*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 unconstrained optimization
problem.
*解とアルゴリズムの基本的性質:Basic Properties of Solutions and Algorithms [#yedf6bd8]

ミニマム点の定義
-Definition:A point x* is said to be a relative minimum point or a local
minimum point of f if there is an ε > 0 such that f(x*) =< f(x) for all
x such that ||x-x*||<ε. If the inequality is strict for all x ><x* then x* is said to be a strict relative minimum point.
-Definition: A point x*  is said to be a global minimum point of f if
f(x*) < f(x) for all x . If the inequality is strict for all x without x*
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 conditions [#fc02fdd1]
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、任意のベクトルdに対して内積f'(x*)・dが正または0である。
-Let f in C1. If x* is a relative minimum, then for any vector d which is feasible at x*, we have f'(x)d >= 0. (f'(x) is the gradient, i.e. the vector of partial derivatives of f at x.)
-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)>=である。

By a one-sided Taylor expansion of the function h(t) = f(x* + td),
which is defined in the interval [0, α], we obtain that h(t) = h(0) + t・h'(0) +o(t). Since 0 is a relative minimum of h we obtain that th'(0) >= 0, for all t small enough, which implies that h'(0) >= 0. The claim now follows from
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 components are the partial derivatives of f. That is:
#ref(http://tokyo.atso-net.jp/wiki/index.php?plugin=ref&page=%E5%BE%AE%E5%88%86%E4%BF%82%E6%95%B0&src=gradiant.png)

Here the gradient is written as a row vector, but it is often taken to be a column vector.

*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*= 2. This is a
global minimum. (Why?)

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