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• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

We conclude this section by giving an economic interpretation of the relation between the optimal basis and the vector $\mathbf{y}^{T}=\mathbf{c}{\mathbf{B}}^{T} \mathbf{B}^{-1}$. This vector is not a feasible solution to the dual unless $\mathbf{B}$ is an optimal basis for the primal, but nevertheless, it has an economic interpretation. Furthermore, as we have seen in the development of the simplex method in the next chapter, this $\mathbf{y}$ vector can be used at every step to calculate the relative cost coefficients or reduced gradients. For this reason $\mathbf{y}^{T}=\mathbf{c}{\mathbf{B}}^{T} \mathbf{B}^{-1}$, corresponding to any basis, is often called the vector of simplex multipliers or shadow prices.

Let us pursue the economic interpretation of these simplex multipliers. As usual, denote the columns of $\mathbf{A}$ by $\mathbf{a}{1}, \mathbf{a}{2}, \ldots, \mathbf{a}{n}$ and denote by $\mathbf{e}{1}, \mathbf{e}{2}, \ldots, \mathbf{e}{m}$ the $m$ unit vectors in $E^{m}$. The components of the $\mathbf{a}{j}$ ‘s and $\mathbf{b}$ tell how to construct these vectors from the $\mathbf{e}{i}$ ‘s.

Given any basis B, however, consisting of $m$ columns of $\mathbf{A}$, any other vector can be constructed (synthetically) as a linear combination of these basis vectors. If there is a unit cost $c_{j}$ associated with each basis vector $\mathbf{a}{j}$, then the cost of a (synthetic) vector constructed from the basis can be calculated as the corresponding linear combination of the $c{j}$ ‘s associated with the basis. In particular, the cost of the $i$ th unit vector, $\mathbf{e}{i}$, when constructed from the basis $\mathbf{B}$, is $y{i}$, the $i$ th component of $\mathbf{y}^{T}=\mathbf{c}{\mathbf{B}}^{T} \mathbf{B}^{-1}$. Thus the $y{i}$ ‘s can be interpreted as synthetic prices of the unit vectors.
Now, any vector can be expressed in terms of the basis $\mathbf{B}$ in two steps: (1) express the unit vectors in terms of the basis, and then (2) express the desired vector as a linear combination of unit vectors. The corresponding synthetic cost of a vector constructed from the basis $\mathbf{B}$ can correspondingly be computed directly by: (1) finding the synthetic price of the unit vectors, and then (2) using these prices to evaluate the cost of the linear combination of unit vectors. Thus, the simplex multipliers can be used to quickly evaluate the synthetic cost of any vector that is expressed in terms of the unit vectors. The difference between the true cost of this vector and the synthetic cost is the relative cost. The process of calculating the synthetic cost of a vector, with respect to a given basis, by using the simplex multipliers is sometimes referred to as pricing out the vector.

Optimality of the primal corresponds to the situation where every vector $\mathbf{a}{1}, \mathbf{a}{2}$, $\ldots, \mathbf{a}{n}$ is cheaper when constructed from the basis than when purchased directly at its own price. Thus we have $\mathbf{y}^{T} \mathbf{a}{j} \leqslant c_{j}$ for $j=1,2, \ldots, n$ or equivalently $\mathbf{y}^{T} \mathbf{A} \leqslant \mathbf{c}^{T}$.

## 数学代写|线性规划作业代写Linear Programming代考|Complementary Slackness

The optimal solutions to primal and dual programs satisfy an additional relation that has an economic interpretation. This relation can be stated for any pair of dual linear programs, but we state it here only for the asymmetric and the symmetric pairs defined in Sect. 3.1.
Theorem (Complementary slackness-asymmetric form) Let $\mathbf{x}$ and $\mathbf{y}$ be feasible solutions for the primal and dual programs, respectively, in the pair (3.2). A necessary and sufficient condition that they both be optimal solutions is that ${ }^{\dagger}$ for all $j$
i) $x_{j}>0 \Rightarrow \mathbf{y}^{T} \mathbf{a}{j}=c{j}$
ii) $x_{j}=0 \Leftarrow \mathbf{y}^{T} \mathbf{a}{j}{j}$.
Proof If the stated conditions hold, then clearly $\left(\mathbf{y}^{T} \mathbf{A}-\mathbf{c}^{T}\right) \mathbf{x}=0$. Thus $\mathbf{y}^{T} \mathbf{b}=$ $\mathbf{c}^{T} \mathbf{x}$, and by the corollary to Lemma 1, Sect. 3.2, the two solutions are optimal. Conversely, if the two solutions are optimal, it must hold, by the Duality Theorem, that $\mathbf{y}^{T} \mathbf{b}=\mathbf{c}^{T} \mathbf{x}$ and hence that $\left(\mathbf{y}^{T} \mathbf{A}-\mathbf{c}^{T}\right) \mathbf{x}=0$. Since each component of $\mathbf{x}$ is nonnegative and each component of $\mathbf{y}^{T} \mathbf{A}-\mathbf{c}^{T}$ is nonpositive, the conditions (i) and (ii) must hold.

We present a stronger version of complementary slackness theorem-strict complementary slackness condition and leave its proof as an exercise.
Theorem (Strict complementary slackness-asymmetric form) Let both the primal and dual problems of (3.2) be feasible. Then there is an optimal solution pair $\mathbf{x}$ and $\mathbf{y}$ such that for all $j$
i) $x_{j}>0 \Leftrightarrow \mathbf{y}^{T} \mathbf{a}{j}=c{j}$
ii) $x_{j}=0 \Leftrightarrow \mathbf{y}^{T} \mathbf{a}{j}{j}$.
Nōtê that āt a strict complemenentåry solutioñ pairs, for all $j, x_{j}=0$ âlso impliês $\mathbf{y}^{T} \mathbf{a}{j}{j}$ and $\mathbf{y}^{T} \mathbf{a}{j}=c{j}$ also implies $x_{j}>0$ (not just “is implied by”).

# 线性规划代写

## 数学代写|线性规划作业代写Linear Programming代考|Complementary Slackness

iii) $x_{j}=0 \Leftrightarrow \mathbf{y}^{T} \mathbf{a} j j$.

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## MATLAB代写

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