Engineering Optimization: Theory and Practice, Fourth Edition

Problems 243 4.14 Express the dual of the following LP problem: Maximizef= 2 x 1 +x 2 subject to x 1 − 2 x 2 ≥ 2 x 1 + 2 x 2 = 8 ...

244 Linear Programming II: Additional Topics and Extensions 4.25 Assume that productsA, B, C, andDrequire, in addition to the st ...

Problems 245 4.38 Transform the following LP problem into the form required by Karmarkar’s method: Minimizef= x 1 +x 2 +x 3 subj ...

246 Linear Programming II: Additional Topics and Extensions Figure 4.6 Plastic hinges in a frame. Figure 4.7 Possible failure me ...

Problems 247 to ensure nonzero reserve strength in each failure mechanism. Also, suggest a suitable technique for solving the pr ...

5 Nonlinear Programming I: One-Dimensional Minimization Methods 5.1 Introduction In Chapter 2 we saw that if the expressions for ...

5.1 Introduction 249 Figure 5.1 Planar truss: (a) nodal and member numbers; (b) nodal degrees of freedom. zero, as they correspo ...

250 Nonlinear Programming I: One-Dimensional Minimization Methods It is important to note that an explicit closed-form solution ...

5.1 Introduction 251 Figure 5.2 Contact stress between two spheres. SOLUTION Forν 1 =ν 2 = 0. 3 , Eq. (E 1 ) educes tor f (λ)= 0 ...

252 Nonlinear Programming I: One-Dimensional Minimization Methods x 2 Figure 5.3 Iterative process of optimization. The iterativ ...

5.2 Unimodal Function 253 Table 5.1 One-dimensional Minimization Methods Elimination methods Unrestricted search Requiring no de ...

254 Nonlinear Programming I: One-Dimensional Minimization Methods Figure 5.5 Outcome of first two experiments: (a) f 1 < f 2 ...

5.3 Unrestricted Search 255 used must be small in relation to the final accuracy desired. Although this method is very simple to ...

256 Nonlinear Programming I: One-Dimensional Minimization Methods i Value ofs xi=x 1 +s fi=f(xi) Isfi>fi− 1? 1 — 0.0 0.0 — 2 ...

5.5 Dichotomous Search 257 is given by Ln=xj+ 1 −xj− 1 = 2 n+ 1 L 0 (5.2) The final interval of uncertainty obtainable for diffe ...

258 Nonlinear Programming I: One-Dimensional Minimization Methods Figure 5.7 Dichotomous search. function at the two points, alm ...

5.5 Dichotomous Search 259 whereδ is a small quantity, say 0.001, andnis the number of experiments. If the middle point of the f ...

260 Nonlinear Programming I: One-Dimensional Minimization Methods The final set of experiments will be conducted at x 5 = ( 0. 7 ...

5.6 Interval Halving Method 261 Figure 5.8 Possibilities in the interval halving method: (a) f 2 >f 0 >f 1 ; (b) f 1 >f ...

262 Nonlinear Programming I: One-Dimensional Minimization Methods Example 5.6 Find the minimum off=x(x− 1 .5) in the interval (0 ...