What Is Optimization Toolbox?
What Is Optimization Toolbox? What Is Optimization Toolbox?
3 Standard AlgorithmsThe line search method attempts to decrease the objective function along theline by repeatedly minimizing polynomial interpolation models of theobjectivefunction.Thelinesearch procedure has two main steps:• The bracketing phase determines the range of points on the lineto be searched. The bracket corresponds to an intervalspecifying the range of values of α.• The sectioning step divides the bracket into subintervals, on whichthe minimum of the objective function is approximated by polynomialinterpolation.The resulting step length α satisfies the Wolfe conditions:where c 1and c 2are constants with 0 < c 1< c 2
Quasi-Newton ImplementationQuasi-Newton ImplementationThis section describes the implementation of the quasi-Newton method in thetoolbox. The algorithm consists of two phases:• “Hessian Update” on page 3-11• “Line Search Procedures” on page 3-11Hessian UpdateMany of the optimization functions determine the direction of search byupdating the Hessian matrix at each iteration, using the BFGS method(Equation 3-6). The function fminunc also provides an option to use theDFP method given in “Quasi-Newton Methods” on page 3-7 (set HessUpdateto 'dfp' in options to select the DFP method). The Hessian, H, isalwaysmaintained to be positive definite so that the direction of search, d, isalwaysin a descent direction. This means that for some arbitrarily small step inthe direction d, the objective function decreases in magnitude. You achievepositive definiteness of H by ensuring that H is initialized to be positivedefinite and thereafter (from Equation 3-11) is always positive. Theterm is a product of the line search step length parameter andacombination of the search direction d with past and present gradientevaluations,(3-11)You always achieve the condition that is positive by performing asufficiently accurate line search. This is because the search direction, d, isadescent direction, so that and negative gradient of are alwayspositive. Thus, the possible negative termcan be made as smallin magnitude as required by increasing the accuracy of the line search.Line Search ProceduresAfter choosing the direction of the search, the optimization function uses aline search procedure to determine how far to move in the search direction.This section describes the line search procedures used by the functionslsqnonlin, lsqcurvefit, andfsolve.3-11
- Page 81 and 82: Large-Scale Examplesans =1.1885e-01
- Page 83 and 84: Large-Scale ExamplesW = Hinfo*Y - V
- Page 85 and 86: Large-Scale Exampleswere not the sa
- Page 87 and 88: Large-Scale Examplestradeoff is ben
- Page 89 and 90: Large-Scale Examplesfunction W = qp
- Page 91 and 92: Large-Scale Examples% RUNQPBOX4PREC
- Page 93 and 94: Large-Scale Examplesalgorithm: 'lar
- Page 95 and 96: Large-Scale Examplescgiterations: 0
- Page 97 and 98: Large-Scale Examplesdoes not give a
- Page 99 and 100: Default Options SettingsDetermining
- Page 101 and 102: Displaying Iterative OutputDisplayi
- Page 103 and 104: Displaying Iterative Outputbintprog
- Page 105 and 106: Displaying Iterative OutputfsolveTh
- Page 107 and 108: Displaying Iterative Outputlsqnonli
- Page 109 and 110: Calling an Output Function Iterativ
- Page 111 and 112: Calling an Output Function Iterativ
- Page 113 and 114: Calling an Output Function Iterativ
- Page 115 and 116: Optimizing Anonymous Functions Inst
- Page 117 and 118: Optimizing Anonymous Functions Inst
- Page 119 and 120: Typical Problems and How to Deal wi
- Page 121 and 122: Typical Problems and How to Deal wi
- 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: Unconstrained Optimizationvariables
- Page 135 and 136: Quasi-Newton Implementationvalues o
- 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
- Page 157 and 158: Constrained Optimizationis updated
- Page 159 and 160: Constrained OptimizationInitializat
- Page 161 and 162: Constrained OptimizationSimplex Alg
- Page 163 and 164: Constrained Optimization3 Updates t
- Page 165 and 166: Multiobjective OptimizationMultiobj
- Page 167 and 168: Multiobjective OptimizationIn the t
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- Page 171 and 172: Multiobjective OptimizationWhat is
- Page 173 and 174: Multiobjective Optimization(3-51)wh
- 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
3 Standard AlgorithmsThe line search method attempts to decrease the objective function along theline by repeatedly minimizing polynomial interpolation models of theobjectivefunction.Thelinesearch procedure has two main steps:• The bracketing phase determines the range of points on the lineto be searched. The bracket corresponds to an intervalspecifying the range of values of α.• The sectioning step divides the bracket into subintervals, on whichthe minimum of the objective function is approximated by polynomialinterpolation.The resulting step length α satisfies the Wolfe conditions:where c 1and c 2are constants with 0 < c 1< c 2