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船舶设计问题上的新全局优化英文文献和中文翻译(2)

时间:2020-11-14 10:55来源:毕业论文
In particular, thefilled function in Lucidi and Piccialli (2002) has the favorable property to be globallyconvexized, but the price that must be paid is the existence of a known local minimumpoint wo


In particular, thefilled function in Lucidi and Piccialli (2002) has the favorable property to be globallyconvexized, but the price that must be paid is the existence of a known local minimumpoint worse than x∗k that has a large basin of attraction, so that the local minimizationin step 3 is often attracted by this local minimum. The filled function we introducehere is not globally convexized, but does not have stationary points at all where theobjective function is higher than f(x∗k ), and the numerical experiments prove that thenew filled function is most reliable in practice to locate the global minimum point.2.1 A new filled functionIn this subsection we introduce a new filled function which has the following expres-sion:Q(x, ˜ x∗) = exp −x −˜ x∗2γ 2 +1 −exp (−τ [f(x)− f( ˜ x∗)+ ]), (3)where ˜ x∗ is a known stationary point of the original objective function f(x), γ> 0is a constant, and  > 0 and τ ≥ 1 are real parameters. This filled function is constituted by two terms; the first one, exp(−x −˜ x  /γ 2), makes point ˜ x∗ a local maximum of the filled function Q(x, ˜ x∗) (drawing ourinspiration from Renpu 1990 and Xu et al. 2001). The second term, 1−exp(τ [f(x)−f( ˜ x∗)+ ]), filters the stationary points of f(x) which have objective values greateror equal to f( ˜ x∗) and ensures that, for right values of the parameters, Q(x, ˜ x∗) has alocal minimum point in a point with lower objective value than f( ˜ x∗).More in detail,it is possible to prove the following theoretical result:Proposition 2.1 There exists a ¯ τ> 0 such that for all τ ≥¯ τ the filled function hasthe following properties:(i) The point ˜ x∗ is an isolated local maximizer of the filled function Q(x, ˜ x∗).(ii) Q(x, ˜ x∗) has no unconstrained stationary point in {x ∈ Lf (f (x0)) : f(x) ≥f( ˜ x∗)} except ˜ x∗.(iii) If ˜ x∗ is not a global minimum of f(x) and   satisfies the condition0 < <f( ˜ x∗)− f(x∗), (4)where x∗ is a global minimum of f(x), then all the global minimum pointsˇ x of the filled function Q(x, ˜ x∗) over Lf (f (x0)) belong to the region {x ∈Lf (f (x0)) : f(x)<f( ˜ x∗)}.Proof First of all we note that the gradient of Q(x, ˜ x∗) has the following expression:∇Q(x, ˜ x∗) =−2(x −˜ x∗)γ 2exp −x −˜ x∗2γ 2 + τ∇f(x) exp (−τ(f(x) −f( ˜ x∗)+  )). (5)We begin by proving point (i). Since the point ˜ x∗ is a stationary point of problem (2),it satisfies ∇f( ˜ x∗) = 0. Therefore, (5) implies that where λmax(∇2f( ˜ x )) is the maximum eigenvalue of the matrix ∇2f( ˜ x ). The aboveinequality implies that there exists a τ1 > 0 such that, for all τ ≥ τ1, the Hessianmatrix ∇2Q( ˜ x∗, ˜ x∗) is negative definite. Therefore the point ˜ x∗ is an isolated localmaximizer of Q(x, ˜ x∗) for all τ ≥ τ1.As for point (ii), recalling the expression (5) of the gradient of Q(x, ˜ x∗) wenote that if there would exists an unconstrained stationary point ˆ x ∈ Lf (f (x0)) ofQ(x, ˜ x∗) such that ˆ x = ˜ x∗ and f( ˆ x) ≥ f( ˜ x∗),itmustsatisfy2ˆ x −˜ x∗γ 2exp −ˆ x −˜ x∗2γ 2 = τ∇f( ˆ x) exp(−τ(f( ˆ x)−f( ˜ x∗)+  )). (9)Point (i) implies the existence of > 0 such that ˆ x −˜ x∗ >. By Assumption 1the compactness of Lf (f (x0)) follows. Therefore there exist two constants D and Lsuch that ˆ x −˜ x∗≤ D and ∇f(x)≤ L for all x ∈ Lf (f (x0)). This implies thefollowing estimates for the two sides of (9):2γ 2exp −D2γ 2 ≤ 2ˆ x −˜ x∗γ 2exp −ˆ x −˜ x∗2γ 2 (10)andτ∇f( ˆ x) exp(−τ(f( ˆ x)−f( ˜ x∗)+  )) ≤ τLexp(−τ ). (11)Therefore (10) and (11) imply that there exists a τ2 ≥ τ1 such that for all τ ≥ τ2condition (9) does not hold.Finally, we prove point (iii). 船舶设计问题上的新全局优化英文文献和中文翻译(2):http://www.youerw.com/fanyi/lunwen_64778.html
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