Preface |
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ix | |
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1 | (14) |
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2 | (2) |
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Engineering Design versus Engineering Analysis |
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4 | (1) |
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Conventional versus Optimum Design Process |
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4 | (2) |
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Optimum Design versus Optimal Control |
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6 | (1) |
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Basic Terminology and Notation |
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7 | (8) |
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7 | (2) |
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9 | (1) |
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Superscripts/Subscripts and Summation Notation |
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9 | (2) |
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11 | (1) |
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11 | (1) |
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U.S.-British versus SI Units |
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12 | (3) |
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Optimum Design Problem Formulation |
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15 | (40) |
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The Problem Formulation Process |
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16 | (2) |
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Step 1: Project/Problem Statement |
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16 | (1) |
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Step 2: Data and Information Collection |
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16 | (1) |
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Step 3: Identification/Definition of Design Variables |
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16 | (1) |
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Step 4: Identification of a Criterion to Be Optimized |
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17 | (1) |
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Step 5: Identification of Constraints |
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17 | (1) |
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18 | (2) |
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Insulated Spherical Tank Design |
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20 | (2) |
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22 | (2) |
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Design of a Two-Bar Bracket |
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24 | (6) |
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30 | (2) |
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Formulation 1 for Cabinet Design |
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30 | (1) |
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Formulation 2 for Cabinet Design |
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31 | (1) |
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Formulation 3 for Cabinet Design |
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31 | (1) |
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Minimum Weight Tubular Column Design |
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32 | (3) |
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Formulation 1 for Column Design |
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33 | (1) |
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Formulation 2 for Column Design |
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34 | (1) |
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Minimum Cost Cylindrical Tank Design |
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35 | (1) |
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36 | (2) |
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Minimum Weight Design of a Symmetric Three-Bar Truss |
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38 | (3) |
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A General Mathematical Model for Optimum Design |
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41 | (14) |
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Standard Design Optimization Model |
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42 | (1) |
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Maximization Problem Treatment |
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43 | (1) |
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Treatment of ``Greater Than Type'' Constraints |
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43 | (1) |
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Discrete and Integer Design Variables |
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44 | (1) |
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45 | (1) |
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Active/Inactive/Violated Constraints |
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45 | (1) |
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46 | (9) |
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55 | (28) |
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Graphical Solution Process |
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55 | (5) |
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Profit Maximization Problem |
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55 | (1) |
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Step-by-Step Graphical Solution Procedure |
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56 | (4) |
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Use of Mathematica for Graphical Optimization |
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60 | (4) |
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61 | (1) |
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Identification and Hatching of Infeasible Region for an Inequality |
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62 | (1) |
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Identification of Feasible Region |
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62 | (1) |
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Plotting of Objective Function Contours |
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63 | (1) |
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Identification of Optimum Solution |
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63 | (1) |
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Use of MATLAB for Graphical Optimization |
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64 | (2) |
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Plotting of Function Contours |
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64 | (1) |
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64 | (2) |
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Design Problem with Multiple Solutions |
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66 | (1) |
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Problem with Unbounded Solution |
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66 | (1) |
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67 | (2) |
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Graphical Solution for Minimum Weight Tubular Column |
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69 | (1) |
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Graphical Solution for a Beam Design Problem |
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69 | (14) |
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72 | (11) |
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83 | (92) |
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Definitions of Global and Local Minima |
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84 | (5) |
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84 | (5) |
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89 | (1) |
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Review of Some Basic Calculus Concepts |
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89 | (14) |
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90 | (2) |
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92 | (1) |
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93 | (3) |
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Quadratic Forms and Definite Matrices |
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96 | (6) |
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Concept of Necessary and Sufficient Conditions |
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102 | (1) |
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Unconstrained Optimum Design Problems |
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103 | (16) |
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Concepts Related to Optimality Conditions |
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103 | (1) |
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Optimality Conditions for Functions of Single Variable |
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104 | (5) |
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Optimality Conditions for Functions of Several Variables |
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109 | (7) |
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Roots of Nonlinear Equations Using Excel |
