fundamental-engineering-optimization-methods.pdf

These move limits represent the maximum allowed

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limits, to bind the LP solution. These move limits represent the maximum allowed change in ݀ in the current iteration. They are generally selected as a percentage (1–100%) of the design variable values. They serve dual purpose of binding the LP solution and obviating the need for line search in the current iteration. Restrictive move limits tend to make the SLP problem infeasible.

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Download free eBooks at bookboon.com Fundamental Engineering Optimization Methods 152 ±umerical Optimization Methods The SLP algorithm is presented below: SLP Algorithm (Arora, p. 508) : Initialize: choose ǡ ߝ ൐ Ͳǡ ߝ ൐ Ͳ ² For ݇ ൌ Ͳǡͳǡʹǡ ǥ 1. Choose move limits ο ௜௟ ǡ ο ௜௨ as some fraction of current design x k 2. Compute ݂ ǡ ࢉǡ ݃ ǡ ݄ ǡ ܾ ǡ ݁ 3. Formulate and solve the LP subproblem for d k 4. If and ݃ ൑ ߝ Ǣ ݅ ൌ ͳǡ ǥ ǡ ݉Ǣ ห݄ ห ൑ ߝ Ǣ ݅ ൌ ͳǡ ǥ ǡ ݌Ǣ DQG ฮࢊ ฮ ൑ ߝ ³ stop 5. Substitute ௞ାଵ ՚ ࢞ ൅ ߙࢊ ǡ ݇ ՚ ݇ ൅ ͳ ² The SLP algorithm is simple to apply, but should be used with caution in engineering design problems as it can easily run into convergence problems. The selection of move limits is one of trial and error and can be best achieved in an interactive mode. An example is presented to explain the SLP method: Example 7.6: Sequential Linear Programming We perform one iteration of the SLP algorithm for the following NLP problem: ǡ௫ ݂ሺݔ ǡ ݔ ሻ ൌ ݔ െ ݔ ݔ ൅ ݔ Subject to: ͳ െ ݔ െ ݔ ൑ ͲǢ െݔ ൑ Ͳǡ െݔ ൑ Ͳ The NLP problem is convex and has a single minimum at כ ൌ ቀ ξଶ ǡ ξଶ ቁǤ The objective and constraint gradients are: ׏݂ ൌ ሾʹݔ െ ݔ ǡ ʹݔ െ ݔ ሿǡ ׏݃ ൌ ሾെʹݔ ǡ െʹݔ ሿǡ ׏݃ ൌ ሾെͳǡͲሿǡ ׏݃ ൌ ሾͲǡ െͳሿ ² Let ൌ ሺͳǡ ͳሻǡ so that ݂ ൌ ͳǡ ࢉ ൌ ሾͳ ͳሿ · further, let ߝ ൌ ߝ ൌ ͲǤͲͲͳ · then, using SLP method, the resulting LP problem at the current step is defined as: ǡௗ ݂ሺݔ ǡ ݔ ሻ ൌ ݀ ൅ ݀ Subject to: െʹ െʹ െͳ Ͳ Ͳ െͳ ൩ ൤ ݀ ݀ ൨ ൑ ൥ ͳ ͳ ͳ
Download free eBooks at bookboon.com Click on the ad to read more Fundamental Engineering Optimization Methods 153 ±umerical Optimization Methods Since the LP problem is unbounded, we may use 50% move limits to bind the solution. The resulting update is given as: כ ൌ ቂെ ǡ െ ³ VR WKDW ൌ ቂ ǡ ³ with resulting constraint violations given as: ݃ ൌ ቄ ǡ Ͳǡ Ͳቅ ² We note that smaller move limits in this step could have avoided resulting constraint violation.

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