# 数学代写|线性规划作业代写Linear Programming代考|Examples of Unconstrained Problems

## 数学代写|线性规划作业代写Linear Programming代考|Examples of Unconstrained Problems

Unconstrained optimization problems occur in a variety of contexts, but most frequently when the problem formulation is simple. More complex formulations often involve explicit functional constraints. However, many problems with constraints are frequently converted to unconstrained problems, such as using the barrier functions, e.g., the analytic center problem for (dual) linear programs. We present a few more examples here that should begin to indicate the wide scope to which the theory applies.

Example 1 (Logistic Regression) Recall the classification problem where we have vectors $\mathbf{a}i \in E^d$ for $i=1,2, \ldots, n_1$ in a class, and vectors $\mathbf{b}_j \in E^d$ for $j=$ $1,2, \ldots, n_2$ not. Then we wish to find $\mathbf{y} \in E^d$ and a number $\beta$ such that $$\frac{\exp \left(\mathbf{a}_i^T \mathbf{y}+\beta\right)}{1+\exp \left(\mathbf{a}_i^T \mathbf{y}+\beta\right)}$$ is close to 1 for all $i$, and $$\frac{\exp \left(\mathbf{b}_j^T \mathbf{y}+\beta\right)}{1+\exp \left(\mathbf{b}_j^T \mathbf{y}+\beta\right)}$$ is close to 0 for all $j$. The problem can be cast as a unconstrained optimization problem, called the max-likelihood, $$\operatorname{maximize}{\mathbf{y}, \beta}\left(\prod_i \frac{\exp \left(\mathbf{a}_i^T \mathbf{y}+\beta\right)}{1+\exp \left(\mathbf{a}_i^T \mathbf{y}+\beta\right)}\right)\left(\prod_j\left(1-\frac{\exp \left(\mathbf{b}_j^T \mathbf{y}+\beta\right)}{1+\exp \left(\mathbf{b}_j^T \mathbf{y}+\beta\right)}\right)\right),$$ which can be also equivalently, using a logarithmic transformation, written as
$$\operatorname{minimize}_{\mathbf{y},} \beta \sum_i \log \left(1+\exp \left(-\mathbf{a}_i^T \mathbf{y}-\beta\right)\right)+\sum_j \log \left(1+\exp \left(\mathbf{b}_j^T \mathbf{y}+\beta\right)\right) .$$
The optimal solution to logistic regression may be infinite (not attainable), so that one typically adds a weighted regularization term, e.g., $\mu|\mathbf{y}|^2$, to the objective for a fixed parameter $\mu \geq 0$.

Example 2 (Utility Maximization) A common problem in economic theory is the determination of the best way to combine various inputs in order to maximize a utility function $f\left(x_1, x_2, \ldots, x_n\right)$ (in the monetary unit) of the amounts $x_j$ of the inputs, $i=1,2, \ldots, n$. The unit prices of the inputs are $p_1, p_2, \ldots, p_n$. The producer wishing to maximize profit must solve the problem
$$\text { maximize } f\left(x_1, x_2, \ldots, x_n\right)-p_1 x_1-p_2 x_2 \ldots-p_n x_n$$

## 数学代写|线性规划作业代写Linear Programming代考|Second-Order Conditions

The proof of Proposition 1 in Sect. 7.1 is based on making a first-order approximation to the function $f$ in the neighborhood of the relative minimum point. Additional conditions can be obtained by considering higher-order approximations. The second-order conditions, which are defined in terms of the Hessian matrix $\nabla^2 f$ of second partial derivatives of $f$ (see Appendix A), are of extreme theoretical importance and dominate much of the analysis presented in later chapters.

Proof The first condition is just Proposition 1, and the second applies only if $\nabla f\left(\mathbf{x}^\right) \mathbf{d}=0$. In this case, introducing $\mathbf{x}(\alpha)=\mathbf{x}^+\alpha \mathbf{d}$ and $g(\alpha)=f(\mathbf{x}(\alpha))$ as before, we have, in view of $g^{\prime}(0)=0$,
$$g(\alpha)-g(0)=\frac{1}{2} g^{\prime \prime}(0) \alpha^2+o\left(\alpha^2\right)$$
If $g^{\prime \prime}(0)<0$ the right side of the above equation is negative for sufficiently small $\alpha$ which contradicts the relative minimum nature of $g(0)$. Thus
$$g^{\prime \prime}(0)=\mathbf{d}^T \nabla^2 f\left(\mathbf{x}^\right) \mathbf{d} \geqslant 0$$ Example 1 For the same problem as Example 2 of Sect. 7.1, we have for $\mathbf{d}=$ $\left(d_1, d_2\right)$ $$\nabla f\left(\mathbf{x}^\right) \mathbf{d}=\frac{3}{2} d_2$$
Thus condition (ii) of Proposition 1 applies only if $d_2=0$. In that case we have $\mathbf{d}^T \nabla^2 f\left(\mathbf{x}^*\right) \mathbf{d}=2 d_1^2 \geqslant 0$, so condition (ii) is satisfied.

Again of special interest is the case where the minimizing point is an interior point of $\Omega$, as, for example, in the case of completely unconstrained problems. We then obtain the following classical result.

# 线性规划代考

## 数学代写|线性规划作业代写Linear Programming代考|Examples of Unconstrained Problems

$$\text { maximize } f\left(x_1, x_2, \ldots, x_n\right)-p_1 x_1-p_2 x_2 \ldots-p_n x_n$$

## 数学代写|线性规划作业代写Linear Programming代考|Second-Order Conditions

$$g(\alpha)-g(0)=\frac{1}{2} g^{\prime \prime}(0) \alpha^2+o\left(\alpha^2\right)$$

$g^{\wedge}{\backslash p r i m e \backslash p r i m e}(0)=\backslash$ mathbf ${d}^{\wedge} T \backslash$ nabla^ 2 f $\backslash$ left $\backslash$ mathbf ${x}^{\wedge} \backslash$ right $) \backslash$ mathbf ${d} \backslash g e q s l a n t$

$$\left.\backslash \text { nabla f } \backslash \text { left } \backslash \text { mathbf }{x}^{\wedge} \backslash \text { right }\right) \backslash \text { mathbf }{d}=\backslash \text { frac }{3}{2} \text { d_2 }$$

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