fundamental-engineering-optimization-methods.pdf

Denote a vector normal to the non negativity

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denote a vector normal to the non-negativity constraint: െݔ ൑ Ͳ ² Then, the optimality requires that there exist real numbers, ݒ ൒ Ͳǡ ݅ ൌ ͳǡ ǥ ǡ ݉ and ݑ ൒ Ͳǡ ݆ ൌ ͳǡ ǥ ǡ ݊ ³ such that the following conditions hold: ࢉ ൌ σ ݒ െ σ ݑ σ ݒ ݏ ൅ σ ݑ ݔ ൌ Ͳ (5.21) Let the Lagrange multipliers be grouped as: ߤ א ൛ݑ ǡ ݒ ³ and let ܰ א ሼࢇ ǡ െࢋ denote the set of active constraint normals, then the complementarity condition is expressed as: ࢉ ൌ െ׏ݖ ൌ σ ߤ ௜אࣣ ܰ ³ where denotes the set of active constraints. The above condition states that at the optimal point the negative of objective function gradient lies in the convex cone spanned by the active constraint normals. When this condition holds, the descent-feasible cone is empty, i.e., we cannot move in a direction that further decreases the objective function without leaving the feasible region. This result is consistent with Farkas Lemma, which for the LP problems is stated as follows:
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Download free eBooks at bookboon.com Click on the ad to read more Fundamental Engineering Optimization Methods 103 Linear Programming Methods Farka’s Lemma (Belegundu and Chandrupatla, p. 204): Given a set of vectors, ǡ ݅ ൌ ͳǡ ǥ ǡ ݉ ³ and a vector c , there is no vector d satisfying the conditions ࢊ ൏ Ͳ and ࢊ ൐ Ͳǡ ݅ ൌ ͳǡ ǥ ǡ ݉ ³ if and only if c can be written as: ࢉ ൌ σ ߤ ௜ୀଵ ǡ ߤ ൒ Ͳ ² An illustrative example for the optimality conditions appears below: Example 5.11: Optimality Conditions for the LP problem We reconsider example 5.1 that was formulated as: ǡ௫ ݖ ൌ ͵ݔ ൅ ʹݔ 6XEMHFW WR± ͵ݔ ൅ ʹݔ ൒ ͳʹǡ ʹݔ ൅ ͵ݔ ൑ ͳ͸ǡ ݔ ൒ Ͳǡ ݔ ൒ Ͳ Application of the optimality conditions results in the following equations: ݔ ሺʹݒ ൅ ʹݒ െ ʹሻ ൅ ݔ ሺݒ ൅ ͵ݒ െ ͵ሻ ൌ Ͳ ݒ ሺʹݔ ൅ ݔ െ ͳʹሻ ൅ ݒ ሺʹݔ ൅ ͵ݔ െ ͳ͸ሻ ൌ Ͳ
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Download free eBooks at bookboon.com Fundamental Engineering Optimization Methods 104 Linear Programming Methods We split these into four equations and use Matlab symbolic toolbox to solve them, which gives the following candidate solutions: ሼݔ ǡ ݔ ǡ ݒ ǡ ݒ ሽ ൌ ሺͲǡͲǡͲǡͲሻǡ ሺ͸ǡͲǡͳǡͲሻǡ ሺͺǡͲǡͲǡͳሻǡ ሺͷǡʹǡͲǡͳሻǡ ሺͲǡͳʹǡ͵ǡͲሻǡ ሺͲǡͷǤ͵͵ǡͲǡͳሻǡ ቀͺ െ ଷ௭ ǡ ݖǡ Ͳǡͳቁ Then, it can be verified that the optimality conditions hold only in the case of: ሼݔ ǡ ݔ ǡ ݒ ǡ ݒ ሽ ൌ ሺͷǡʹǡͲͳሻǤ The optimum value of the objective function is: z* = 17.
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