fundamental-engineering-optimization-methods.pdf

Ͷሻ? zkhuh ? ሺ? ? ݔ ? ʹߣሺݔ ଵ ? ݔ ଵ

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െ ͷሻǡ ZKHUH ׏ ࣦሺ࢞ǡ ࣅሻ ൌ ൤ െݔ ൅ ʹߣሺݔ െ ͵ሻ െݔ ൅ ʹߣݔ ൨ ൌ ቂ ʹߣ െͳ െͳ ʹߣ ቃ ቂ ݔ ݔ ቃ െ ቂ ͸ߣ Ͳ
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Download free eBooks at bookboon.com Fundamental Engineering Optimization Methods 66 Mathematical Optimization Solving the FONC together with the equality constraint, the optimum solution to the problem is given as: כ ൌ ሺͶǡʹሻǡ ߣ כ ൌ ͳǡ ݂ כ ൌ െͺ ² The Hessian at the optimum point is computed as: ׏ ሺ࢞ כ ǡ ߣ כ ሻ ൌ ቂ ʹ െͳ െͳ ʹ ³ and is positive definite. Therefore, using local duality theory, ׏ ࣦሺ࢞ǡ ࣅሻ ൌ Ͳ is solved for x, which gives: ݔ ݔ ቃ ൌ ସఒ ିଵ ͳʹߣ ͸ߣ ² Next, the dual problem defined by ࣅஹ૙ ࣦሺ࢞ሺߣሻǡ ߣሻ ൌ ସ൫ఒାఒ ିଶ଴ఒ ሺସఒ ିଵሻ ǡ ߣ ് േ ³ which has a local solution at ߣ כ ൌ ͳ with ࣦሺ࢞ሺߣ כ ሻǡ ߣ כ ሻ ൌ െͺ ³ which matches the primal solution. 4.6.4 Separable Problems Dual methods are particularly powerful when applied to separable problems that are structured as: ݂ሺ࢞ሻ ൌ ෍ ݂ ሺݔ 6XEMHFW WR± σ ݃ ሺݔ ൑ Ͳǡ σ ݄ ሺݔ ሻ ൌ Ͳ (4.37) The dual function for the separable problem is formulated as: ߮ሺ࢛ǡ ࢜ሻ ൌ ൭෍ ݂ ሺݔ ൅ ݑ ෍ ݃ ሺݔ ൅ ݒ ෍ ݄ ሺݔ which decomposes into m separate single-variable problems given as: ݂ ሺݔ ሻ ൅ ݑ݃ ሺݔ ሻ ൅ ݒ݄ ሺݔ ³ EO E L O which can be relatively easy to solve. Thus, the formulation of a dual problem becomes simple. The next example shows how local duality can be applied to engineering problems that are separable. Example 4.8: truss design problem (Belegundu and Chandrupatla, p. 274) A truss contains a total of 16 elements of length ܮ ൌ ͵Ͳ ǡ ݅ ൌ ͳǡ ǥ ǡͳʹǢ ܮ ൌ ͵Ͳξʹ ǡ ݅ ൌ ͳʹǡ ǥ ǡͳ͸ E O G ܲ ʹͷ ͲͲͲ OE K L 7K L K I and cross-sectional area D ݔ (design variable). The truss bears a load ܲ ൌ ʹͷǡͲͲͲ OE at the tip. The weight of the structure is to be minimized with a bound on tip deflection, ߜ ൑ ͳ in. The problem is formulated as: ܮ ݔ ଵ଺ ௜ୀଵ 6XEMHFW WR± σ െ ߙቁ ଵ଺ ௜ୀଵ ൑ Ͳ ³ ݔ ൒ ݔ ZKHUH ܿ ௉௅ ாఋ ǡ ߙ ൌ ଵ଺ ǡ ݔ ൌ ͳͲ ି଺ LQ²
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Download free eBooks at bookboon.com Click on the ad to read more Fundamental Engineering Optimization Methods 67 Mathematical Optimization The dual function is defined as: ߮ሺߤሻ ൌ ஹ௫ σ ܮ ݔ ଵ଺ ௜ୀଵ ൅ ߤ ቀ െ ߙቁ which leads to individual problems of the form: ஹ௫ ߰ ൌ ܮ ݔ ൅ ߤ ቀ െ ߙቁ ²
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