Chapter v. Strong Minima and the Weierstrass Condition A. Classifying Local Minima
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چکیده
Remark. The notation in condition (iii) reflects the possibility that ḣ might fail to exist at finitely many points. (Various alternative ways to deal with this possibility exist; none is completely satisfactory.) There is an obvious heierarchy in these concepts: every global minimizer also satisfies the criteria to lie in each of the local categories listed above. Any strong local minimizer is certainly also a weak local minimizer, and any weak local minimizer clearly gives a directional local minimum. [Indeed, if x̂ gives a WLM, and any fixed arc h ∈ VII is given, then the number M(h) = supt∈[a,b) { |h(t)|, ∣∣ḣ(t+) ∣∣ } is finite because h is piecewise smooth. Choosing εd = εw/M(h) will show that whenever |λ| < εd, the arc λh satisfies both inequalities in condition (iii), so Λ[x̂] ≤ Λ[x̂+ λh]. This establishes the desired statement defining a DLM.]
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