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116 | (3) |
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Constrained Optimum Design Problems |
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119 | (24) |
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119 | (2) |
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Necessary Conditions: Equality Constraints |
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121 | (7) |
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Necessary Conditions: Inequality Constraints---Karush-Kuhn-Tucker (KKT) Conditions |
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128 | (12) |
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Solution of KKT Conditions Using Excel |
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140 | (1) |
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Solution of KKT Conditions Using MATLAB |
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141 | (2) |
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Postoptimality Analysis: Physical Meaning of Lagrange Multipliers |
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143 | (6) |
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Effect of Changing Constraint Limits |
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143 | (3) |
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Effect of Cost Function Scaling on Lagrange Multipliers |
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146 | (1) |
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Effect of Scaling a Constraint on Its Lagrange Multiplier |
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147 | (1) |
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Generalization of Constraint Variation Sensitivity Result |
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148 | (1) |
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149 | (9) |
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149 | (2) |
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151 | (2) |
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Convex Programming Problem |
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153 | (3) |
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Transformation of a Constraint |
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156 | (1) |
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Sufficient Conditions for Convex Programming Problems |
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157 | (1) |
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Engineering Design Examples |
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158 | (17) |
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158 | (4) |
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Design of a Rectangular Beam |
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162 | (4) |
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166 | (9) |
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More on Optimum Design Concepts |
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175 | (16) |
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Alternate Form of KKT Necessary Conditions |
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175 | (3) |
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178 | (1) |
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Second-Order Conditions for Constrained Optimization |
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179 | (5) |
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Sufficiency Check for Rectangular Beam Design Problem |
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184 | (7) |
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185 | (6) |
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Linear Programming Methods for Optimum Design |
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191 | (68) |
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Definition of a Standard Linear Programming Problem |
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192 | (3) |
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192 | (1) |
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193 | (1) |
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193 | (2) |
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Basic Concepts Related to Linear Programming Problems |
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195 | (6) |
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195 | (3) |
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198 | (3) |
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Optimum Solution for LP Problems |
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201 | (1) |
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Basic Ideas and Steps of the Simplex Method |
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201 | (17) |
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202 | (1) |
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Canonical Form/General Solution of Ax = b |
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202 | (1) |
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203 | (2) |
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205 | (1) |
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Basic Steps of the Simplex Method |
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206 | (5) |
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211 | (7) |
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Two-Phase Simplex Method--Artificial Variables |
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218 | (10) |
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219 | (1) |
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219 | (1) |
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Definition of Phase I Problem |
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220 | (1) |
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220 | (1) |
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221 | (5) |
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Degenerate Basic Feasible Solution |
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226 | (2) |
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228 | (15) |
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Changes in Resource Limits |
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229 | (6) |
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Ranging Right Side Parameters |
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235 | (4) |
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Ranging Cost Coefficients |
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239 | (2) |
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Changes in the Coefficient Matrix |
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241 | (2) |
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Solution of LP Problems Using Excel Solver |
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243 | (16) |
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246 | (13) |
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More on Linear Programming Methods for Optimum Design |
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259 | (18) |
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Derivation of the Simplex Method |
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259 | (3) |
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Selection of a Basic Variable That Should Become Nonbasic |
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259 | (1) |
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Selection of a Nonbasic Variable That Should Become Basic |
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260 | (2) |
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262 | (1) |
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Duality in Linear Programming |
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263 | (14) |
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263 | (1) |
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264 | (1) |
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Treatment of Equality Constraints |
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265 | (1) |
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Alternate Treatment of Equality Constraints |
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266 | (1) |
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Determination of Primal Solution from Dual Solution |
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267 | (4) |
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Use of Dual Tableau to Recover Primal Solution |
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271 | (2) |
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Dual Variables as Lagrange Multipliers |
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273 | (2) |
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275 | (2) |
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Numerical Methods for Unconstrained Optimum Design |
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277 | (28) |
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General Concepts Related to Numerical Algorithms |
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278 | (4) |
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279 | (1) |
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Descent Direction and Descent Step |
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280 | (2) |
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Convergence of Algorithms |
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282 | (1) |
