The trial solution is required to satisfy the

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denote the penalty function at the trial solution. The trial solution is required to satisfy the following descent condition: Ȱ ௞ାଵǡ௝ ൅ ݐ ߛฮࢊ ൑ Ȱ ௞ǡ௝ ǡ Ͳ ൏ ߛ ൏ ͳ (7.54) where a common choice for γ is: ߛ ൌ ² Further, ݐ ൌ ߤ ǡ ߤ ൌ ǡ ݆ ൌ Ͳǡͳǡʹǡ ǥ ² The above descent condition ensures that the constraint violation decreases at each step of the method. The following example illustrates the application of approximate line search algorithm. Example 7.7: Sequential Quadratic Programming with Approximate Line Search We consider the above NL problem, given as: ǡ௫ ݂ሺݔ ǡ ݔ ሻ ൌ ݔ െ ݔ ݔ ൅ ݔ VXEMHFW WR ݃ ǣ ͳ െ ݔ െ ݔ ൑ Ͳǡ ݃ ǣ െ ݔ ൑ Ͳǡ ݃ ǣ െ ݔ ൑ Ͳ ²
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Download free eBooks at Fundamental Engineering Optimization Methods 157 ±umerical Optimization Methods where the gradient functions are computed as: ׏݂ ൌ ሾʹݔ െ ݔ ǡ ʹݔ െ ݔ ሿǡ ׏݃ ൌ ሾെʹݔ ǡ െʹݔ ሿǡ ׏݃ ൌ ሾെͳǡͲሿǡ ׏݃ ൌ ሾͲǡ െͳሿ ² Let ݔ ൌ ሺͳǡ ͳሻ · then, ݂ ൌ ͳǡ ࢉ ൌ ሾͳǡ ͳሿ ǡ ݃ ሺͳǡͳሻ ൌ ݃ ሺͳǡͳሻ ൌ ݃ ሺͳǡͳሻ ൌ െͳ ² Since, at this point, there are no active constraints, ܸ ൌ Ͳ ³ the preferred search direction is: ࢊ ൌ െࢉ ൌ ሾെͳǡ െͳሿ · the line search problem is defined as: Ȱሺߙሻ ൌ ݂ሺ࢞ ൅ ߙࢊ ሻ ൌ ሺͳ െ ߙሻ ² This problem can be analytically solved by setting Ȱ ሺߙሻ ൌ Ͳ ³ with the solution: ߙ ൌ ͳ ³ resulting in ݔ ൌ ሺͲǡ Ͳሻ · however, this analytical solution results in a large constraint violation that is undesired. Use of the approximate line search method for the problem results in the following computations: Let ݐ ൌ ͳǡ ܴ ൌ ͳͲǡ ߛ ൌ ߤ ൌ · WKHQ ଵǡ଴ ൌ ሺͲǡͲሻǡ ԡࢊ ԡ ൌ ʹǡ ݂ ଵǡ଴ ൌ Ͳǡ ܸ ଵǡ଴ ൌ ͳǡ Ȱ ଵǡ଴ ൌ ͳͲǡ and the descent condition Ȱ ଵǡ଴ ԡࢊ ԡ ൑ Ȱ ൌ ͳ is not met. We then try ݐ WR REWDLQ± ଵǡଵ ൌ ቀ ǡ ቁ ǡ ± ଵǡଵ ǡ Ȱ ଵǡଵ ൌ ͷ ³ and the descent condition fails again; next, for ݐ ³ ZH JHW± ଵǡଶ ൌ ቀ ǡ ቁ ǡ ± ଵǡଶ ൌ Ͳǡ Ȱ ଵǡଶ ଵ଺ ³ and the descent condition checks as: Ȱ ଵǡଶ ԡࢊ ԡ ൑ Ȱ ² Therefore, we set: ߙ ൌ ݐ ǡ ࢞ ൌ ࢞ ଵǡଶ ൌ ቀ ǡ with no constraint violation.
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