Engineering Optimization: Theory and Practice, Fourth Edition

Problems 483 7.14 Consider the problem: Minimizef=(x 1 − 1 )^2 +(x 2 − 5 )^2 subject to g 1 = −x^21 +x 2 − 4 ≤ 0 g 2 = −(x 1 − 2 ...

484 Nonlinear Programming III: Constrained Optimization Techniques 7.19 Approximate the following problem as a quadratic program ...

Problems 485 subject to 0 ≤x 1 0 ≤x 2 ≤ x 1 √ 3 0 ≤x 1 + √ 3 x 2 ≤ 6 7.23 Construct theφkfunction, according to(a)interior and(b ...

486 Nonlinear Programming III: Constrained Optimization Techniques 7.28 Solve the following problem using an interior penalty fu ...

Problems 487 Starting point for Unconstrained Value of minimization of minimum of k rk φ (X, rk) φ (X, rk)=X∗k f (X∗k)=fk∗ 1 1 ( ...

488 Nonlinear Programming III: Constrained Optimization Techniques Figure 7.26 Two-bar truss subjected to a parametric load. 7.4 ...

Problems 489 Determine whether the solution X=    √^0 √^2 2    is optimum by finding the values of the Lagrange multiplier ...

490 Nonlinear Programming III: Constrained Optimization Techniques 7.45 Find the extrapolated solution of Problem 7.44 by using ...

Problems 491 7.55 Find the solution of the following problem using the MATLAB functionfminconwith the starting point:X 1 = { 1. ...

8 Geometric Programming 8.1 Introduction Geometric programming is a relatively new method of solving a class of nonlinear progra ...

8.4 Solution Using Differential Calculus 493 is a second-degree polynomial in the variables,x 1 , x 2 , andx 3 (coefficients of ...

494 Geometric Programming By multiplying Eq. (8.4) byxk, we can rewrite it as xk ∂f ∂xk = ∑N j= 1 akj(cj x a 1 j 1 x a 2 j 2 · · ...

8.4 Solution Using Differential Calculus 495 Equations (8.7) are called theorthogonality conditions and Eq. (8.9) is called the ...

496 Geometric Programming Degree of Difficulty. The quantityN−n−1 is termed adegree of difficulty in geometric programming. In t ...

8.4 Solution Using Differential Calculus 497 These equations, in the case of problems with a zero degree of difficulty, give a u ...

498 Geometric Programming Figure 8.1 Open rectangular box. that is,  1 + 3 − 4 = 0 (E 2 )  1 + 2 − 4 = 0 (E 3 )  2 + 3 − ...

8.4 Solution Using Differential Calculus 499 It can be seen that the minimum total cost has been obtained before finding the opt ...

500 Geometric Programming or x 1 ∗= 1 Finally, we can obtainx 3 ∗by adding Eqs. (E 14 ) (E, 15 ) and (E, 16 ) sa w 3 = n 1l +ln ...

8.6 Primal–Dual Relationship and Sufficiency Conditions in the Unconstrained Case 501 = ( c 1  1 ) 1 ( c 2  2 ) 2 ·· · ( CN ...

502 Geometric Programming In this section we prove thatf∗=v∗ and also thatf∗corresponds to the global minimum off (X). For conve ...