fundamental-engineering-optimization-methods.pdf

ሺ ? the modified kkt conditions are given as

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ሺ࡯࢞ െ ࢊሻ The modified KKT conditions are given as: Feasibility: ࡭࢞ െ ࢈ െ ࢙ ൌ ૙ǡ ࡯࢞ ൌ ࢊ Optimality: ࡽ࢞ ൅ ࢉ െ ࢛ െ ࡭ ࢜ ൅ ࡯ ࢝ ൌ ૙ Complementarity: ࢞ ൅ ࢜ ࢙ ൌ ૙ Non-negativity: ࢞ ൒ ૙ǡ ࢙ ൒ ૙ǡ ࢛ ൒ ૙ǡ ࢜ ൒ ૙ where the Lagrange multipliers w for the equality constraints are not restricted in sign. By introducing: ࢝ ൌ ܡ െ ܢǢ ܡǡ ܢ ൒ ૙ ³ we can represent the combined optimality and feasibility conditions as: െ࡭ ൩ ቂ ቃ െ ൥ ൩ ቂ ቃ ൅ ൥ െ࡯ ൩ ቂ ቃ ൅ ቈ െ࢈ െࢊ ቉ ൌ ቈ (5.28) The above problem can be similarly solved via LCP framework, which is introduced in Sec. 5.8.
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Download free eBooks at bookboon.com Click on the ad to read more Fundamental Engineering Optimization Methods 106 Linear Programming Methods 5.8.2 The Dual QP Problem We reconsider the QP problem (5.22) and observe that the Lagrangian function (5.23) is stationary at the optimum with respect to x , u , v . Then, as per Lagrangian duality (Sec. 4.5), it can be used to define the following dual QP problem (called Wolfe’s dual): ࢞ǡ࢛ǡ࢜ ࣦሺ࢞ǡ ࢛ǡ ࢜ሻ ൌ ࡽ࢞ ൅ ࢉ ࢞ െ ࢛ ࢞ ൅ ࢜ ሺ࡭࢞ െ ࢈ሻ Subject to: ׏ࣦሺ࢞ǡ ࢛ǡ ࢜ሻ ൌ ࡽ࢞ ൅ ࢉ െ ࢛ ൅ ࡭ ࢜ ൌ ૙ǡ ࢞ ൒ ૙ǡ ࢜ ൒ ૙ (5.29) Further, by relaxing the non-negativity condition on the design variable x , we can eliminate u from the formulation, which results in a simpler dual problem defined as: ࢞ǡ࢜ஹ૙ ࣦሺ࢞ǡ ࢜ሻ ൌ ࡽ࢞ ൅ ࢉ ࢞ ൅ ࢜ ሺ࡭࢞ െ ࢈ሻ Subject to: ࡽ࢞ ൅ ࢉ ൅ ࡭ ࢜ ൌ ૙ (5.30) The implicit function theorem allows us to express the solution vector x in the vicinity of the optimum point as a function of the Lagrange multipliers DV± ࢞ ൌ ࢞ሺ࢜ሻ ² Next, the Lagrangian is expressed as an implicit function Ȱሺ࢜ሻ of the multipliers, termed as the dual function. Further, the dual function is obtained as a solution to the following minimization problem: Ȱሺ࢜ሻ ൌ ࣦሺ࢞ǡ ࢜ሻ ൌ ࡽ࢞ ൅ ࢉ ࢞ ൅ ࢜ ሺ࡭࢞ െ ࢈ሻ (5.31) We do not reinvent the wheel we reinvent light. Fascinating lighting offers an infinite spectrum of possibilities: Innovative technologies and new markets provide both opportunities and challenges. An environment in which your expertise is in high demand. Enjoy the supportive working atmosphere within our global group and benefit from international career paths. Implement sustainable ideas in close cooperation with other specialists and contribute to influencing our future. Come and join us in reinventing light every day.
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