A typical antenna optimization design problem is presented, and various issues involved in the de... more A typical antenna optimization design problem is presented, and various issues involved in the design process are discussed. Defining a suitable objective function is a central question, as is the type of optimization algorithm that should be used, stochastic versus deterministic. These questions are addressed by way of an example. A single-resistor loaded broadband HF monopole design is considered in detail, and the resulting antenna compared to published results for similar continuously loaded and discrete resistor loaded designs.
Dynamic Threshold Optimization (DTO) adaptively "compresses" the decision space (DS) in a global ... more Dynamic Threshold Optimization (DTO) adaptively "compresses" the decision space (DS) in a global search and optimization problem by bounding the objective function from below. This approach is different from "shrinking" DS by reducing bounds on the decision variables. DTO is applied to Schwefel's Problem 2.26 in 2 and 30 dimensions with good results. DTO is universally applicable, and the author believes it may be a novel approach to global search and optimization.
This note describes a parameter-free implementation of Central Force Optimization for determinist... more This note describes a parameter-free implementation of Central Force Optimization for deterministic multidimensional search and optimization. The user supplies only one input: the objective function to be maximized, nothing more. The CFO equations of motion are simplified by assigning specific values to CFO's basic parameters, and this particular algorithmic implementation also includes hardwired internal parameters so that none is user-specified. The algorithm's performance is tested against a widely used suite of twenty three benchmark functions and compared to other state-of-the-art algorithms. CFO performs very well indeed. Includes important update 20 March 2010 addressing the issue of different probes coalescing into one.
A dipole-loaded monopole antenna is optimized for uniform hemispherical coverage using VSO, a new... more A dipole-loaded monopole antenna is optimized for uniform hemispherical coverage using VSO, a new global search design and optimization algorithm. The antenna's performance is compared to genetic algorithm and hill-climber optimized loaded monopoles, and VSO is tested against two suites of benchmark functions and several other algorithms.
Design and optimization (D&O) problems in applied electromagnetics, in particular antenna D&O, of... more Design and optimization (D&O) problems in applied electromagnetics, in particular antenna D&O, often rely on global search and optimization metaheuristics based on Nature-inspired metaphors. Most are stochastic in nature because the underlying process is, for example the swarming behavior of fish and birds (Particle Swarm Optimization) or the foraging behavior of ants (Ant Colony Optimization). While these algorithms generally work well (enough), their inherent randomness frequently limits utility in solving real-world antenna problems. A better approach is to use an inherently deterministic optimizer or to implement a stochastic algorithm so that it is effectively deterministic. This article discusses in detail Central Force Optimization (CFO) which is a deterministic algorithm analogizing gravitational kinematics. It provides an overview and emphasizes CFO's utility in solving the real-world problems routinely encountered by practicing engineers with examples drawn from antenna optimization. Because CFO is inherently deterministic only a single run is required, which can be a major advantage in addressing electromagnetic problems. Even in cases where the balance between decision space exploration and exploitation is better achieved using a stochastic approach, CFO offers the advantage of being easily hybridized to include pure or pseudo randomness. One method of injecting pseudo randomness is using π fractions which are discussed in the companion paper (Part II).
Design and optimization problems in applied electromagnetics, in particular antenna D&O, often re... more Design and optimization problems in applied electromagnetics, in particular antenna D&O, often rely on global search and optimization metaheuristics based on Nature-inspired metaphors. Because the algorithms are inherently stochastic each run produces different results. These uncertain outcomes can be mitigated by pseudo randomly sampling the decision space. This article describes π fractions as a method to create deterministic uniformly distributed decision space sample points. These fractions appear to be uniformly distributed on [0,1) and can be used in any stochastic algorithm rendering it effectively deterministic without compromising its ability to explore the decision space. π fractions are generated using the BBP π digit extraction algorithm. This approach is tested using genetic algorithm πGASR with good results.
This note compares the performance of two multidimensional search and optimization algorithms: Gr... more This note compares the performance of two multidimensional search and optimization algorithms: Group Search Optimizer and Central Force Optimization. GSO is a new state-of-the-art algorithm that has gained some notoriety, consequently providing an excellent yardstick for measuring the performance of other algorithms. CFO is a novel deterministic metaheuristic that has performed well against GSO in previous tests. The CFO implementation reported here includes architectural improvements in errant probe retrieval and decision space adaptation that result in even better performance. Detailed results are provided for the twenty-three function benchmark suite used to evaluate GSO. CFO performs better than or essentially as well as GSO on twenty functions and nearly as well on one of the remaining three. Includes update 24 February 2010. Comment: Includes detailed numerical results and source code in appendices. Update 02-24-10: Replaces Fig. A2(b) for improved visualization; corrects mino...
