# Algorithms for Variable-Size Optimization: Applications in Space Systems and Renewable Energy

Section I: Background and MotivationIntroduction and BackgroundMathematical BackgroundDefinitionsDifferentiability and Taylor’s TheoremOrthogonal VectorsGram-Schmidt OrthogonalizationQ-Orthogonal (Q-Conjugate) DirectionsConvergence RatesSystems Architecture OptimizationInterplanetary Trajectory OptimizationMicrogrid OptimizationTraffic Network Signal Coordination PlanningOptimal Grouping ProblemsSystems Design OptimizationStructural Topology OptimizationPixel Classification ProblemsModeling Examples of Variable-Size Design Space ProblemsSatellite Orbit Design OptimizationInterplanetary Trajectory OptimizationOptimization of Wave Energy ConvertersSection II: Classical Optimization AlgorithmsFundamentals and Core AlgorithmsEqual Interval Search AlgorithmGolden Section MethodLinear versus Nonlinear OptimizationLinear ProgrammingNonlinear ProgrammingUnconstrained OptimizationNon-Gradient AlgorithmsCyclic Coordinated Descent MethodPattern Search MethodPowell’s MethodGradient-Based OptimizationSteepest Descent MethodConjugate Gradient MethodVariable Metric MethodsSecond Order MethodsNumerical Algorithms for Constrained OptimizationIndirect MethodsBarrier MethodsExterior Penalty Function MethodsAugmented Lagrange Multiplier MethodAlgorithm for Indirect MethodsDirect MethodsSequential Linear ProgrammingQuadratic ProgrammingSequential Quadratic ProgrammingSection III: Variable-Size Design Space OptimizationHidden Genes Genetic AlgorithmsIntroduction to Global OptimizationGenetic AlgorithmsSimilarity Templates (schemata)Markov Chain ModelFundamental Concepts of Hidden Genes Genetic AlgorithmsThe Hidden Genes Concept in BiologyConcept of Optimization using Hidden Genes Genetic AlgorithmsOutline of a Simple HGGAThe Schema Theorem and the Simple HGGAReproductionCrossoverMutationHidden Genes Assignment MethodsLogical Evolution of TagsStochastic Evolution of TagsExamples: VSDS Mathematical FunctionsExamples using Stochastically Evolving TagsExamples using Logically Evolving TagsStatistical AnalysisMarkov Chain Model of HGGAFinal RemarksStructured Chromosome Genetic AlgorithmsStructured-Chromosome Evolutionary Algorithms (SCEAs)Crossover in SCGAMutation in SCGATransformation in SCDENiching in SCGA and SCDETrajectory Optimization using SCEAEarth-Mars MissionEarth-Saturn Mission (Cassini 2-like Mission)Jupiter Europa Orbiter MissionComparisons and DiscussionDynamic-Size Multiple Population Genetic AlgorithmsThe Concept of DSMPGAApplication: Space Trajectory OptimizationNumerical ExamplesDiscussionSpace Trajectory OptimizationBackgroundA Simple Implementation of HGGAOptimizationNumerical ResultsDiscussionTrajectory Optimization using HGGA with Binary TagsEarth–Jupiter Mission using HGGAEarth–Jupiter Mission: Numerical Results and ComparisonsControl and Shape Optimization of Wave Energy ConvertersA Conical Buoy in Regular WaveGeneral Shape Buoys in Regular WavesWECs in Irregular WavesSimultaneous Optimization of Shape and ControlDiscussionBibliography