ሺ ? the modified kkt conditions are given as

Info icon This preview shows pages 105–107. Sign up to view the full content.

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.
Image of page 105

Info icon This preview has intentionally blurred sections. Sign up to view the full version.

Download free eBooks at 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.
Image of page 106
Image of page 107
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern