Pi Fractions for Generating Uniformly Distributed Sampling Points in Global Search and Optimization Algorithms

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.Comment: Discussion of bidimensional correlation has been adde

On the Utility of Directional Information for Repositioning Errant Probes in Central Force Optimization

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. 1 improved for clarity; minor typos corrected

Central Force Optimization Applied to the PBM Suite of Antenna Benchmarks

Central Force Optimization (CFO) is a new nature-inspired deterministic multi-dimensional search and optimization metaheuristic based on the metaphor of gravitational kinematics. CFO is applied to the PBM antenna benchmark suite and the results compared to published performance data for other optimization algorithms. CFO acquits itself quite well. CFO's gradient-like nature is discussed, and it is speculated that a "generalized hyperspace derivative" might be defined for optimization problems as a new mathematical construct based on the Unit Step function. What appears to be a sufficient but not necessary condition for local trapping, oscillation in the probe average distance curve, is discussed in the context of the theory of gravitational "resonant returns" that gives rise to strikingly similar oscillatory curves. It is suggested that the theory may be applicable to CFO as an aid to understanding trapping and to developing effective mitigation techniques, possibly based on a concept of "energy" in CFO space. It also is suggested that CFO may be re-formulated as a "total energy" model by analogizing conservation of energy for orbiting masses in physical space

Are Near Earth Objects the Key to Optimization Theory?

This note suggests that near earth objects and Central Force Optimization have something in common, that NEO theory may hold the key to solving some vexing problems in deterministic optimization: local trapping and proof of convergence. CFO analogizes Newton's laws to locate the global maxima of a function. The NEO-CFO nexus is the striking similarity between CFO's Davg and an NEO's Delta-V curves. Both exhibit oscillatory plateau-like regions connected by jumps, suggesting that CFO's metaphorical "gravity" indeed behaves like real gravity, thereby connecting NEOs and CFO and being the basis for speculating that NEO theory may address difficult issues in optimization

Pseudorandomness in Central Force Optimization

Central Force Optimization is a deterministic metaheuristic for an evolutionary algorithm that searches a decision space by flying probes whose trajectories are computed using a gravitational metaphor. CFO benefits substantially from the inclusion of a pseudorandom component (a numerical sequence that is precisely known by specification or calculation but otherwise arbitrary). The essential requirement is that the sequence is uncorrelated with the decision space topology, so that its effect is to pseudorandomly distribute probes throughout the landscape. While this process may appear to be similar to the randomness in an inherently stochastic algorithm, it is in fact fundamentally different because CFO remains deterministic at every step. Three pseudorandom methods are discussed (initial probe distribution, repositioning factor, and decision space adaptation). A sample problem is presented in detail and summary data included for a 23-function benchmark suite. CFO's performance is quite good compared to other highly developed, state-of-the-art algorithms. Includes corrections 02-03-2010.Comment: Includes Source Code and Corrections 02-03-201

Dipole-Loaded Monopole Optimized Using VSO, v.3

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.Comment: arXiv admin note: substantial text overlap with arXiv:1107.1437, arXiv:1103.5629, arXiv:1108.0901, arXiv:1003.1039. Version 2, 02 Jul 2013: minor typos corrected; hill climber material added; source code listing updated. Version 3, 06 Jul 2013: replaces VSO diagram/pseudocode to clarify algorithm's elitist nature; other minor change

A novel methodology for antenna design and optimization: Variable Zo

This paper describes "Variable Zo," a novel and proprietary approach to antenna design and optimization. The new methodology is illustrated by applying it to the design of a resistively-loaded bowtie antenna and to two broadband Yagi-Uda arrays. Variable Zo is applicable to any antenna design or optimization methodology. Using it will result in generally better antenna designs across any user-specified set of performance objectives.Comment: Ver. 2 (14 July 2011). Adds Yagi-Uda array design example. Updates source cod

Dynamic Threshold Optimization - A New Approach?

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.Comment: Rev. 05 June 2012: Typos & reference [1] corrected. Material adde

Comparative Results: Group Search Optimizer and Central Force Optimization

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 minor typos (note that trajectory plots were removed to meet file size restrictions - see Ver. 1 for complete set

Issues in Antenna Optimization - A Monopole Case Study

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.Comment: Ver. 2. Corrects formatting in Table 1, updates references, and adds source code in Appendi