,India under Grant No.09/013(0474)/2012-EMR-1
文摘This paper considers a nonsmooth semi-infinite minimax fractional programming problem（SIMFP） involving locally Lipschitz invex functions. The authors establish necessary optimality conditions for SIMFP. The authors establish the relationship between an optimal solution of SIMFP and saddle point of scalar Lagrange function for SIMFP. Further, the authors study saddle point criteria of a vector Lagrange function defined for SIMFP.
基金National Natural Science Foundation of China(51479032,51579044)Yangtze River Scholars in Universities of Heilongjiang Province and Water Conservancy Science and Technology Project of Heilongjiang Province(201318,201503)The Outstanding Youth Fund of Heilongjiang Province(JC201402).
文摘For optimizing the water-use structure and increasing plantation benefit of unit water consumption,a multi-objective model for water resources utilization was established based on fractional programming(FP).Meanwhile,considering the stochasticity of water availability in the study area,the impact of the risk factor(λ)from a quantitative and qualitative perspective was analyzed.The chance-constrained programming(CCP)and conditional value-at-risk(CVaR)models were introduced into five important major grain production areas in Sanjiang Plain,and the crop planting structure under this condition was optimized.The results showed that,after optimization,overall benefit of cultivation increased from 42.07 billion Yuan to 42.47 billion Yuan,water consumption decreased from 15.90 billion m3 to 11.95 billion m3,the plantation benefit of unit water consumption increased from 2.65 Yuan/m3 to 3.55 Yuan/m3.Furthermore,the index of water consumption,benefit of cultivation and plantation benefit of unit water consumption showed an increasing trend with the increase of violation likelihood.However,through the quantification ofλfrom an economic perspective,the increasing ofλcould not enhance plantation benefit of unit water consumption significantly.
文摘In this paper, we present several parametric duality results under various generalized （a,v,p）-V- invexity assumptions for a semiinfinite multiobjective fractional programming problem.
基金supported by Natural Science Foundation of China(Nos.11861002 and 12171601)the Key Project of North Minzu University(No.ZDZX201804)+1 种基金the Construction Project of First-Class Disciplines in Ningxia Higher Education(NXYLXK2017B09)the Postgraduate Innovation Project of North Minzu Universit(No.YCX21157)..
文摘This paper aims at studying optimality conditions of robust weak efficient solutions for a nonsmooth uncertain multi-objective fractional programming problem(NUMFP).The concepts of two types of generalized convex function pairs,called type-I functions and pseudo-quasi-type-I functions,are introduced in this paper for(NUMFP).Under the assumption that(NUMFP)satisfies the robust constraint qualification with respect to Clarke subdifferential,necessary optimality conditions of the robust weak efficient solution are given.Sufficient optimality conditions are obtained under pseudo-quasi-type-I generalized convexity assumption.Furthermore,we introduce the concept of robust weak saddle points to(NUMFP),and prove two theorems about robust weak saddle points.The main results in the present paper are verified by concrete examples.
文摘Abstract In this paper, we discuss numerous sets of global parametric sufficient efficiency conditions under various generalized （a,n, p）-V-invexity assumptions for a semiinfinite multiobjective fractional programming problem.
基金Foundation item： Supported by Hunan Provincial Natural Science Foundation of China（05JJ40103） Supported by Soft Science Research Fund of Hunan Province（2006ZK3028） Supported by Scientific Research Fund of Hunan Provincial Education Department（105B0707, 08C470）
文摘In this paper, some necessary and sufficient optimality conditions are obtained for a fractional multiple objective programming involving semilocal E-convex and related functions. Also, some dual results are established under this kind of generalized convex functions. Our results generalize the ones obtained by Preda[J Math Anal Appl, 288（2003） 365-382].
基金the financial support from the National Key Research and Development Program of China(No.2023YFB3907001)the National Natural Science Foundation of China(Nos.U2233217,62371029)the UK Engineering and Physical Sciences Research Council(EPSRC),China(Nos.EP/M026981/1,EP/T021063/1 and EP/T024917/)。
文摘Interference significantly impacts the performance of the Global Navigation Satellite Systems(GNSS),highlighting the need for advanced interference localization technology to bolster anti-interference and defense capabilities.The Uniform Circular Array(UCA)enables concurrent estimation of the Direction of Arrival(DOA)in both azimuth and elevation.Given the paramount importance of stability and real-time performance in interference localization,this work proposes an innovative approach to reduce the complexity and increase the robustness of the DOA estimation.The proposed method reduces computational complexity by selecting a reduced number of array elements to reconstruct a non-uniform sparse array from a UCA.To ensure DOA estimation accuracy,minimizing the Cramér-Rao Bound(CRB)is the objective,and the Spatial Correlation Coefficient(SCC)is incorporated as a constraint to mitigate side-lobe.The optimization model is a quadratic fractional model,which is solved by Semi-Definite Relaxation(SDR).When the array has perturbations,the mathematical expressions for CRB and SCC are re-derived to enhance the robustness of the reconstructed array.Simulation and hardware experiments validate the effectiveness of the proposed method in estimating interference DOA,showing high robustness and reductions in hardware and computational costs associated with DOA estimation.
文摘Integrated sensing and communication(ISAC) is considered an effective technique to solve spectrum congestion in the future. In this paper, we consider a hybrid reconfigurable intelligent surface(RIS)-assisted downlink ISAC system that simultaneously serves multiple single-antenna communication users and senses multiple targets. Hybrid RIS differs from fully passive RIS in that it is composed of both active and passive elements, with the active elements having the effect of amplifying the signal in addition to phase-shifting. We maximize the achievable sum rate of communication users by collaboratively improving the beamforming matrix at the dual function base station(DFBS) and the phase-shifting matrix of the hybrid RIS, subject to the transmit power constraint at the DFBS, the signal-to-interference-plus-noise-ratio(SINR) constraint of the radar echo signal and the RIS constraint are satisfied at the same time. The builtin RIS-assisted ISAC design problem model is significantly non-convex due to the fractional objective function of this optimization problem and the coupling of the optimization variables in the objective function and constraints. As a result, we provide an effective alternating optimization approach based on fractional programming(FP) with block coordinate descent(BCD)to solve the optimization variables. Results from simulations show that the hybrid RIS-assisted ISAC system outperforms the other benchmark solutions.
