fundamental-engineering-optimization-methods.pdf

We however note that convexity is a sufficient but

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sufficient for a global minimum in the case of convex optimization problems. We, however, note that convexity is a sufficient but not necessary condition for a global minimum, i.e., nonexistence of convexity does not preclude the existence of a global minimum. An example of a convex optimization problem is presented below. Example 4.6: We consider the following optimization problem: ǡ௫ ݂ሺݔ ǡ ݔ ሻ ൌ ݔ ൅ ݔ െ ݔ ݔ ³ subject to ݃ሺݔ ǡ ݔ ሻǣ ݔ ൅ ݔ െ ͳ ൑ ͲǢ ݄ሺݔ ǡ ݔ ሻǣ ݔ ൅ ݔ െ ܿ ൌ Ͳ As was done in Example 4.5, we convert the inequality constraint to equality via: ݔ ൅ ݔ െ ͳ ൅ ݏ ൌ Ͳ ² We then use Lagrange multipliers to formulate a Lagrangian function given as: ࣦሺݔ ǡ ݔ ǡ ݑǡ ݒǡ ݏሻ ൌ ݔ ൅ ݔ െ ݔ ݔ ൅ ݑሺݔ ൅ ݔ ൅ ݏ െ ͳሻ ൅ ݒሺݔ ൅ ݔ െ ܿሻ ²
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Download free eBooks at bookboon.com Fundamental Engineering Optimization Methods 57 Mathematical Optimization The resulting KKT conditions evaluate as: ሺʹݑ ൅ ͳሻݔ ൅ ݒ െ ݔ ൌ Ͳǡ ሺʹݑ ൅ ͳሻݔ ൅ ݒ െ ݔ ൌ Ͳǡ ݔ ൅ ݔ െ ܿ ൌ Ͳǡ ݔ ൅ ݔ ൅ ݏ െ ͳ ൌ Ͳǡ ݑݏ ൌ Ͳ ² From the switching condition: ݑ כ ൌ Ͳ RU ݏ כ ൌ Ͳ ² Similar to Example 4.5, the former condition evaluates as: ሺݔ כ ǡ ݔ כ ሻ ൌ ቀ ǡ ቁ ǡ ݏ כ ൌ േටͳ െ ǡ ݒ כ Ǣ while the latter condition has no feasible solution. Function evaluation at the sole candidate points results in: ݂ሺݔ כ ǡ ݔ כ ሻ ൌ െ ² 4.4.2 A Geometric Viewpoint The optimality criteria for constrained optimization problems have geometrical connotations. The following definitions help understand the geometrical viewpoint associated with the KKT conditions. Active constraint set. The set of active constraints at x is defined as: ࣣ ൌ ൛݅ ׫ ݆ǣ ݄ ሺ࢞ሻ ൌ Ͳǡ ݃ ሺ࢞ሻ ൌ Ͳൟ ² The set of active constraint normals is given as: ࣭ ൌ ሼ׏݄ ሺ࢞ሻǡ ׏݃ ሺ࢞ሻǡ ݆ א ࣣሽ ² Constraint tangent hyperplane. The constraint tangent hyperplane is defined by the set of vectors ൌ ൛ࢊǣ ׏݄ ሺ࢞ሻ ࢊ ൌ Ͳǡ ׏݃ ሺ࢞ሻ ࢊ ൌ Ͳǡ ݆ א ࣣൟ ² Regular point. Assume x is a feasible point. Then, x is a regular point if the vectors in the active constraint set are linearly independent. Feasible direction. Assume that x is a regular point. A vector d is a feasible direction if ݄ ሺ࢞ሻ ࢊ ൌ Ͳǡ ׏݃ ሺ࢞ሻ ࢊ ൏ Ͳǡ ݆ א ࣣǢ where the feasibility condition for each active inequality constraint defines a half space. The intersection of those half spaces is a feasible cone within which a feasible vector d should lie.
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