fundamental-engineering-optimization-methods.pdf

In order to develop more general optimality criteria

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and may not always be feasible. In order to develop more general optimality criteria, we follow Lagrange’s approach to the problem and consider the variation in the objective and constraint functions at a stationary point, given as: ݂݀ ൌ ߲݂ ߲ݔ ݀ݔ ൅ ڮ ൅ ߲݂ ߲ݔ ݀ݔ ൌ Ͳ (4.5) ݄݀ ൌ ߲݄ ߲ݔ ݀ݔ ൅ ڮ ൅ ߲݄ ߲ݔ ݀ݔ ൌ Ͳ We now combine these two conditions via a scalar weight (Lagrange multiplier, λ ) to write: ߲݂ ߲ݔ ൅ ߣ ߲݄ ߲ݔ ቇ ݀ݔ ௝ୀଵ ൌ Ͳ (4.6) Since variations ݀ݔ are independent, the above condition implies that: డ௙ డ௫ ൅ ߣ డ௛ డ௫ ൌ Ͳǡ ݆ ൌ ͳǡ ǥ ǡ ݊ .
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Download free eBooks at bookboon.com Fundamental Engineering Optimization Methods 50 Mathematical Optimization We further note that application of FONC to a Lagrangian function defined as: ࣦሺ࢞ǡ ߣሻ ൌ ݂ሺ࢞ሻ ൅ ߣ݄ሺ࢞ሻ also gives rise to the above condition. For multiple equality constraints, the Lagrangian function is similarly formulated as: ࣦሺ࢞ǡ ࣅሻ ൌ ݂ሺ࢞ሻ ൅ ෍ ߣ ݄ ሺ࢞ሻ ௜ୀଵ (4.7) Then, in order for ݂ሺ࢞ሻ to have a local minimum at x *, the following FONC must be satisfied: ߲ࣦ ߲ݔ ߲݂ ߲ݔ ൅ ෍ ߣ ߲݄ ߲ݔ ݔ ௜ୀଵ ൌ ͲǢ ݆ ൌ ͳǡ ǥ ǡ ݊ (4.8) ݄ ሺ࢞ሻ ൌ Ͳǡ ݅ ൌ ͳǡ ǥ ǡ ݈ The above FONC can be equivalently stated as: ׏ ࣦሺ࢞ כ ǡ ࣅ כ ሻ ൌ Ͳǡ ׏ ࣦሺ࢞ כ ǡ ࣅ כ ሻ ൌ Ͳ These conditions suggest that ࣦሺ࢞ כ ǡ ࣅ כ is stationary with respect to both x and λ ; therefore, minimization of ࣦሺ࢞ǡ ࣅሻ amounts to an unconstrained optimization problem. Further, the Lagrange Multiplier Theorem (Arora, p.135) states that if x * is a regular point (defined below) then the FONC result in a unique solution to R ߣ כ ² We note that the above FONC further imply: ׏ ݂ሺ࢞ כ ሻ ൌ െ σ ߣ ׏݄ ሺ࢞ כ ௜ୀଵ ² Algebraically, it means that the cost function gradient is a linear combination of the constraint gradients. Geometrically, it means that the negative of the cost function gradient lies in the convex cone spanned by the constraint normals (Sec. 4.3.5). SOSC for equality constrained problems are given as: ׏ ࣦሺ࢞ כ ǡ ࣅ כ ሻ ൌ ׏ ݂ሺ࢞ כ ሻ ൐ Ͳ ² Further discussion on SOC for constrained optimization problems is delayed till Sec. 4.4.3. An example is now presented to explain the optimization process for equality constrained problems. Example 4.3: We consider the following optimization problem: ǡ௫ ݂ሺݔ ǡ ݔ ሻ ൌ െݔ ݔ Subject to: ݄ሺݔ ǡ ݔ ሻ ൌ ݔ ൅ ݔ െ ͳ ൌ Ͳ
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Download free eBooks at bookboon.com Fundamental Engineering Optimization Methods 51 Mathematical Optimization We first note that the equality constraint can be used to develop an unconstrained problem in one variable, given as: ݂ሺݔ ሻ ൌ െݔ
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