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282 | (1) |
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Basic Ideas and Algorithms for Step Size Determination |
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282 | (11) |
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Definition of One-Dimensional Minimization Subproblem |
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282 | (1) |
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Analytical Method to Compute Step Size |
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283 | (2) |
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Concepts Related to Numerical Methods to Compute Step Size |
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285 | (1) |
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286 | (2) |
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Alternate Equal Interval Search |
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288 | (1) |
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289 | (4) |
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Search Direction Determination: Steepest Descent Method |
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293 | (3) |
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Search Direction Determination: Conjugate Gradient Method |
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296 | (9) |
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300 | (5) |
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More on Numerical Methods for Unconstrained Optimum Design |
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305 | (34) |
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More on Step Size Determination |
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305 | (5) |
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306 | (3) |
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309 | (1) |
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More on Steepest Descent Method |
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310 | (5) |
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Properties of the Gradient Vector |
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310 | (4) |
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Orthogonality of Steepest Descent Directions |
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314 | (1) |
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Scaling of Design Variables |
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315 | (3) |
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Search Direction Determination: Newton's Method |
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318 | (6) |
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Classical Newton's Method |
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318 | (1) |
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319 | (4) |
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323 | (1) |
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Search Direction Determination: Quasi-Newton Methods |
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324 | (5) |
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Inverse Hessian Updating: DFP Method |
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324 | (3) |
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Direct Hessian Updating: BFGS Method |
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327 | (2) |
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Engineering Applications of Unconstrained Methods |
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329 | (3) |
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Minimization of Total Potential Energy |
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329 | (2) |
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Solution of Nonlinear Equations |
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331 | (1) |
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Solution of Constrained Problems Using Unconstrained Optimization Methods |
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332 | (7) |
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Sequential Unconstrained Minimization Techniques |
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333 | (1) |
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Multiplier (Augmented Lagrangian) Methods |
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334 | (1) |
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335 | (4) |
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Numerical Methods for Constrained Optimum Design |
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339 | (40) |
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340 | (6) |
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Basic Concepts Related to Algorithms for Constrained Problems |
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340 | (2) |
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Constraint Status at a Design Point |
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342 | (1) |
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343 | (2) |
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345 | (1) |
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Convergence of an Algorithm |
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345 | (1) |
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Linearization of Constrained Problem |
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346 | (6) |
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Sequential Linear Programming Algorithm |
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352 | (6) |
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The Basic Idea---Move Limits |
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352 | (1) |
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353 | (4) |
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SLP Algorithm: Some Observations |
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357 | (1) |
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Quadratic Programming Subproblem |
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358 | (5) |
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Definition of QP Subproblem |
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358 | (3) |
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Solution of QP Subproblem |
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361 | (2) |
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Constrained Steepest Descent Method |
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363 | (6) |
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364 | (2) |
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366 | (2) |
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368 | (1) |
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CSD Algorithm: Some Observations |
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368 | (1) |
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Engineering Design Optimization Using Excel Solver |
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369 | (10) |
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373 | (6) |
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More on Numerical Methods for Constrained Optimum Design |
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379 | (34) |
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Potential Constraint Strategy |
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379 | (4) |
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Quadratic Programming Problem |
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383 | (5) |
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383 | (1) |
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KKT Necessary Conditions for the QP Problem |
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384 | (1) |
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Transformation of KKT Conditions |
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384 | (1) |
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Simplex Method for Solving QP Problem |
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385 | (3) |
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Approximate Step Size Determination |
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388 | (12) |
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388 | (1) |
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389 | (4) |
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CSD Algorithm with Approximate Step Size |
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393 | (7) |
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Constrained Quasi-Newton Methods |
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400 | (7) |
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Derivation of Quadratic Programming Subproblem |
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400 | (3) |
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Quasi-Newton Hessian Approximation |
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403 | (1) |
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Modified Constrained Steepest Descent Algorithm |
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404 | (2) |
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Observations on the Constrained Quasi-Newton Methods |
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406 | (1) |
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406 | (1) |
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Other Numerical Optimization Methods |
