本文主要目的為針對傳統 CMA-ES 的突變機制做改良，由於傳統的突變機制 上，主要是以自適應機制來決定其突變的行為，雖然能夠在演化過程中適度地調 整策略參數以決定突變方向，但常態分佈的突變行為仍然限制了個體所能夠突變 的程度。在本論文第三章提出了一個分群突變的機制，其主要精神為額取混合機 率模型具有極大分佈彈性的概念，來經由更有彈性的突變進行對於各種方向上的 搜尋，進而降低區域搜尋的限制。由視覺化的實驗觀察，本論文分群突變的概念 將有效地使個體在突變的過程中，達到更全面的搜尋能力，因此能有效找到最佳 解位置。 本論文的未來工作主要是解決收斂速度過慢的問題，這問題在多漏 斗型函數實驗上面更顯得嚴重，以下列出了幾點可行方法：
z 從理論上著手，[39]在最近得出了部分 CMA-ES 更新公式是一種「期望適應 值上的自然梯度下降法(natural gradient decent on expectation of fitness)」的理 論證明。倘若從這種下降法推廣到常態混和機率分佈，或許可以從裡面看出 一些 CMA-ES 混合模型版本的重要的端倪。這種完全由常態混合機率模型推 倒獲得的理論結果，可能會隱藏著更加自然的組件數量更新機制，其重要性 將 會 大 於 每 一 代 人 為 刻 意 使 用 某 種 分 群 法 再 硬 套 入 混 合 機 率 模 型 的 MS-CMA-ES。
z 更改混和機率參數更新機制，在簡單的球形測試函式上，MS-CMA-ES 收斂 速度慢的原因除了是取樣數不同也有可能是不正常分群造成的，不必要的分 群是否造成收斂速度什麼影響是另一個未來研究方向。在複雜的多漏斗多峰 函式上，收斂速度慢的原因本論文認為是由於混和機率參數更新機制不良所 造成的，或許使用其他指標例如適應值進步程度會更好。另外混和機率更新 參數也需要重新設計，不應該是個跟取樣數量、維度、和組件數無關的固定 常數。
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附錄一 測試函式定義
D: dimensions,o = [o ,o ,...,o ]: the shifted global optimim.
F2: Elliptic Function
( )
i-1 M: orthogonal matrix.F3: Rosenbrock’s Function
( ) ( )
D: dimensions,o = [o ,o ,...,o ]: the shifted global optimim.
F4:Rastrigin’s Function
( )
D 2
4 i
i=1
F (x) =
∑
z −10cos(2πzi) 10 + f_bias ,+1 2 D
1 1 D
z = x - o, x = [x ,x ,...,x ], D: dimensions,
o = [o ,o ,...,o ]: the shifted global optimim. D: dimensions,
o = [o ,o ,...,o ]: the shifted global optimim,
M': linear transformation matrix, condition number = 3, M = M'(1 + 0.3 N(0,1) ).
F6: Schwefel’s Function
D 2
A, B are two D*D matrix, a ,b are integer random number in the range [-100,100],
= [ , ,..., ], α α α α
∑ ∑
j are random numbers in the range [- , ].
α π π
F7: Double-Rastrigin’s Function
( )
1 2 D
1 1 D
z = x - o, x = [x ,x ,...,x ], D: dimensions,
o = [o ,o ,...,o ]: the shifted global optimim.
F8: Weierstrass’s Function
65
( )
8D max max
k k
8 i
i=1 0 0
F (x) = k kcos 2 b (z +0.5) -Dk kcos(2 b 0.5) +f_bias ,
k k
a π a π
= =
⎡ ⎤ ⎡ ⋅ ⎤
⎢ ⎥ ⎣ ⎦
⎣ ⎦
∑ ∑ ∑
a=0.5, b=3, kmax=20, z=(x-o)*M, x = [x ,x ,...,x ],1 2 D
1 1 D
D: dimensions,
o = [o ,o ,...,o ]: the shifted global optimim,
M: linear transformation matrix, condition number = 5.