fundamental-engineering-optimization-methods.pdf

ݔ ڮ ݔ ሻ൯ሺݔ ? ଵ ͳ for ͳ ³ they result

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ݔ ൅ ڮ ൅ ݌ ݔ ሻ൯ሺെݔ ௜ୀଵ ൌ Ͳ For ݊ ൌ ͳ ³ they result in the following equations: ͳ σ ݔ σ ݔ σ ݔ ቍ ቀ ݌ ݌ ቁ ൌ ቌ σ ݕ σ ݔ ݕ
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Download free eBooks at bookboon.com Fundamental Engineering Optimization Methods 48 Mathematical Optimization As డ௣ σ ݔ ௝ା௞ ³ the SONC for the problem are evaluated as: ͳ ڮ σ ݔ ڭ ڰ ڭ σ ݔ ڮ σ ݔ ଶ௡ ൲ ൒ Ͳ ² For ݊ ൌ ͳ ³ the determinant of the Hessian evaluates as: σ ݔ െ ቀ σ ݔ ³ which defines the variance in the case of independent and identically distributed random variables. Finally, we note that since the data-fitting problem is convex, FONC are both necessary and sufficient for a minimum. Example 4.2: Open box problem We wish to determine the dimensions of an open box of maximum volume that can be constructed form a sheet of paper (8.5 ×11 in) by cutting squares from the corners and folding the sides upwards. Let x denote the width of the paper that is folded up, then the problem is formulated as: ݂ሺݔሻ ൌ ሺͳͳ െ ʹݔሻሺͺǤͷ െ ʹݔሻݔ The FONC for the problem evaluate as: ݂ ሺݔሻ ൌ ʹݔሺͳͻǤͷ െ Ͷݔሻ െ ሺͳͳ െ ʹݔሻሺͺǤͷ െ ʹݔሻ ൌ Ͳ ² Using Matlab Symbolic toolbox ‘solve’ command, we obtain two candidate solutions: ݔ כ ൌ ͳǤͷͺͷǡ ͶǤͻͳͷ ² Application of SOC results in: ݂ ᇱᇱ ሺݔሻ ൌ െ͵ͻǤͻͷǡ͵ͻǤͻͷ ³ respectively, indicating a maximum of ݂ሺ࢞ሻ at ݔ כ ൌ ͳǤͷͺͷ ZLWK ݂ሺݔ כ ሻ ൌ ͸͸Ǥͳͷ FX²LQ² 4.3 Optimality Criteria for the Constrained Problems The majority of engineering design problems involve constraints (LE, GE, EQ) that are modeled as functions of optimization variables. In this section, we explore how constraints affect the optimality criteria. An important consideration when applying the optimality criteria to problems involving constraints is whether x* lies on a constraint boundary. This is implied in the case for problems involving only equality constraints, which are discussed first. 4.3.3 Equality Constrained Problems The optimality criteria for equality constrained problems involve the use of Lagrange multipliers. To develop this concept, we consider a problem with a single equality constraint, stated as: ݂ሺ࢞ሻ VXEMHFW WR ݄ሺ࢞ሻ ൌ Ͳ (4.4)
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Download free eBooks at bookboon.com Click on the ad to read more Fundamental Engineering Optimization Methods 49 Mathematical Optimization We first note that the constraint equation can be used to solve for and substitute one of the variables (say ݔ ) in the objective function, and hence develop an unconstrained optimization problem in variables. n -1 This, however, depends on the form of ݄ሺ࢞ሻ and may not always be feasible. In order to develop more
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