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407 | (6) |
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Method of Feasible Directions |
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407 | (2) |
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Gradient Projection Method |
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409 | (1) |
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Generalized Reduced Gradient Method |
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410 | (1) |
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411 | (2) |
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Introduction to Optimum Design with MATLAB |
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413 | (20) |
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Introduction to Optimization Toolbox |
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413 | (2) |
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Variables and Expressions |
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413 | (1) |
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Scalar, Array, and Matrix Operations |
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414 | (1) |
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414 | (1) |
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Unconstrained Optimum Design Problems |
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415 | (3) |
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Constrained Optimum Design Problems |
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418 | (2) |
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Optimum Design Examples with MATLAB |
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420 | (13) |
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Location of Maximum Shear Stress for Two Spherical Bodies in Contact |
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420 | (1) |
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Column Design for Minimum Mass |
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421 | (4) |
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Flywheel Design for Minimum Mass |
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425 | (4) |
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429 | (4) |
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Interactive Design Optimization |
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433 | (32) |
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Role of Interaction in Design Optimization |
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434 | (2) |
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What Is Interactive Design Optimization? |
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434 | (1) |
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Role of Computers in Interactive Design Optimization |
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434 | (1) |
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Why Interactive Design Optimization? |
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435 | (1) |
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Interactive Design Optimization Algorithms |
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436 | (12) |
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436 | (4) |
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Constraint Correction Algorithm |
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440 | (2) |
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Algorithm for Constraint Correction at Constant Cost |
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442 | (3) |
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Algorithm for Constraint Correction at Specified Increase in Cost |
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445 | (1) |
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Constraint Correction with Minimum Increase in Cost |
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446 | (1) |
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Observations on Interactive Algorithms |
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447 | (1) |
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Desired Interactive Capabilities |
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448 | (2) |
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Interactive Data Preparation |
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448 | (1) |
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448 | (1) |
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Interactive Decision Making |
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449 | (1) |
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450 | (1) |
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Interactive Design Optimization Software |
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450 | (4) |
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User Interface for IDESIGN |
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451 | (2) |
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453 | (1) |
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Examples of Interactive Design Optimization |
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454 | (11) |
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Formulation of Spring Design Problem |
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454 | (1) |
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Optimum Solution for the Spring Design Problem |
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455 | (1) |
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Interactive Solution for Spring Design Problem |
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455 | (2) |
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Use of Interactive Graphics |
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457 | (5) |
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462 | (3) |
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Design Optimization Applications with Implicit Functions |
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465 | (48) |
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Formulation of Practical Design Optimization Problems |
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466 | (7) |
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466 | (1) |
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Example of a Practical Design Optimization Problem |
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467 | (6) |
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Gradient Evaluation for Implicit Functions |
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473 | (5) |
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Issues in Practical Design Optimization |
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478 | (1) |
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Selection of an Algorithm |
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478 | (1) |
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Attributes of a Good Optimization Algorithm |
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478 | (1) |
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Use of General-Purpose Software |
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479 | (2) |
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480 | (1) |
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Integration of an Application into General-Purpose Software |
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480 | (1) |
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Optimum Design of Two-Member Frame with Out-of-Plane Loads |
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481 | (2) |
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Optimum Design of a Three-Bar Structure for Multiple Performance Requirements |
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483 | (8) |
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Symmetric Three-Bar Structure |
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483 | (1) |
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Asymmetric Three-Bar Structure |
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484 | (6) |
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490 | (1) |
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Discrete Variable Optimum Design |
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491 | (2) |
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Continuous Variable Optimization |
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492 | (1) |
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Discrete Variable Optimization |
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492 | (1) |
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Optimal Control of Systems by Nonlinear Programming |
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493 | (20) |
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A Prototype Optimal Control Problem |
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493 | (4) |
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Minimization of Error in State Variable |
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497 | (6) |
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Minimum Control Effort Problem |
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503 | (2) |
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Minimum Time Control Problem |
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505 | (3) |
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Comparison of Three Formulations for Optimal Control of System Motion |
