What Is Optimization Toolbox?
What Is Optimization Toolbox? What Is Optimization Toolbox?
3 Standard AlgorithmsFigure 3-12:Geometrical Representation of the Goal Attainment MethodSpecification of the goals, , defines the goal point, P. The weightingvector defines the direction of search from P to the feasible function space,. During the optimization is varied, which changes the size of thefeasible region. The constraint boundaries converge to the unique solutionpoint .Algorithm Improvements for the Goal AttainmentMethodThe goal attainment method has the advantage that it can be posed as anonlinear programming problem. Characteristics of the problem can also beexploited in a nonlinear programming algorithm. In sequential quadraticprogramming (SQP), the choice of merit function for the line search is noteasy because, in many cases, it is difficult to “define” the relative importancebetween improving the objective function and reducing constraint violations.This has resulted in a number of different schemes for constructing themerit function (see, for example, Schittkowski [38]). In goal attainmentprogramming there might be a more appropriate merit function, which youcan achieve by posing Equation 3-50 as the minimax problem3-50
Multiobjective Optimization(3-51)whereFollowing the argument of Brayton et. al. [2]forminimaxoptimization usingSQP, using the merit function of Equation 3-41 for the goal attainmentproblem of Equation 3-51 gives(3-52)When the merit function of Equation 3-52 is used as the basis of a line searchprocedure, then, although might decrease for a step in a given searchdirection, the function max might paradoxically increase. This is acceptinga degradation in the worst case objective. Sincetheworstcaseobjectiveisresponsible for the value of the objective function ,thisisacceptingastepthat ultimately increases the objective function to be minimized. Conversely,might increase when max decreases, implying a rejection of a stepthat improves the worst case objective.Following the lines of Brayton et. al. [2], a solution is therefore to setequal to the worst case objective, i.e.,A problem in the goal attainment method is that it is common to use aweighting coefficient equal to 0 to incorporate hard constraints. The meritfunction of Equation 3-53 then becomes infinite for arbitrary violations ofthe constraints.(3-53)3-51
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- Page 123 and 124: 3Standard AlgorithmsStandard Algori
- Page 125 and 126: Multiobjective Optimization (p. 3-4
- Page 127 and 128: Demos of Medium-Scale MethodsDemos
- Page 129 and 130: Unconstrained OptimizationFigure 3-
- Page 131 and 132: Unconstrained Optimizationvariables
- Page 133 and 134: Quasi-Newton ImplementationQuasi-Ne
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- Page 137 and 138: Quasi-Newton ImplementationMixed Cu
- Page 139 and 140: Quasi-Newton ImplementationCase 4.3
- Page 141 and 142: Least-Squares OptimizationIn proble
- Page 143 and 144: Least-Squares OptimizationLevenberg
- Page 145 and 146: Least-Squares OptimizationThe linea
- Page 147 and 148: Nonlinear Systems of EquationsNonli
- Page 149 and 150: Nonlinear Systems of Equationsand s
- Page 151 and 152: Constrained OptimizationConstrained
- Page 153 and 154: Constrained OptimizationGiven the p
- Page 155 and 156: Constrained OptimizationUpdating th
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- Page 159 and 160: Constrained OptimizationInitializat
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- Page 163 and 164: Constrained Optimization3 Updates t
- Page 165 and 166: Multiobjective OptimizationMultiobj
- Page 167 and 168: Multiobjective OptimizationIn the t
- Page 169 and 170: Multiobjective OptimizationThe afor
- Page 171: Multiobjective OptimizationWhat is
- Page 175 and 176: Selected BibliographySelected Bibli
- Page 177 and 178: Selected Bibliography[24] Han, S.P.
- Page 179 and 180: 4Large-Scale AlgorithmsLarge-Scale
- Page 181 and 182: Trust-Region Methods for Nonlinear
- Page 183 and 184: Trust-Region Methods for Nonlinear
- Page 185 and 186: Preconditioned Conjugate GradientsP
- Page 187 and 188: Linearly Constrained ProblemsLinear
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- Page 191 and 192: Quadratic ProgrammingQuadratic Prog
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- Page 199 and 200: Selected Bibliography[11] Zhang, Y.
- Page 201 and 202: 5Optimization ToolGetting Started w
- Page 203 and 204: Getting Started with the Optimizati
- Page 205 and 206: Selecting a SolverSelecting a Solve
- Page 207 and 208: Defining the ProblemDefining the Pr
- Page 209 and 210: Defining the ProblemFunction to Min
- Page 211 and 212: Defining the ProblemConstraintsLine
- Page 213 and 214: Defining the ProblemM-file, or as a
- Page 215 and 216: Defining the ProblemBounds are lowe
- Page 217 and 218: Defining the Problemfseminf Problem
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3 Standard AlgorithmsFigure 3-12:Geometrical Representation of the Goal Attainment MethodSpecification of the goals, , defines the goal point, P. The weightingvector defines the direction of search from P to the feasible function space,. During the optimization is varied, which changes the size of thefeasible region. The constraint boundaries converge to the unique solutionpoint .Algorithm Improvements for the Goal AttainmentMethodThe goal attainment method has the advantage that it can be posed as anonlinear programming problem. Characteristics of the problem can also beexploited in a nonlinear programming algorithm. In sequential quadraticprogramming (SQP), the choice of merit function for the line search is noteasy because, in many cases, it is difficult to “define” the relative importancebetween improving the objective function and reducing constraint violations.This has resulted in a number of different schemes for constructing themerit function (see, for example, Schittkowski [38]). In goal attainmentprogramming there might be a more appropriate merit function, which youcan achieve by posing Equation 3-50 as the minimax problem3-50
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