Satellite launch vehicle lies at the cross-road of multiple challenging technologies and its design and optimization present a typical example of multidisciplinary design and optimization（MDO） process.The complexity...Satellite launch vehicle lies at the cross-road of multiple challenging technologies and its design and optimization present a typical example of multidisciplinary design and optimization（MDO） process.The complexity of problem demands highly effi-cient and effective algorithm that can optimize the design.Hyper heuristic approach（HHA） based on meta-heuristics is applied to the optimization of air launched satellite launch vehicle（ASLV）.A non-learning random function（NLRF） is proposed to con-trol low-level meta-heuristics（LLMHs） that increases certainty of global solution,an essential ingredient required in product conceptual design phase of aerospace systems.Comprehensive empirical study is performed to evaluate the performance advan-tages of proposed approach over popular non-gradient based optimization methods.Design of ASLV encompasses aerodynamics,propulsion,structure,stages layout,mass distribution,and trajectory modules connected by multidisciplinary feasible design approach.This approach formulates explicit system-level goals and then forwards the design optimization process entirely over to optimizer.This distinctive approach for launch vehicle system design relieves engineers from tedious,iterative task and en-ables them to improve their component level models.Mass is an impetus on vehicle performance and cost,and so it is considered as the core of vehicle design process.Therefore,gross launch mass is to be minimized in HHA.展开更多
This paper studies the solution technique to solve the DRAMA spares allocation optimization problem. DRAMA model is an analytic spare optimization model of a multi-item, multi-location, and two-echelon inventory syste...This paper studies the solution technique to solve the DRAMA spares allocation optimization problem. DRAMA model is an analytic spare optimization model of a multi-item, multi-location, and two-echelon inventory system. The computation of its system spares availability is much complicated. The objective function and constraint functions of DRAMA model could be written as the separable forms. A new bound heuristic algorithm has been presented by improving the bound heuristic algorithm for solving the reliability redundancy optimization problem (BHA in short). With the results, the proposed algorithm has been found to be more economical and effective than BHA to obtain the solutions of large DRAMA model. The new algorithm could be used to solve reliability redundancy optimization problems with the separable forms.展开更多
Damage identification of the offshore floating wind turbine by vibration/dynamic signals is one of the important and new research fields in the Structural Health Monitoring（SHM）. In this paper a new damage identific...Damage identification of the offshore floating wind turbine by vibration/dynamic signals is one of the important and new research fields in the Structural Health Monitoring（SHM）. In this paper a new damage identification method is proposed based on meta-heuristic algorithms using the dynamic response of the TLP（Tension-Leg Platform） floating wind turbine structure. The Genetic Algorithms（GA）, Artificial Immune System（AIS）, Particle Swarm Optimization（PSO）, and Artificial Bee Colony（ABC） are chosen for minimizing the object function, defined properly for damage identification purpose. In addition to studying the capability of mentioned algorithms in correctly identifying the damage, the effect of the response type on the results of identification is studied. Also, the results of proposed damage identification are investigated with considering possible uncertainties of the structure. Finally, for evaluating the proposed method in real condition, a 1/100 scaled experimental setup of TLP Floating Wind Turbine（TLPFWT） is provided in a laboratory scale and the proposed damage identification method is applied to the scaled turbine.展开更多
In this study, Imperialistic Competitive Algorithm（ICA） is utilized for locating the critical failure surface and computing the factor of safety（FOS） in a slope stability analysis based on the limit equilibrium ap...In this study, Imperialistic Competitive Algorithm（ICA） is utilized for locating the critical failure surface and computing the factor of safety（FOS） in a slope stability analysis based on the limit equilibrium approach. The factor of safety relating to each trial slip surface is calculated using a simplified algorithm of the Morgenstern-Price method, which satisfies both the force and the moment equilibriums. General slip surface is considered non-circular in this study that is constituted by linking random straight lines.To explore the performance of the proposed algorithm, four benchmark test problems are analyzed. The results demonstrate that the present techniques can provide reliable, accurate and efficient solutions for locating the critical failure surface and relating FOS. Moreover, in contrast with previous studies the present algorithm could reach the lower value of FOS and reached more exact solutions.展开更多
