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# 数学代写|凸优化代写Convex Optimization代考|Linear-Conic Problems

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## 数学代写|凸优化代写Convex Optimization代考|Linear-Conic Problems

An important special case of conic programming, called linear-conic problem, arises when $\operatorname{dom}(f)$ is an affine set and $f$ is linear over $\operatorname{dom}(f)$, i.e.,
$$f(x)= \begin{cases}c^{\prime} x & \text { if } x \in b+S, \ \infty & \text { if } x \notin b+S\end{cases}$$
where $b$ and $c$ are given vectors, and $S$ is a subspace. Then the primal problem can be written as
\begin{aligned} & \text { minimize } c^{\prime} x \ & \text { subject to } x-b \in S, \quad x \in C \end{aligned}
see Fig. 1.2.1.
To derive the dual problem, we note that
\begin{aligned} f^{\star}(\lambda) & =\sup {x-b \in S}(\lambda-c)^{\prime} x \ & =\sup {y \in S}(\lambda-c)^{\prime}(y+b) \ & = \begin{cases}(\lambda-c)^{\prime} b & \text { if } \lambda-c \in S^{\perp}, \ \infty & \text { if } \lambda-c \notin S^{\perp} .\end{cases} \end{aligned}
It can be seen that the dual problem $\min _{\lambda \in \dot{C}} f^*(\lambda)$ [cf. Eq. (1.18)], after discarding the superfluous term $c^{\prime} b$ from the cost, can be written as
\begin{aligned} & \operatorname{minimize} b^{\prime} \lambda \ & \text { subject to } \lambda-c \in S^{\perp}, \quad \lambda \in \hat{C}, \end{aligned}
where $\hat{C}$ is the dual cone:
$$\hat{C}=\left{\lambda \mid \lambda^{\prime} x \geq 0, \forall x \in C\right}$$
By specializing the conditions of the Conic Duality Theorem (Prop. 1.2.2) to the linear-conic duality context, we obtain the following.

## 数学代写|凸优化代写Convex Optimization代考|Special Forms of Linear-Conic Problems

The primal and dual linear-conic problems (1.19) and (1.20) have been placed in an elegant symmetric form. There are also other useful formats that parallel and generalize similar formats in linear programming. For example, we have the following dual problem pairs:
\begin{aligned} & \min {A x=b, x \in C^{\prime} x} c^{\prime} x \Longleftrightarrow \max {c-A^{\prime} \lambda \in C} b^{\prime} \lambda, \ & \min {A x-b \in C} c^{\prime} x \Longleftrightarrow \max {A^{\prime} \lambda=c, \lambda \in C} b^{\prime} \lambda, \end{aligned}
where $A$ is an $m \times n$ matrix, and $x \in \Re^n, \lambda \in \Re^m, c \in \Re^n, b \in \Re^m$.
To verify the duality relation (1.21), let $\bar{x}$ be any vector such that $A \bar{x}=b$, and let us write the primal problem on the left in the primal conic form (1.19) as
\begin{aligned} & \text { minimize } c^{\prime} x \ & \text { subject to } x-\bar{x} \in \mathrm{N}(A), \quad x \in C, \end{aligned}
where $\mathrm{N}(A)$ is the nullspace of $A$. The corresponding dual conic problem (1.20) is to solve for $\mu$ the problem
\begin{aligned} & \text { minimize } \bar{x}^{\prime} \mu \ & \text { subject to } \mu-c \in \mathrm{N}(A)^{\perp}, \quad \mu \in C \text {. } \end{aligned}
Since $\mathrm{N}(A)^{\perp}$ is equal to $\mathrm{Ra}\left(A^{\prime}\right)$, the range of $A^{\prime}$, the constraints of problem (1.23) can be equivalently written as $c-\mu \epsilon-\operatorname{Ra}\left(A^{\prime}\right)=\operatorname{Ra}\left(A^{\prime}\right), \mu \in \hat{C}$, or
$$c-\mu=A^{\prime} \lambda_{,} \quad \mu \in \tilde{C},$$
for some $\lambda \in \Re^m$. Making the change of variables $\mu=c-A^{\prime} \lambda$, the dual problem (1.23) can be written as
$$\begin{array}{ll} \operatorname{minimize} & \bar{x}^{\prime}\left(c-A^{\prime} \lambda\right) \ \text { subject to } & c-A^{\prime} \lambda \in \hat{C} . \end{array}$$
By discarding the constant $\bar{x}^{\prime} c$ from the cost function, using the fact $A \bar{x}=$ $b$, and changing from minimization to maximization, we see that this dual problem is equivalent to the one in the right-hand side of the duality pair (1.21). The duality relation (1.22) is proved similarly.

We next discuss two important special cases of conic programming: second order cone programming and semidefinite programming. These problems involve two different special cones, and an explicit definition of the affine set constraint. They arise in a variety of applications, and their computational difficulty in practice tends to lie between that of linear and quadratic programming on one hand, and general convex programming on the other hand.

## 数学代写|凸优化代写Convex Optimization代考|Linear-Conic Problems

$$f(x)= \begin{cases}c^{\prime} x & \text { if } x \in b+S, \ \infty & \text { if } x \notin b+S\end{cases}$$

\begin{aligned} & \text { minimize } c^{\prime} x \ & \text { subject to } x-b \in S, \quad x \in C \end{aligned}

\begin{aligned} f^{\star}(\lambda) & =\sup {x-b \in S}(\lambda-c)^{\prime} x \ & =\sup {y \in S}(\lambda-c)^{\prime}(y+b) \ & = \begin{cases}(\lambda-c)^{\prime} b & \text { if } \lambda-c \in S^{\perp}, \ \infty & \text { if } \lambda-c \notin S^{\perp} .\end{cases} \end{aligned}

\begin{aligned} & \operatorname{minimize} b^{\prime} \lambda \ & \text { subject to } \lambda-c \in S^{\perp}, \quad \lambda \in \hat{C}, \end{aligned}

$$\hat{C}=\left{\lambda \mid \lambda^{\prime} x \geq 0, \forall x \in C\right}$$

## 数学代写|凸优化代写Convex Optimization代考|Special Forms of Linear-Conic Problems

\begin{aligned} & \min {A x=b, x \in C^{\prime} x} c^{\prime} x \Longleftrightarrow \max {c-A^{\prime} \lambda \in C} b^{\prime} \lambda, \ & \min {A x-b \in C} c^{\prime} x \Longleftrightarrow \max {A^{\prime} \lambda=c, \lambda \in C} b^{\prime} \lambda, \end{aligned}

\begin{aligned} & \text { minimize } c^{\prime} x \ & \text { subject to } x-\bar{x} \in \mathrm{N}(A), \quad x \in C, \end{aligned}

\begin{aligned} & \text { minimize } \bar{x}^{\prime} \mu \ & \text { subject to } \mu-c \in \mathrm{N}(A)^{\perp}, \quad \mu \in C \text {. } \end{aligned}

$$c-\mu=A^{\prime} \lambda_{,} \quad \mu \in \tilde{C},$$

$$\begin{array}{ll} \operatorname{minimize} & \bar{x}^{\prime}\left(c-A^{\prime} \lambda\right) \ \text { subject to } & c-A^{\prime} \lambda \in \hat{C} . \end{array}$$

## MATLAB代写

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