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fundamental-engineering-optimization-methods.pdf

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Download free eBooks at bookboon.com Fundamental Engineering Optimization Methods 160 ±umerical Optimization Methods The resulting KKT conditions for an optimum are given as: ׏݂ ൅ ׏ ࣦ ο࢞ ൅ ࡺ࢜ ൌ ૙ǡ ࢎ ൅ ࡺο࢞ ൌ ૙ ² In matrix form, these KKT conditions are similar to those used in the Newton-Raphson update. 7.6.5 SQP with Hessian Update The above Newton’s implementation of SQP algorithm uses Hessian of the Lagrangian function for the update. Since Hessian computation is relatively costly, an approximate to the Hessian may instead be used. Towards that end, let ࡴ ൌ ׏ ³ then the modified QP subproblem is defined as (Arora, p. 557): ݂ ҧ ൌ ࢉ ࢊ ൅ ͳ ʹ ࡴࢊ 6XEMHFW WR³ ࢊ ൑ ࢈ǡ ࡺ ࢊ ൌ ࢋ (7.64) We note that quasi-Newton methods (Sec. 7.3.4) solve the unconstrained minimization problem by solving a set of linear equations given as: ൌ െࢉ IRU ³ ZKHUH where represents an approximation to the Hessian matrix. In particular, the popular BFGS method uses the following Hessian update: ௞ାଵ ൌ ࡴ ൅ ࡰ ൅ ࡱ (7.65) where ǡ ࡱ ǡ ࢙ ൌ ߙ ǡ ࢟ ൌ ࢉ ௞ାଵ െ ࢉ ǡ ࢉ ൌ ׏݂൫࢞ ² Next, the BFGS Hessian update is modified to apply to the constrained optimization problems as follows: let ൌ ߙ ǡ ࢠ ൌ ࡴ ǡ ࢟ ൌ ׏ࣦ൫࢞ ௞ାଵ ൯ െ ׏ࣦ൫࢞ ൯ǡ ࢙ ൌ ߦ ǡ ࢙ ൌ ߦ · further, define: ൌ ߠ࢟ ൅ ሺͳ െ ߠሻࢠ ³ where ߠ ൌ ቄͳǡ ଴Ǥ଼క ିక ³ ൌ ߦ Ǣ then, the Hessian update is given as: ௞ାଵ ൌ ࡴ ൅ ࡰ െ ࡱ ǡ ࡰ ǡ ࡱ ² The modified SQP algorithm is given as follows: Modified SQP Algorithm (Arora, p. 558) : Initialize: choose ǡ ܴ ൌ ͳǡ ࡴ ൌ ܫǢ ߝ ǡ ߝ ൐ Ͳ ² For ݇ ൌ Ͳǡͳǡʹǡ ǥ 1. Compute ݂ ǡ ݃ ǡ ݄ ǡ ࢉǡ ܾ ǡ ݁ ǡ DQG ܸ ² ,I ݇ ൐ Ͳǡ compute H k 2. Formulate and solve the modified QP subproblem for search direction d k and the Lagrange multipliers DQG ² 3. If ܸ ൑ ߝ DQG ฮࢊ ฮ ൑ ߝ ³ stop.
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