A new meta-heuristic method is proposed to enhance current meta-heuristic methods for global optimization and test scheduling for three-dimensional （3D） stacked system-on-chip （SoC） by hybridizing grey wolf optimi...A new meta-heuristic method is proposed to enhance current meta-heuristic methods for global optimization and test scheduling for three-dimensional （3D） stacked system-on-chip （SoC） by hybridizing grey wolf optimization with differential evo- lution （HGWO）. Because basic grey wolf optimization （GWO） is easy to fall into stagnation when it carries out the operation of at- tacking prey, and differential evolution （DE） is integrated into GWO to update the previous best position of grey wolf Alpha, Beta and Delta, in order to force GWO to jump out of the stagnation with DE＇s strong searching ability. The proposed algorithm can accele- rate the convergence speed of GWO and improve its performance. Twenty-three well-known benchmark functions and an NP hard problem of test scheduling for 3D SoC are employed to verify the performance of the proposed algorithm. Experimental results show the superior performance of the proposed algorithm for exploiting the optimum and it has advantages in terms of exploration.展开更多
We extended an improved version of the discrete particle swarm optimization (DPSO) algorithm proposed by Liao et al.(2007) to solve the dynamic facility layout problem (DFLP). A computational study was performed with ...We extended an improved version of the discrete particle swarm optimization (DPSO) algorithm proposed by Liao et al.(2007) to solve the dynamic facility layout problem (DFLP). A computational study was performed with the existing heuristic algorithms, including the dynamic programming (DP), genetic algorithm (GA), simulated annealing (SA), hybrid ant system (HAS), hybrid simulated annealing (SA-EG), hybrid genetic algorithms (NLGA and CONGA). The proposed DPSO algorithm, SA, HAS, GA, DP, SA-EG, NLGA, and CONGA obtained the best solutions for 33, 24, 20, 10, 12, 20, 5, and 2 of the 48 problems from (Balakrishnan and Cheng, 2000), respectively. These results show that the DPSO is very effective in dealing with the DFLP. The extended DPSO also has very good computational efficiency when the problem size increases.展开更多
Moth-flame optimization(MFO)is a novel metaheuristic algorithm inspired by the characteristics of a moth’s navigation method in nature called transverse orientation.Like other metaheuristic algorithms,it is easy to f...Moth-flame optimization(MFO)is a novel metaheuristic algorithm inspired by the characteristics of a moth’s navigation method in nature called transverse orientation.Like other metaheuristic algorithms,it is easy to fall into local optimum and leads to slow convergence speed.The chaotic map is one of the best methods to improve exploration and exploitation of the metaheuristic algorithms.In the present study,we propose a chaos-enhanced MFO(CMFO)by incorporating chaos maps into the MFO algorithm to enhance its performance.The chaotic map is utilized to initialize the moths’population,handle the boundary overstepping,and tune the distance parameter.The CMFO is benchmarked on three groups of benchmark functions to find out the most efficient one.The performance of the CMFO is also verified by using two real engineering problems.The statistical results clearly demonstrate that the appropriate chaotic map(singer map)embedded in the appropriate component of MFO can significantly improve the performance of MFO.展开更多
The optimization of discrete problems is largely encountered in engineering and information domains. Solving these problems with continuous-variables approach then convert the continuous variables to discrete ones doe...The optimization of discrete problems is largely encountered in engineering and information domains. Solving these problems with continuous-variables approach then convert the continuous variables to discrete ones does not guarantee the optimal global solution. Evolutionary Algorithms (EAs) have been applied successfully in combinatorial discrete optimization. Here, the mathematical basics of real-coding Genetic Algorithm are presented in addition to three other Evolutionary Algorithms: Particle Swarm Optimization (PSO), Ant Colony Algorithms (ACOA) and Harmony Search (HS). The EAs are presented in as unifying notations as possible in order to facilitate understanding and comparison. Our combinatorial discrete problem example is the famous benchmark case of New-York Water Supply System WSS network. The mathematical construction in addition to the obtained results of Real-coding GA applied to this case study (authors), are compared with those of the three other algorithms available in literature. The real representation of GA, with its two operators: mutation and crossover, functions significantly faster than binary and other coding and illustrates its potential as a substitute to the traditional optimization methods for water systems design and planning. The real (actual) representation is very effective and provides two near-optimal feasible solutions to the New York tunnels problem. We found that the four EAs are capable to afford hydraulically-feasible solutions with reasonable cost but our real-coding GA takes more evaluations to reach the optimal or near-optimal solutions compared to other EAs namely the HS. HS approach discovers efficiently the research space because of the random generation of solutions in every iteration, and the ability of choosing neighbor values of solution elements “changing the diameter of the pipe to the next greater or smaller commercial diameter” beside keeping good current solutions. Our proposed promising point to improve the performance of GA is by introducing completely new individuals in every generation in GA using a new “immigration” operator beside “mutation” and “crossover”.展开更多
