fundamental-engineering-optimization-methods.pdf

278 for inequality constrained problems the al may be

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Chandrupatla, p. 278). For inequality constrained problems, the AL may be defined as (Arora, p. 480): ࣪ሺ࢞ǡ ࢛ǡ ݎሻ ൌ ݂ሺ࢞ሻ ൅ ෍ ൞ ݑ ݃ ሺ࢞ሻ ൅ ͳ ʹ ݎ݃ ሺ࢞ሻǡ ݃ ݑ ݎ ൒ Ͳ ͳ ʹݎ ݑ ǡ ݃ ݑ ݎ ൏ Ͳ (7.43) The AL algorithm is given below. The Augmented Lagrangian Algorithm (Arora, p. 480) Initialize: estimate ݔ ǡ ݑ ൒ Ͳǡ ݒ ǡ ݎ ൐ ͲǢ FKRRVH ߙ ൐ Ͳǡ ߚ ൐ ͳǡ ߳ ൐ Ͳǡ ߢ ൐ Ͳǡ ܭ ൌ λ For ݇ ൌ ͳǡʹǡ ǥ 1. Solve ࣪ሺ࢞ǡ ࢛ǡ ࢜ǡ ݎ 2. Evaluate ݄ ൫࢞ ൯ǡ ݅ ൌ ͳǡ Ǥ Ǥ ǡ ݈Ǣ ݃ ൫࢞ ൯ǡ ݆ ൌ ͳǡ ǥ ǡ ݉Ǣ compute ܭ ൌ ݉ܽݔ ቄȁ݄ ȁǡ ݅ ൌ ͳǡ ǥ ǡ ݈Ǣ ቀ݃ ǡ െ ቁ ǡ ݆ ൌ ͳǡ ǥ ǡ ݉ቅ 3. Check termination: If ܭ ൑ ߢ DQG ฮ׏࣪൫࢞ ൯ฮ ൑ ߳ ݉ܽݔ൛ͳǡ ฮ࢞ ฮൟ ³ quit 4. If ܭ ൏ ܭ (i.e., constraint violations have improved), set ܭ ൌ ܭ Set ݒ ௞ାଵ ൌ ݒ ൅ ݎ ݄ ൫࢞ ൯Ǣ ݅ ൌ ͳǡ ǥ ǡ ݈ ² 6HW ݑ ௞ାଵ ൌ ݑ ൅ ݎ ݉ܽݔ ൜݃ ൫࢞ ൯ǡ െ ൠ Ǣ ݆ ൌ ͳǡ ǥ ǡ ݉ ² If ܭ ǡ (i.e., constraint violations did not improve by a factor ߙ ), set ݎ ௞ାଵ ൌ ߚݎ An example for the AL method is now presented. Example 7.5: Design of cylindrical water tank (Belegundu and Chandrupatla, p. 278) We consider the design of an open-top cylindrical water tank. We wish to maximize the volume of the tank for a given surface area ܣ ² Let d be the diameter and h be the height; then, the optimization problem is formulated as: ௗǡ௟ ݂ሺ݀ǡ ݈ሻ ൌ ߨ݀ ݈ Ͷ subject to ݄ǣ గௗ ൅ ߨ݈݀ െ ܣ ൌ Ͳ
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Download free eBooks at bookboon.com Click on the ad to read more Fundamental Engineering Optimization Methods 150 ±umerical Optimization Methods We drop the constant ǡ convert to a minimization problem, assume ସ஺ ൌ ͳǡ and redefine the problem as: ௗǡ௟ ݂ ҧ ሺ݀ǡ ݈ሻ ൌ െ݀ ݈ subject to ݄ǣ ݀ ൅ Ͷ݈݀ െ ͳ ൌ Ͳ A Lagrangian function for the problem is formulated as: ࣦሺ݀ǡ ݈ǡ ߣሻ ൌ െ݀ ݈ ൅ ߣሺ݀ ൅ Ͷ݈݀ െ ͳሻ The FONC for the problem are: െʹ݈݀ ൅ ʹߣሺ݀ ൅ ʹ݈ሻ ൌ Ͳǡ െ݀ ൅ Ͷ݀ߣ ൌ Ͳǡ ݀ ൅ Ͷ݈݀ െ ͳ ൌ Ͳ ² Using FONC, the optimal solution is given as: ݀ כ ൌ ʹ݈ כ ൌ Ͷߣ כ ξଷ ² The Hessian at the optimum point is given as: ׏ ࣦሺ݀ כ ǡ ݈ כ ǡ ߣ כ ሻ ൌ ቂ െʹߣ െͶߣ െͶߣ Ͳ ² , It is evident that the Hessian is not positive definite. Next, the AL for the problem is formed as: ࣪ሺ݀ǡ ݈ǡ ߣǡ ݎሻ ൌ െ݀ ݈ ൅ ߣሺ݀ ൅ Ͷ݈݀ െ ͳሻ ൅ ͳ ʹ ݎሺ݀ ൅ Ͷ݈݀ െ ͳሻ The dual function is defined as: ߰ሺߣሻ ൌ ௗǡ௟ ࣪ሺ݀ǡ ݈ǡ ߣǡ ݎሻ ² The dual optimization problem is then formulated as: ௗǡ௟ ߰ሺߣሻ ²
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