Dynamic Threshold Optimization (DTO) adaptively "compresses" the decision space (DS) in... more Dynamic Threshold Optimization (DTO) adaptively "compresses" the decision space (DS) in a global search and optimization problem by bounding the objective function from below. This approach is different from "shrinking" DS by reducing bounds on the decision variables. DTO is applied to Schwefel's Problem 2.26 in 2 and 30 dimensions with good results. DTO is universally applicable, and the author believes it may be a novel approach to global search and optimization.
Central Force Optimization is a global search and optimization algorithm that searches a decision... more Central Force Optimization is a global search and optimization algorithm that searches a decision space by flying "probes" whose trajectories are deterministically computed using two equations of motion. Because it is possible for a probe to fly outside the domain of feasible solutions, a simple errant probe retrieval method has been used previously that does not include the directional information contained in a probe's acceleration vector. This note investigates the effect of adding directionality to the "repositioning factor" approach. As a general proposition, it appears that doing so does not improve convergence speed or accuracy. In fact, adding directionality to the original errant probe retrieval scheme appears to be highly inadvisable. Nevertheless, there may be alternative probe retrieval schemes that do benefit from directional information, and the results reported here may assist in or encourage their development. Comment: Ver. 2, 6 June 2010 (Fig...
Pi Fractions are used to create deterministic uniformly distributed pseudorandom decision space s... more Pi Fractions are used to create deterministic uniformly distributed pseudorandom decision space sample points for a global search and optimization algorithm. These fractions appear to be uniformly distributed on [0,1] and can be used in any stochastic algorithm rendering it effectively deterministic without compromising its ability to explore the decision space. Pi Fractions are generated using the BBP Pi digit extraction algorithm. The Pi Fraction approach is tested using genetic algorithm Pi-GASR with very good results. A Pi Fraction data file is available upon request.
This paper presents Central Force Optimization, a novel, nature inspired, deterministic search me... more This paper presents Central Force Optimization, a novel, nature inspired, deterministic search metaheuristic for constrained multi-dimensional optimization. CFO is based on the metaphor of gravitational kinematics. Equations are presented for the positions and ...
A typical antenna optimization design problem is presented, and various issues involved in the de... more A typical antenna optimization design problem is presented, and various issues involved in the design process are discussed. Defining a suitable objective function is a central question, as is the type of optimization algorithm that should be used, stochastic versus deterministic. These questions are addressed by way of an example. A single-resistor loaded broadband HF monopole design is considered in detail, and the resulting antenna compared to published results for similar continuously loaded and discrete resistor loaded designs.
Dynamic Threshold Optimization (DTO) adaptively "compresses" the decision space (DS) in a global ... more Dynamic Threshold Optimization (DTO) adaptively "compresses" the decision space (DS) in a global search and optimization problem by bounding the objective function from below. This approach is different from "shrinking" DS by reducing bounds on the decision variables. DTO is applied to Schwefel's Problem 2.26 in 2 and 30 dimensions with good results. DTO is universally applicable, and the author believes it may be a novel approach to global search and optimization.
This note describes a parameter-free implementation of Central Force Optimization for determinist... more This note describes a parameter-free implementation of Central Force Optimization for deterministic multidimensional search and optimization. The user supplies only one input: the objective function to be maximized, nothing more. The CFO equations of motion are simplified by assigning specific values to CFO's basic parameters, and this particular algorithmic implementation also includes hardwired internal parameters so that none is user-specified. The algorithm's performance is tested against a widely used suite of twenty three benchmark functions and compared to other state-of-the-art algorithms. CFO performs very well indeed. Includes important update 20 March 2010 addressing the issue of different probes coalescing into one.
A dipole-loaded monopole antenna is optimized for uniform hemispherical coverage using VSO, a new... more A dipole-loaded monopole antenna is optimized for uniform hemispherical coverage using VSO, a new global search design and optimization algorithm. The antenna's performance is compared to genetic algorithm and hill-climber optimized loaded monopoles, and VSO is tested against two suites of benchmark functions and several other algorithms.
