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# 数学代写|凸优化代写Convex Optimization代考|Constrained minimization

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## 数学代写|凸优化代写Convex Optimization代考|Constrained minimization

Let us demonstrate how we can use the models for solving constrained minimization problems. Consider the problem
$$\begin{array}{ll} & \min f(x), \ \text { s.t. } & f_j(x) \leq 0, j=1 \ldots m, \ & x \in Q, \end{array}$$
where $Q$ is a bounded closed convex set, and functions $f(x), f_j(x)$ are Lipschitz continuous on $Q$.

Let us rewrite this problem as a problem with a single functional constraint. Denote $\bar{f}(x)=\max {1 \leq j \leq m} f_j(x)$. Then we obtain the equivalent problem $$\begin{array}{ll} & \min f(x), \ \text { s.t. } & \bar{f}(x) \leq 0, \ & x \in Q . \end{array}$$ Note that $f(x)$ and $\bar{f}(x)$ are convex and Lipschitz continuous. In this section we will try to solve (3.3.5) using the models for both of them. Let us define the corresponding models. Consider a sequence $X=$ $\left{x_k\right}{k=0}^{\infty}$. Denote
$$\begin{gathered} \hat{f}k(X ; x)=\max {0 \leq j \leq k}\left[f\left(x_j\right)+\left\langle g\left(x_j\right), x-x_j\right\rangle\right] \leq f(x), \ \check{f}k(X ; x)=\max {0 \leq j \leq k}\left[\bar{f}\left(x_j\right)+\left\langle\bar{g}\left(x_j\right), x-x_j\right\rangle\right] \leq \bar{f}(x), \end{gathered}$$
where $g\left(x_j\right) \in \partial f\left(x_j\right)$ and $\bar{g}\left(x_j\right) \in \partial \bar{f}\left(x_j\right)$.

## 数学代写|凸优化代写Convex Optimization代考|Black box concept in convex optimization

In this chapter we are going to present the main ideas underlying the modern polynomial-time interior-point methods in nonlinear optimization. In order to start, let us look first at the traditional formulation of a minimization problem.

Suppose we want to solve a minimization problem in the following form:
$$\min _{x \in R^n}\left{f_0(x) \mid f_j(x) \leq 0, j=1 \ldots m\right} .$$
We assume that the functional components of this problem are convex. Note that all standard convex optimization schemes for solving this problem are based on the black-box concept. This means that we assume our problem to be equipped with an oracle, which provides us with some information on the functional components of the problem at some test point $x$. This oracle is local: If we change the shape of a component far enough from the test point, the answer of the oracle does not change. These answers comprise the only information available for numerical methods. ${ }^1$

However, if we look carefully at the above situation, we can see a certain contradiction. Indeed, in order to apply the convex optimization methods, we need to be sure that our functional components are convex. However, we can check convexity only by analyzing the structure of these functions ${ }^2$ : If our function is obtained from the basic convex functions by convex operations (summation, maximum, etc.), we conclude that it is convex.

Thus, the functional components of the problem are not in a black box at the moment we check their convexity and choose a minimization scheme. But we put them in a black box for numerical methods. That is the main conceptual contradiction of the standard convex optimization theory. $^3$

## 数学代写|凸优化代写Convex Optimization代考|Constrained minimization

$$\begin{array}{ll} & \min f(x), \ \text { s.t. } & f_j(x) \leq 0, j=1 \ldots m, \ & x \in Q, \end{array}$$

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