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開始行:
*最適化問題 [#k0fe967b]
最適化問題は、一般に次のように定式化される。
min f(x) xはベクトル
gi(x) = 0 i = 1, . . . ,me
gi(x) < 0 i = me + 1, . . . ,m
xl < x < xu
-制約式のない場合は、制約のない最適化問題とよぶ
The general optimization problem has the form:
min f(x)
subject to:
gi(x) = 0 i = 1, . . . ,me
gi(x) < 0 i = me + 1, . . . ,m
xl < x < xu
In particular, if m = 0, the problem is called an unconst...
problem.
*解とアルゴリズムの基本的性質:Basic Properties of Soluti...
ミニマム点の定義
-Definition:A point x* is said to be a relative minimum p...
minimum point of f if there is an ε > 0 such that f(x*) =...
x such that ||x-x*||<ε. If the inequality is strict for ...
-Definition: A point x* is said to be a global minimum p...
f(x*) < f(x) for all x . If the inequality is strict for ...
then x* is said to be a strict global minimum point.
*定理1:1次元の最少値の必要条件 First-order necessary c...
fがスカラー関数の場合、もしx*がローカルに最少であるならば...
-Let f in C1. If x* is a relative minimum, then for any v...
-f'(x)は勾配ベクトル
[証明]
h(t) = f(x* + td)をテーラー展開すると
h(t) = h(0) + t・h'(0) +o(t)
となる。t=0がミニマム値であるから任意のtに対してt・h'(0)...
By a one-sided Taylor expansion of the function h(t) = f(...
which is defined in the interval [0, α], we obtain that h...
the chain rule.
*補題 [#g84334a9]
もしx*が最少の極値を与えるならば、f'(x*) = 0が成り立つ。
If x* is a local minimum then f'(x*) = 0.
*勾配:ベクトル [#g9a3fa2e]
The gradient of f is defined to be the vector field whose...
#ref(http://tokyo.atso-net.jp/wiki/index.php?plugin=ref&p...
Here the gradient is written as a row vector, but it is o...
*例題 [#u270e7c9]
Example: Consider the function f(x, y) = x2 −xy +y2 −3y.
From the first order conditions we get that x* = 1 and y*...
global minimum. (Why?)
終了行:
*最適化問題 [#k0fe967b]
最適化問題は、一般に次のように定式化される。
min f(x) xはベクトル
gi(x) = 0 i = 1, . . . ,me
gi(x) < 0 i = me + 1, . . . ,m
xl < x < xu
-制約式のない場合は、制約のない最適化問題とよぶ
The general optimization problem has the form:
min f(x)
subject to:
gi(x) = 0 i = 1, . . . ,me
gi(x) < 0 i = me + 1, . . . ,m
xl < x < xu
In particular, if m = 0, the problem is called an unconst...
problem.
*解とアルゴリズムの基本的性質:Basic Properties of Soluti...
ミニマム点の定義
-Definition:A point x* is said to be a relative minimum p...
minimum point of f if there is an ε > 0 such that f(x*) =...
x such that ||x-x*||<ε. If the inequality is strict for ...
-Definition: A point x* is said to be a global minimum p...
f(x*) < f(x) for all x . If the inequality is strict for ...
then x* is said to be a strict global minimum point.
*定理1:1次元の最少値の必要条件 First-order necessary c...
fがスカラー関数の場合、もしx*がローカルに最少であるならば...
-Let f in C1. If x* is a relative minimum, then for any v...
-f'(x)は勾配ベクトル
[証明]
h(t) = f(x* + td)をテーラー展開すると
h(t) = h(0) + t・h'(0) +o(t)
となる。t=0がミニマム値であるから任意のtに対してt・h'(0)...
By a one-sided Taylor expansion of the function h(t) = f(...
which is defined in the interval [0, α], we obtain that h...
the chain rule.
*補題 [#g84334a9]
もしx*が最少の極値を与えるならば、f'(x*) = 0が成り立つ。
If x* is a local minimum then f'(x*) = 0.
*勾配:ベクトル [#g9a3fa2e]
The gradient of f is defined to be the vector field whose...
#ref(http://tokyo.atso-net.jp/wiki/index.php?plugin=ref&p...
Here the gradient is written as a row vector, but it is o...
*例題 [#u270e7c9]
Example: Consider the function f(x, y) = x2 −xy +y2 −3y.
From the first order conditions we get that x* = 1 and y*...
global minimum. (Why?)
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