As part of a research activity at Politecnico di Torino, aiming to develop multi-disciplinary design procedures implementing nature inspired meta-heuristic algorithms, a performance design optimization procedure for h...As part of a research activity at Politecnico di Torino, aiming to develop multi-disciplinary design procedures implementing nature inspired meta-heuristic algorithms, a performance design optimization procedure for helicopter rotors has been developed and tested. The procedure optimizes the aerodynamic performance of blades by selecting the point of taper initiation, the root chord, the taper ratio, and the maximum twist which minimize horsepower for different flight regimes. Satisfactory aerodynamic performance is defined by the requirements which must hold for any flight condition: the required power must be minimized, both the section drag divergence Mach number on the advancing side of the rotor disc and the maximum section lift coefficient on the retreating side of the rotor disc must be avoided and, even more important, the rotor must be trimmed. The procedure uses a comprehensive mathematical model to estimate the trim states of the helicopter and the optimization algorithm consists of a repulsive particle swarm optimization program. A comparison with an evolutionary micro-genetic algorithm is also presented.展开更多
Electromagnetism-like (EML) algorithm is a new evolutionary algorithm that bases on the electromagnetic attraction and repulsion among particles. It was originally proposed to solve optimization problems with bounded ...Electromagnetism-like (EML) algorithm is a new evolutionary algorithm that bases on the electromagnetic attraction and repulsion among particles. It was originally proposed to solve optimization problems with bounded variables. Since its inception, many variants of the EML algorithm have been proposed in the literature. However, it remains unclear how to simulate the electromagnetic heuristics in an EML algorithm effectively to achieve the best performance. This study surveys and compares the EML algorithms in the literature. Furthermore, local search and perturbed point are two techniques commonly used in an EML algorithm to fine tune the solution and to help escaping from local optimums, respectively. Performance study is conducted to understand their impact on an EML algorithm.展开更多
Combinatorial optimization problems are found in many application fields such as computer science, engineering and economy. In this paper, a new efficient meta-heuristic, Intersection-Based Scaling (IBS for abbreviati...Combinatorial optimization problems are found in many application fields such as computer science, engineering and economy. In this paper, a new efficient meta-heuristic, Intersection-Based Scaling (IBS for abbreviation), is proposed and it can be applied to the combinatorial optimization problems. The main idea of IBS is to scale the size of the instance based on the intersection of some local optima, and to simplify the search space by extracting the intersection from the instance, which makes the search more efficient. The combination of IBS with some local search heuristics of different combinatorial optimization problems such as Traveling Salesman Problem (TSP) and Graph Partitioning Problem (GPP) is studied, and comparisons are made with some of the best heuristic algorithms and meta-heuristic algorithms. It is found that it has significantly improved the performance of existing local search heuristics and significantly outperforms the known best algorithms. Keywords combinatorial optimization - TSP (Traveling Salesman Problem) - GPP (Graph Partitioning Problem) - IBS (Intersection-Based Scaling) - meta heuristic Regular PaperThis work is supported by the National Basic Research 973 Program of China (Grant No.TG1998030401).Peng Zou was born in 1979. He received the B.S. degree in computer software from University of Science and Technology of China (USTC) in 2000. Now he is a Ph.D. candidate in computer science of USTC. His current research interests include algorithms for NP-hard problems and parallel computing.Zhi Zhou was born in 1976. He received the B.S. degree in computer software from USTC in 1995. Now he is a Ph.D. candidate in computer science of USTC. His current research interests include combinatorial problem and parallel computing.Ying-Yu Wan was born in 1976. He received the B.S. degree in computer software from USTC in 1997, and the Ph.D. degree from USTC in 2002. His current research interests include parallel computing and combinatorial problem.Guo-Liang Chen was born in 1938. Now he is an Academician of CAS and Ph.D. supervisor in Department of Computer Science at USTC, director of the National High Performance Computing Center at Hefei. His current research interests include parallel computing, computer architecture and combinatorial optimization.Jun Gu was born in 1956. He received the B.S. degree in electronic engineering from USTC in 1982, and the Ph.D. degree in computer science from University of Utah. Now he is a professor and Ph.D. supervisor in computer science at USTC and Hong Kong University of Science and Technology. His main research interrests include algorithms for NP-hard problems.展开更多
