A plot of ሺߣ? yv² ߣ vs shows a concave function

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A plot of ߰ሺߣሻ YV² ߣ vs. shows a concave function with ߣ כ ൌ ߣ ௠௔௫ ൌ ͲǤͳͶͶ ² The optimum values for the design variables are the same as above: ݀ כ ൌ ʹ݈ כ ൌ ͲǤͷ͹͹Ǥ
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Download free eBooks at Fundamental Engineering Optimization Methods 151 ±umerical Optimization Methods 7.5 Sequential Linear Programming The sequential linear programming (SLP) method aims to sequentially solve the nonlinear optimization problem as a series of linear programs. In particular, we employ the first order Taylor series expansion to iteratively develop and solve a new LP subprogram to solve the KKT conditions associated with the NP problem. SLP methods are generally not robust, and have been mostly replaced by SQP methods. To develop the SLP method, let x k denote the current estimate of design variables and let d denote the change in variable; then, we express the first order expansion of the objective and constraint functions in the neighborhood of x k as: ݂൫࢞ ൅ ࢊ൯ ൌ ݂൫࢞ ൯ ൅ ׏݂൫࢞ ݃ ൫࢞ ൅ ࢊ൯ ൌ ݃ ൫࢞ ൯ ൅ ׏݃ ൫࢞ ࢊǡ ݅ ൌ ͳǡ ǥ ǡ ݉ ݄ ൫࢞ ൅ ࢊ൯ ൌ ݄ ൫࢞ ൯ ൅ ׏݄ ൫࢞ ࢊǡ ݆ ൌ ͳǡ ǥ ǡ ݈ (7.44) To proceed further, let: ݂ ൌ ݂൫࢞ ൯ǡ ݃ ൌ ݃ ൫࢞ ൯ǡ ݄ ൌ ݄ ൫࢞ · and define: ܾ ൌ െ݃ ǡ ݁ ൌ െ݄ ǡ ࢉ ൌ ׏݂൫࢞ ൯ǡ ൌ ׏݃ ൫࢞ ൯ǡ ࢔ ൌ ׏݄ ൫࢞ ൯ǡ ࡭ ൌ ሾࢇ ǡ ࢇ ǡ ǥ ǡ ࢇ ሿǡ ࡺ ൌ ሾ࢔ ǡ ࢔ ǡ ǥ ǡ ࢔ ² Then, after dropping the constant term ݂ from the objective function, we define the following LP subprogram for the current iteration of the NP problem (Arora, p. 498): ݂ ҧ ൌ ࢉ Subject to: ࢊ ൑ ࢈ǡ ࡺ ࢊ ൌ ࢋ (7.45) where ݂ ҧ represents the linearized change in the original cost function and the columns of A and N represent, respectively, the gradients of inequality and equality constraints. Since the objective and constraint functions are now linear, the resulting LP subproblem can be converted to standard form and solved via the Simplex method. Problems with a small number of variables can also be solved graphically or by application of KKT conditions to the LP problem. The following points regarding the SLP method should be noted: 1. Since both positive and negative changes to design variables x k are allowed, the variables ݀ are unrestricted in sign and, therefore, must be replaced by ݀ ൌ ݀ െ ݀ ି in the Simplex algorithm. 2. In order to apply the simplex method to the problem, the rhs parameters ܾ ǡ ݁ are assumed non-negative, or else, the respective constraint must be multiplied with െͳ ² 3. SLP methods require additional constraints of the form, െο ௜௟ ൑ ݀ ൑ ο ௜௨ ³ termed as move limits, to bind the LP solution. These move limits represent the maximum allowed change in
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