Design and optimization (D&O) problems in applied electromagnetics, in particular antenna D&O, of... more Design and optimization (D&O) problems in applied electromagnetics, in particular antenna D&O, often rely on global search and optimization metaheuristics based on Nature-inspired metaphors. Most are stochastic in nature because the underlying process is, for example the swarming behavior of fish and birds (Particle Swarm Optimization) or the foraging behavior of ants (Ant Colony Optimization). While these algorithms generally work well (enough), their inherent randomness frequently limits utility in solving real-world antenna problems. A better approach is to use an inherently deterministic optimizer or to implement a stochastic algorithm so that it is effectively deterministic. This article discusses in detail Central Force Optimization (CFO) which is a deterministic algorithm analogizing gravitational kinematics. It provides an overview and emphasizes CFO's utility in solving the real-world problems routinely encountered by practicing engineers with examples drawn from antenna optimization. Because CFO is inherently deterministic only a single run is required, which can be a major advantage in addressing electromagnetic problems. Even in cases where the balance between decision space exploration and exploitation is better achieved using a stochastic approach, CFO offers the advantage of being easily hybridized to include pure or pseudo randomness. One method of injecting pseudo randomness is using π fractions which are discussed in the companion paper (Part II).
Design and optimization problems in applied electromagnetics, in particular antenna D&O, often re... more Design and optimization problems in applied electromagnetics, in particular antenna D&O, often rely on global search and optimization metaheuristics based on Nature-inspired metaphors. Because the algorithms are inherently stochastic each run produces different results. These uncertain outcomes can be mitigated by pseudo randomly sampling the decision space. This article describes π fractions as a method to create deterministic uniformly distributed decision space sample points. These fractions appear to be uniformly distributed on [0,1) and can be used in any stochastic algorithm rendering it effectively deterministic without compromising its ability to explore the decision space. π fractions are generated using the BBP π digit extraction algorithm. This approach is tested using genetic algorithm πGASR with good results.
This note compares the performance of two multidimensional search and optimization algorithms: Gr... more This note compares the performance of two multidimensional search and optimization algorithms: Group Search Optimizer and Central Force Optimization. GSO is a new state-of-the-art algorithm that has gained some notoriety, consequently providing an excellent yardstick for measuring the performance of other algorithms. CFO is a novel deterministic metaheuristic that has performed well against GSO in previous tests. The CFO implementation reported here includes architectural improvements in errant probe retrieval and decision space adaptation that result in even better performance. Detailed results are provided for the twenty-three function benchmark suite used to evaluate GSO. CFO performs better than or essentially as well as GSO on twenty functions and nearly as well on one of the remaining three. Includes update 24 February 2010. Comment: Includes detailed numerical results and source code in appendices. Update 02-24-10: Replaces Fig. A2(b) for improved visualization; corrects mino...
Dynamic Threshold Optimization (DTO) adaptively "compresses" the decision space (DS) in... more Dynamic Threshold Optimization (DTO) adaptively "compresses" the decision space (DS) in a global search and optimization problem by bounding the objective function from below. This approach is different from "shrinking" DS by reducing bounds on the decision variables. DTO is applied to Schwefel's Problem 2.26 in 2 and 30 dimensions with good results. DTO is universally applicable, and the author believes it may be a novel approach to global search and optimization.
Central Force Optimization is a global search and optimization algorithm that searches a decision... more Central Force Optimization is a global search and optimization algorithm that searches a decision space by flying "probes" whose trajectories are deterministically computed using two equations of motion. Because it is possible for a probe to fly outside the domain of feasible solutions, a simple errant probe retrieval method has been used previously that does not include the directional information contained in a probe's acceleration vector. This note investigates the effect of adding directionality to the "repositioning factor" approach. As a general proposition, it appears that doing so does not improve convergence speed or accuracy. In fact, adding directionality to the original errant probe retrieval scheme appears to be highly inadvisable. Nevertheless, there may be alternative probe retrieval schemes that do benefit from directional information, and the results reported here may assist in or encourage their development. Comment: Ver. 2, 6 June 2010 (Fig...
Pi Fractions are used to create deterministic uniformly distributed pseudorandom decision space s... more Pi Fractions are used to create deterministic uniformly distributed pseudorandom decision space sample points for a global search and optimization algorithm. These fractions appear to be uniformly distributed on [0,1] and can be used in any stochastic algorithm rendering it effectively deterministic without compromising its ability to explore the decision space. Pi Fractions are generated using the BBP Pi digit extraction algorithm. The Pi Fraction approach is tested using genetic algorithm Pi-GASR with very good results. A Pi Fraction data file is available upon request.
This paper presents Central Force Optimization, a novel, nature inspired, deterministic search me... more This paper presents Central Force Optimization, a novel, nature inspired, deterministic search metaheuristic for constrained multi-dimensional optimization. CFO is based on the metaphor of gravitational kinematics. Equations are presented for the positions and ...
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