Assume there are several states, and the objective function f\+s(x) is linked with each state s. Robust optimization is to solve the following problem: min x∈X max s∈Sf\+s(x)where X is the feasible s...Assume there are several states, and the objective function f\+s(x) is linked with each state s. Robust optimization is to solve the following problem: min x∈X max s∈Sf\+s(x)where X is the feasible solution set, and S is the collection of states.\;It has been showed that most of robust combinatorial optimization problems are NP\|hard in strong sense. In this paper, we will discuss the borderline between the ′easy′ and the ′hard′ cases of robust combinatorial optimization problems, and further present a heuristic frame work to solve the ′hard′ problems and discuss their concrete implementation of the heuristic method.展开更多
文摘Satellite launch vehicle lies at the cross-road of multiple challenging technologies and its design and optimization present a typical example of multidisciplinary design and optimization（MDO） process.The complexity of problem demands highly effi-cient and effective algorithm that can optimize the design.Hyper heuristic approach（HHA） based on meta-heuristics is applied to the optimization of air launched satellite launch vehicle（ASLV）.A non-learning random function（NLRF） is proposed to con-trol low-level meta-heuristics（LLMHs） that increases certainty of global solution,an essential ingredient required in product conceptual design phase of aerospace systems.Comprehensive empirical study is performed to evaluate the performance advan-tages of proposed approach over popular non-gradient based optimization methods.Design of ASLV encompasses aerodynamics,propulsion,structure,stages layout,mass distribution,and trajectory modules connected by multidisciplinary feasible design approach.This approach formulates explicit system-level goals and then forwards the design optimization process entirely over to optimizer.This distinctive approach for launch vehicle system design relieves engineers from tedious,iterative task and en-ables them to improve their component level models.Mass is an impetus on vehicle performance and cost,and so it is considered as the core of vehicle design process.Therefore,gross launch mass is to be minimized in HHA.
文摘This paper studies the solution technique to solve the DRAMA spares allocation optimization problem. DRAMA model is an analytic spare optimization model of a multi-item, multi-location, and two-echelon inventory system. The computation of its system spares availability is much complicated. The objective function and constraint functions of DRAMA model could be written as the separable forms. A new bound heuristic algorithm has been presented by improving the bound heuristic algorithm for solving the reliability redundancy optimization problem (BHA in short). With the results, the proposed algorithm has been found to be more economical and effective than BHA to obtain the solutions of large DRAMA model. The new algorithm could be used to solve reliability redundancy optimization problems with the separable forms.
文摘Damage identification of the offshore floating wind turbine by vibration/dynamic signals is one of the important and new research fields in the Structural Health Monitoring（SHM）. In this paper a new damage identification method is proposed based on meta-heuristic algorithms using the dynamic response of the TLP（Tension-Leg Platform） floating wind turbine structure. The Genetic Algorithms（GA）, Artificial Immune System（AIS）, Particle Swarm Optimization（PSO）, and Artificial Bee Colony（ABC） are chosen for minimizing the object function, defined properly for damage identification purpose. In addition to studying the capability of mentioned algorithms in correctly identifying the damage, the effect of the response type on the results of identification is studied. Also, the results of proposed damage identification are investigated with considering possible uncertainties of the structure. Finally, for evaluating the proposed method in real condition, a 1/100 scaled experimental setup of TLP Floating Wind Turbine（TLPFWT） is provided in a laboratory scale and the proposed damage identification method is applied to the scaled turbine.
文摘In this study, Imperialistic Competitive Algorithm（ICA） is utilized for locating the critical failure surface and computing the factor of safety（FOS） in a slope stability analysis based on the limit equilibrium approach. The factor of safety relating to each trial slip surface is calculated using a simplified algorithm of the Morgenstern-Price method, which satisfies both the force and the moment equilibriums. General slip surface is considered non-circular in this study that is constituted by linking random straight lines.To explore the performance of the proposed algorithm, four benchmark test problems are analyzed. The results demonstrate that the present techniques can provide reliable, accurate and efficient solutions for locating the critical failure surface and relating FOS. Moreover, in contrast with previous studies the present algorithm could reach the lower value of FOS and reached more exact solutions.
