# 数据分析代考_introduction to data science代考_Constrained optimization: optimality conditions

## 数据分析代考_introduction to data science代考_Constrained optimization: optimality conditions

minx∈Rnf(x)  such that ci(x)=0i∈E ci(x)≤0i∈IFirst order necessary conditions: Karush-Kuhn-Tucker (KKT) conditions THEOREM Suppose that f:Rn→R and ci,i∈E∪I are all continuously differentiable. If x∗ is a local minimizer of f and some regularity conditions are satisfied, then there exist real scalars λi, i∈E and μi≥0,i∈I such that∇f(x⋆)+∑i∈Eλi∇ci(x⋆)+∑i∈Iμi∇ci(x⋆)=0 ci(x⋆)=0∀i∈E ci(x⋆)≤0∀i∈I μici(x⋆)=0∀i∈I

First order necessary conditions: Karush-Kuhn-Tucker (KKT) conditions
THEOREM Suppose that f:Rn→R and ci,i∈E∪I are all continuously differentiable. If Missing superscript or subscript argumentMissing superscript or subscript argument is a local minimizer of f and some regularity conditions are satisfied, then there exist real scalars λi, i∈E and μi≥0,i∈I such thatMissing \left or extra \rightMissing \left or extra \rightRegularity conditions: Many different forms. Also called constraint qualification. Most common is the Linear Independence Constraint Qualification (LICQ): Let J(x⋆)⊆l index those inequality constraints that are satisfied at equality at x⋆, i.e., Missing \left or extra \rightMissing \left or extra \right for all Missing \left or extra \rightMissing \left or extra \right. Then LICQ demands that Missing \left or extra \rightMissing \left or extra \right are linearly independent.

## 数据分析代考_introduction to data science代考_linear programming

Another regularity condition: the objective f and all constraints ci, i∈E∪I are all affine functions (special case of convex), i.e.,
ci(x)=⟨ai,x⟩+bi,i∈E∪I
for some vectors ai∈Rn and scalars bi∈R, and similarly
f(x)=⟨d,x⟩
for some vector d∈Rn.
This special case is called Linear Programming/Optimization.

Another regularity condition: the objective f and all constraints ci, i∈E∪I are all affine functions (special case of convex), i.e.,
ci(x)=⟨ai,x⟩+bi,i∈E∪I
for some vectors ai∈Rn and scalars bi∈R, and similarly
f(x)=⟨d,x⟩
for some vector d∈Rn.
This special case is called Linear Programming/Optimization.

1. Most well studied optimization problem.
2. Main building block in more sophisticated algorithms.
3. KKT conditions are necessary and sufficient (assuming problem is feasible).
4. Highly efficient, specialized algorithms developed: Simplex method, Interior Point methods, Ellipsoid method.

# 数据分析代考

## 数据分析代考_introduction to data science代考_Constrained optimization: optimality conditions

minx∈Rnf(x) such that ci(x)=0i∈Eci(x)≤0i∈I

∇f(x⋆)+∑i∈Eλi∇ci(x⋆)+∑i∈Iμi∇ci(x⋆)=0ci(x⋆)=0∀i∈Eci(x⋆)

THEOREM 假设 f:Rn→R 和 ci,i∈E∪I 都是连续可微的。如果

㞊少 \1eft 或额外的 ∖ right

(LICQ): 让 J(x⋆)⊆l 索引那些在平等时满足的不平等约束 x⋆ ，那是，

LICQ 要求缺少 〈1eft 或额外的 \right } \text { 是线性独立的。 }

## 数据分析代考_introduction to data science代考_linear programming

ci(x)=⟨ai,x⟩+bi,i∈E∪I

f(x)=⟨d,x⟩

ci(x)=⟨ai,x⟩+bi,i∈E∪I对于一些向量 ai∈Rn 和标量 bi∈R, 同样地
f(x)=⟨d,x⟩

1. 研究最多的优化问题。
2. 更复毠算法的主要构建块。
3. KKT 条件充分必要（假设问题可行）。
4. 开发了高效的专用算法: 单纯形法、内点法、椭球法。

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