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Topic 15 - Intro to KTT Conditions

Topic 15 - Intro to KTT Conditions - ISE 5406 Topic#15...

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174 ISE 5406 Topic #15: Intro to KKT Conditions BSS Textbook Reading: Section 4.2 We now present the KKT conditions for inequality constrained problems. KKT points are a subset of Fritz John points where the Lagrange multiplier for the objective gradient is required to be positive. Motivation Recall that we are dealing with a problem of the following form, where , , and all functions are assumed to be continuous and differentiable. minimize subject to for We expressed the Fritz John conditions as follows, where ̅ is the set of binding constraints at ̅ . ̅ ∑ ̅ ̅ with ( ̅ ) , As we discussed, there are a number of Fritz John points that do not correspond to local minima. In particular, one of the drawbacks of the Fritz John conditions is that the gradients of the binding constraints can cancel each other out. This allows the Lagrange multiplier for objective gradient to be zero, and effectively makes the objective function irrelevant to the dual feasibility conditions. We would like to ignore the Fritz John points for which is necessarily equal to zero. Instead, we will focus on the Fritz John points where there exists a set of Lagrangian multipliers with . Without loss of generality, we will require .
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