基金supported by the National Natural Science Foundation of China(6076600161105004)+1 种基金the Guangxi Key Laboratory of Automatic Detecting Technology and Instruments(YQ14110)the Program for Innovative Research Team of Guilin University of Electronic Technology(IRTGUET)
文摘A new meta-heuristic method is proposed to enhance current meta-heuristic methods for global optimization and test scheduling for three-dimensional （3D） stacked system-on-chip （SoC） by hybridizing grey wolf optimization with differential evo- lution （HGWO）. Because basic grey wolf optimization （GWO） is easy to fall into stagnation when it carries out the operation of at- tacking prey, and differential evolution （DE） is integrated into GWO to update the previous best position of grey wolf Alpha, Beta and Delta, in order to force GWO to jump out of the stagnation with DE＇s strong searching ability. The proposed algorithm can accele- rate the convergence speed of GWO and improve its performance. Twenty-three well-known benchmark functions and an NP hard problem of test scheduling for 3D SoC are employed to verify the performance of the proposed algorithm. Experimental results show the superior performance of the proposed algorithm for exploiting the optimum and it has advantages in terms of exploration.
文摘We extended an improved version of the discrete particle swarm optimization (DPSO) algorithm proposed by Liao et al.(2007) to solve the dynamic facility layout problem (DFLP). A computational study was performed with the existing heuristic algorithms, including the dynamic programming (DP), genetic algorithm (GA), simulated annealing (SA), hybrid ant system (HAS), hybrid simulated annealing (SA-EG), hybrid genetic algorithms (NLGA and CONGA). The proposed DPSO algorithm, SA, HAS, GA, DP, SA-EG, NLGA, and CONGA obtained the best solutions for 33, 24, 20, 10, 12, 20, 5, and 2 of the 48 problems from (Balakrishnan and Cheng, 2000), respectively. These results show that the DPSO is very effective in dealing with the DFLP. The extended DPSO also has very good computational efficiency when the problem size increases.
基金supported by the Military Science Project of the National Social Science Foundation of China(15GJ003-141)
文摘Moth-flame optimization(MFO)is a novel metaheuristic algorithm inspired by the characteristics of a moth’s navigation method in nature called transverse orientation.Like other metaheuristic algorithms,it is easy to fall into local optimum and leads to slow convergence speed.The chaotic map is one of the best methods to improve exploration and exploitation of the metaheuristic algorithms.In the present study,we propose a chaos-enhanced MFO(CMFO)by incorporating chaos maps into the MFO algorithm to enhance its performance.The chaotic map is utilized to initialize the moths’population,handle the boundary overstepping,and tune the distance parameter.The CMFO is benchmarked on three groups of benchmark functions to find out the most efficient one.The performance of the CMFO is also verified by using two real engineering problems.The statistical results clearly demonstrate that the appropriate chaotic map(singer map)embedded in the appropriate component of MFO can significantly improve the performance of MFO.
文摘The optimization of discrete problems is largely encountered in engineering and information domains. Solving these problems with continuous-variables approach then convert the continuous variables to discrete ones does not guarantee the optimal global solution. Evolutionary Algorithms (EAs) have been applied successfully in combinatorial discrete optimization. Here, the mathematical basics of real-coding Genetic Algorithm are presented in addition to three other Evolutionary Algorithms: Particle Swarm Optimization (PSO), Ant Colony Algorithms (ACOA) and Harmony Search (HS). The EAs are presented in as unifying notations as possible in order to facilitate understanding and comparison. Our combinatorial discrete problem example is the famous benchmark case of New-York Water Supply System WSS network. The mathematical construction in addition to the obtained results of Real-coding GA applied to this case study (authors), are compared with those of the three other algorithms available in literature. The real representation of GA, with its two operators: mutation and crossover, functions significantly faster than binary and other coding and illustrates its potential as a substitute to the traditional optimization methods for water systems design and planning. The real (actual) representation is very effective and provides two near-optimal feasible solutions to the New York tunnels problem. We found that the four EAs are capable to afford hydraulically-feasible solutions with reasonable cost but our real-coding GA takes more evaluations to reach the optimal or near-optimal solutions compared to other EAs namely the HS. HS approach discovers efficiently the research space because of the random generation of solutions in every iteration, and the ability of choosing neighbor values of solution elements “changing the diameter of the pipe to the next greater or smaller commercial diameter” beside keeping good current solutions. Our proposed promising point to improve the performance of GA is by introducing completely new individuals in every generation in GA using a new “immigration” operator beside “mutation” and “crossover”.
