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First-order optimality condition Theorem (Optimality condition) Suppose f0 is difierentiable and the feasible set X is convex ▻ If x∗ is a local minimum of f0
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Lecture Note #07 KKT Optimality Condition Illustration in One-Dimension x a / 4+(x2) 2 − 1=0 v v v Figure 3: FONC and SONC for Constrained Minimization
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A necessary condition for local optimality is a statement of the form: “if ¯ x must satisfy ” Such a condition helps x is a local minimum of (P), then ¯ us identify all candidates for local optima Corollary 4 Suppose f(x) is differentiable at ¯ x is a local minimum, x If ¯ then ∇f(¯x)=0
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2 Necessary Optimality Conditions x is the minimizing point if and only if (y − ¯ Proof of Theorem 3: Let ¯x ∈ S be the point minimizing the distance from
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In this section we consider first–order optimality conditions using the optimality conditions for minimizing L(x, y, u) to eliminate x from the definition of L Since
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2 Necessary Optimality Conditions Theorem 2 1 (Geometric First-order Necessary Conditions) If ¯x is Furthermore, ¯x is the minimizing point if and only if
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difference of two convex functions A minimization problem in n with the constraints x g C, C closed convex, and an additional finite number of Ž
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