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508 | (1) |
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508 | (5) |
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Discrete Variable Optimum Design Concepts and Methods |
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513 | (18) |
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Basic Concepts and Definitions |
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514 | (2) |
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Definition of Mixed Variable Optimum Design Problem: MV-OPT |
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514 | (1) |
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Classification of Mixed Variable Optimum Design Problems |
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514 | (1) |
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Overview of Solution Concepts |
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515 | (1) |
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Branch and Bound Methods (BBM) |
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516 | (5) |
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517 | (2) |
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BBM with Local Minimization |
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519 | (1) |
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520 | (1) |
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521 | (1) |
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Sequential Linearization Methods |
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522 | (1) |
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522 | (2) |
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Dynamic Rounding-off Method |
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524 | (1) |
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Neighborhood Search Method |
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525 | (1) |
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Methods for Linked Discrete Variables |
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525 | (1) |
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526 | (5) |
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527 | (4) |
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Genetic Algorithms for Optimum Design |
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531 | (12) |
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Basic Concepts and Definitions |
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532 | (2) |
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Fundamentals of Genetic Algorithms |
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534 | (4) |
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Genetic Algorithm for Sequencing-Type Problems |
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538 | (1) |
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539 | (4) |
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540 | (3) |
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Multiobjective Optimum Design Concepts and Methods |
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543 | (22) |
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543 | (3) |
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Terminology and Basic Concepts |
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546 | (6) |
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Criterion Space and Design Space |
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546 | (2) |
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548 | (3) |
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Preferences and Utility Functions |
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551 | (1) |
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Vector Methods and Scalarization Methods |
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551 | (1) |
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Generation of Pareto Optimal Set |
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551 | (1) |
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Normalization of Objective Functions |
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552 | (1) |
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552 | (1) |
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Multiobjective Genetic Algorithms |
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552 | (3) |
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555 | (1) |
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556 | (1) |
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Weighted Global Criterion Method |
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556 | (2) |
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558 | (1) |
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Bounded Objective Function Method |
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558 | (1) |
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559 | (1) |
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559 | (6) |
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560 | (5) |
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Global Optimization Concepts and Methods for Optimum Design |
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565 | (28) |
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Basic Concepts of Solution Methods |
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565 | (2) |
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565 | (2) |
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567 | (1) |
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Overview of Deterministic Methods |
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567 | (5) |
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568 | (1) |
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568 | (1) |
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Methods of Generalized Descent |
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569 | (2) |
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571 | (1) |
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Overview of Stochastic Methods |
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572 | (7) |
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573 | (1) |
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573 | (1) |
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573 | (2) |
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575 | (3) |
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Acceptance-Rejection Methods |
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578 | (1) |
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579 | (1) |
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Two Local-Global Stochastic Methods |
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|
579 | (6) |
|
A Conceptual Local-Global Algorithm |
|
|
579 | (1) |
|
Domain Elimination Method |
|
|
580 | (2) |
|
Stochastic Zooming Method |
|
|
582 | (1) |
|
Operations Analysis of the Methods |
|
|
583 | (2) |
|
Numerical Performance of Methods |
|
|
585 | (8) |
|
Summary of Features of Methods |
|
|
585 | (1) |
|
Performance of Some Methods Using Unconstrained Problems |
|
|
586 | (1) |
|
Performance of Stochastic Zooming and Domain Elimination Methods |
|
|
586 | (1) |
|
Global Optimization of Structural Design Problems |
|
|
587 | (1) |
|
|
588 | (5) |
|
Appendix A Economic Analysis |
|
|
593 | (18) |
|
|
593 | (5) |
|
|
594 | (1) |
|
A.1.2 Basic Economic Formulas |
|
|
594 | (4) |
|
A.2 Economic Bases for Comparison |
|
|
598 | (13) |
|
A.2.1 Annual Base Comparisons |
|
|
599 | (2) |
|
A.2.2 Present Worth Comparisons |
|
|
601 | (3) |
|
|
604 | (7) |
|
Appendix B Vector and Matrix Algebra |
|
|
611 | (36) |
|
B.1 Definition of Matrices |
|
|
611 | (2) |
|
B.2 Type of Matrices and Their Operations |
|
|
613 | (5) |
|
|
613 | (1) |
|
|
613 | (1) |
|
B.2.3 Addition of Matrices |
|
|
613 | (1) |
|
B.2.4 Multiplication of Matrices |
|
|
613 | (2) |
|
B.2.5 Transpose of a Matrix |
|
|
615 | (1) |
|
B.2.6 Elementary Row--Column Operations |
|
|
616 | (1) |
|
B.2.7 Equivalence of Matrices |
|
|
616 | (1) |
|
B.2.8 Scalar Product--Dot Product of Vectors |
|
|
616 | (1) |
|
|
616 | (1) |
|
B.2.10 Partitioning of Matrices |
|
|
617 | (1) |
|
B.3 Solution of n Linear Equations in n Unknowns |
|
|
618 | (10) |
|
|
618 | (1) |
|
|
619 | (2) |
|
B.3.3 Gaussian Elimination Procedure |
|
|
621 | (4) |
|
B.3.4 Inverse of a Matrix: Gauss-Jordan Elimination |
|
|
625 | (3) |
|
B.4 Solution of m Linear Equations in n Unknowns |
|
|
628 | (7) |
|
|
628 | (1) |
|
B.4.2 General Solution of m x n Linear Equations |
|
|
629 | (6) |
|
B.5 Concepts Related to a Set of Vectors |
|
|
635 | (7) |
|
B.5.1 Linear Independence of a Set of Vectors |
|
|
635 | (4) |
|
|
639 | (3) |
|
B.6 Eigenvalues and Eigenvectors |
|
|
642 | (1) |
|
B.7 Norm and Condition Number of a Matrix |
|
|
643 | (4) |
|
B.7.1 Norm of Vectors and Matrices |
|
|
643 | (1) |
|
B.7.2 Condition Number of a Matrix |
|
|
644 | (1) |
|
|
645 | (2) |
|
Appendix C A Numerical Method for Solution of Nonlinear Equations |
|
|
647 | (10) |
|
C.1 Single Nonlinear Equation |
|
|
647 | (3) |
|
C.2 Multiple Nonlinear Equations |
|
|
650 | (7) |
|
|
655 | (2) |
|
Appendix D Sample Computer Programs |
|
|
657 | (18) |
|
D.1 Equal Interval Search |
|
|
657 | (3) |
|
D.2 Golden Section Search |
|
|
660 | (1) |
|
D.3 Steepest Descent Method |
|
|
660 | (9) |
|
D.4 Modified Newton's Method |
|
|
669 | (6) |
References |
|
675 | (8) |
Bibliography |
|
683 | (4) |
Answers to Selected Problems |
|
687 | (8) |
Index |
|
695 | |