文摘As part of a research activity at Politecnico di Torino, aiming to develop multi-disciplinary design procedures implementing nature inspired meta-heuristic algorithms, a performance design optimization procedure for helicopter rotors has been developed and tested. The procedure optimizes the aerodynamic performance of blades by selecting the point of taper initiation, the root chord, the taper ratio, and the maximum twist which minimize horsepower for different flight regimes. Satisfactory aerodynamic performance is defined by the requirements which must hold for any flight condition: the required power must be minimized, both the section drag divergence Mach number on the advancing side of the rotor disc and the maximum section lift coefficient on the retreating side of the rotor disc must be avoided and, even more important, the rotor must be trimmed. The procedure uses a comprehensive mathematical model to estimate the trim states of the helicopter and the optimization algorithm consists of a repulsive particle swarm optimization program. A comparison with an evolutionary micro-genetic algorithm is also presented.
文摘Electromagnetism-like (EML) algorithm is a new evolutionary algorithm that bases on the electromagnetic attraction and repulsion among particles. It was originally proposed to solve optimization problems with bounded variables. Since its inception, many variants of the EML algorithm have been proposed in the literature. However, it remains unclear how to simulate the electromagnetic heuristics in an EML algorithm effectively to achieve the best performance. This study surveys and compares the EML algorithms in the literature. Furthermore, local search and perturbed point are two techniques commonly used in an EML algorithm to fine tune the solution and to help escaping from local optimums, respectively. Performance study is conducted to understand their impact on an EML algorithm.
文摘Combinatorial optimization problems are found in many application fields such as computer science, engineering and economy. In this paper, a new efficient meta-heuristic, Intersection-Based Scaling (IBS for abbreviation), is proposed and it can be applied to the combinatorial optimization problems. The main idea of IBS is to scale the size of the instance based on the intersection of some local optima, and to simplify the search space by extracting the intersection from the instance, which makes the search more efficient. The combination of IBS with some local search heuristics of different combinatorial optimization problems such as Traveling Salesman Problem (TSP) and Graph Partitioning Problem (GPP) is studied, and comparisons are made with some of the best heuristic algorithms and meta-heuristic algorithms. It is found that it has significantly improved the performance of existing local search heuristics and significantly outperforms the known best algorithms. Keywords combinatorial optimization - TSP (Traveling Salesman Problem) - GPP (Graph Partitioning Problem) - IBS (Intersection-Based Scaling) - meta heuristic Regular PaperThis work is supported by the National Basic Research 973 Program of China (Grant No.TG1998030401).Peng Zou was born in 1979. He received the B.S. degree in computer software from University of Science and Technology of China (USTC) in 2000. Now he is a Ph.D. candidate in computer science of USTC. His current research interests include algorithms for NP-hard problems and parallel computing.Zhi Zhou was born in 1976. He received the B.S. degree in computer software from USTC in 1995. Now he is a Ph.D. candidate in computer science of USTC. His current research interests include combinatorial problem and parallel computing.Ying-Yu Wan was born in 1976. He received the B.S. degree in computer software from USTC in 1997, and the Ph.D. degree from USTC in 2002. His current research interests include parallel computing and combinatorial problem.Guo-Liang Chen was born in 1938. Now he is an Academician of CAS and Ph.D. supervisor in Department of Computer Science at USTC, director of the National High Performance Computing Center at Hefei. His current research interests include parallel computing, computer architecture and combinatorial optimization.Jun Gu was born in 1956. He received the B.S. degree in electronic engineering from USTC in 1982, and the Ph.D. degree in computer science from University of Utah. Now he is a professor and Ph.D. supervisor in computer science at USTC and Hong Kong University of Science and Technology. His main research interrests include algorithms for NP-hard problems.
基金Research is supported by the National 863 Program ( No.863- 306- Z T
文摘Assume there are several states, and the objective function f\+s(x) is linked with each state s. Robust optimization is to solve the following problem: min x∈X max s∈Sf\+s(x)where X is the feasible solution set, and S is the collection of states.\;It has been showed that most of robust combinatorial optimization problems are NP\|hard in strong sense. In this paper, we will discuss the borderline between the ′easy′ and the ′hard′ cases of robust combinatorial optimization problems, and further present a heuristic frame work to solve the ′hard′ problems and discuss their concrete implementation of the heuristic method.
