Lectures on Optimization - Theory and Algorithms by J Cea

By J Cea

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We shall therefore write J(uk − ρwk ) with ρ > 0 so that J(uk − ρwk ) ց as k increases for ρ > 0 small enough. Choice of ρ(= ρk ). Once the direction of descent wk is chosen then the iterative procedure can be done with a constant ρ > 0. It is however more suitable to do this with a variable ρ. We shall therefore choose ρ = ρk > 0 in a small interval with the property J(uk − ρk wk ) < J(uk ) and set uk+1 = uk − ρk wk . We do this in several steps. Since, j = inf J(v) ≤ J(uk+1 ) ≤ J(uk ) v∈V 53 we have J(uk ) − J(uk+1 ) ≥ 0 and lim (J(uk ) − J(uk+1 )) = 0 k→+∞ because J(uk ) is decreasing and bounded below.

1. 2)) this method is equivalent to making a change of variables and taking as the direction of descent the direction of the gradient of J in the new variables and then choosing wk as the inverse image of this direction in the original coordinates. Consider the functional J : V = R2 → R of our model problem of Chapter 1, §7: 1 1 R2 ∋ v → J(v) = a(v, v) − L(v) = (Av, v)R2 − ( f, v)R2 ǫR. 2 2 Since a(·, ·) is a positive definite quadratic form, {vǫR2 , J(v) = constant } represents an ellipse. e. the direction of the radial vector through uk (in the new coordinates).

Vdσ = 0. Γ We note that this formula remains valide if uǫH 2 (Ω) ∩ V for any vǫV. First we choose vǫD(Ω) ⊂ V (enough to take vǫC01 (Ω)(Ω) ⊂ V) then the boundary integral vanishes so that we get (−△u + u − f )vdx = 0 ∀vǫD(Ω). 7) −△u + u − f = o in Ω (in the sense of L2 (Ω)). 5. 8) −△u + u − f = 0 in the sense of distributions in Ω. Next we choose vǫV arbitrary. 9) Γ2 whcih means that ∂u/∂n = 0 on Γ in some generalized sense. In fact, by 1 1 trace theorem γvǫH 2 (Γ) and hence ∂u/∂n = 0 in H − 2 (Γ) (see Lions and Magenese [32]).

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