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这是夏威夷大学最优化的代写成功案例。
MATH414课程简介
Introduction to theory and methods for optimization. Topics may include least square analysis, search methods, conjugate direction methods, linear programming, integer programming, and constrained optimization. Pre: 243 or 253A, and 307 or 311; or consent.
Mathematical optimization (alternatively spelled optimisation) or mathematical programming is the selection of a best element, with regard to some criterion, from some set of available alternatives.[1] It is generally divided into two subfields: discrete optimization and continuous optimization. Optimization problems arise in all quantitative disciplines from computer science and engineering[2] to operations research and economics, and the development of solution methods has been of interest in mathematics for centuries
Prerequisites
In the more general approach, an optimization problem consists of maximizing or minimizing a real function by systematically choosing input values from within an allowed set and computing the value of the function. The generalization of optimization theory and techniques to other formulations constitutes a large area of applied mathematics. More generally, optimization includes finding “best available” values of some objective function given a defined domain (or input), including a variety of different types of objective functions and different types of domains.
MATH414 Optimization Theory HELP（EXAM HELP， ONLINE TUTOR）
Consider the linear problem
$$
\begin{aligned}
\operatorname{minimize} \quad 3 x_1-x_2-x_3 & \
\text { subject to } \quad x_1+x_2+x_3 & \leq 4 \
x_1 & \leq 2 \
x_3 & \leq 3 \
3 x_2+x_3 & \leq 6 \
x_1, x_2, x_3 & \geq 0 .
\end{aligned}
$$
In standard form $A x=b, x \geq 0$, the feasible set can be written as follows
$$
\left(\begin{array}{cccccccc}
A_1 & A_2 & A_3 & A_4 & A_5 & A_6 & A_7 & A_8 \
1 & 1 & 1 & 1 & 0 & 0 & 0 & 0 \
1 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \
0 & 0 & 1 & 0 & 0 & 1 & 0 & 0 \
0 & 3 & 1 & 0 & 0 & 0 & 1 & 0 \
3 & -1 & -1 & 0 & 0 & 0 & 0 & 1
\end{array}\right)\left(\begin{array}{l}
x_1 \
x_2 \
x_3 \
x_4 \
x_5 \
x_6 \
x_7 \
x_8
\end{array}\right)=\left(\begin{array}{l}
4 \
2 \
3 \
6 \
0
\end{array}\right), x \geq 0 .
$$
(i) Is the index vector $J_1={1,2,3,5,8}$ a basis? Identify the associated vertex of the polyhedron representing the feasible set.
(ii) Is the index vector $J_1={1,2,3,5,8}$ a feasible basis? If so, is it nondegenerate?
(iii) Is the index vector $J_2={4,2,3,5,8}$ a basis? Identify the associated vertex of the polyhedron representing the feasible set.
(iv) Is the index vector $J_2={4,2,3,5,8}$ a feasible basis? If so, is it nondegenerate?
(v) Consider the bases $J_1=(1,2,3,5,8)$ and $J_2=(4,2,3,5,8)$. Are the optimality conditions satisfied?
(vi) Use the simplex method to compute the solution of the LP starting from the feasible basis $J_1$.
Consider the linear program
$$
\begin{array}{r}
\text { minimize } \quad c^T x \
\text { subject to } A x=b, x \geq 0
\end{array}
$$
where $c, x \in \mathbb{R}^n, b \in \mathbb{R}^m$, and $A \in \mathbb{R}^{m \times n}$. Assume that $d \in \mathbb{R}^n$ is a direction for which
$$
c^T d<0, A d=0, d_i \geq 0 \text { for each } i \text { with } x_i=0 $$ Show that if $x$ is a feasible vector for the LP, then $x+\alpha d$ is also feasible and $$ c^T(x+\alpha d)0$. Deduce that if such a vector $d$ exists, the objective functional $c^T x$ is unbounded from below on its feasible region and hence, the LP has no finite solution.
Consider the following linear program in nonstandard form
$$
\begin{array}{r}
\text { minimize } c^T w+d^T z \quad \text { over }(w, z) \
\text { subject to } A_{11} w+A_{12} z \geq b_1, A_{21} w+A_{22} z=b_2, w \geq 0
\end{array}
$$
(i) By splitting and introducing slack variables, convert this problem to standard form.
(ii) Write down the dual problem for the standard form version, and express it in a cleaner form by eliminating the slack variables.
Assume that $(x, \lambda, s)$ satisfies the KKT conditions from Theorem 1.2.
(i) Show that any primal feasible vector $\bar{x}$ has $c^T \bar{x} \geq c^T x$.
(ii) Show that any dual feasible vector $(\bar{\lambda}, \bar{s})$ has $b^T \bar{\lambda} \leq b^T \lambda$.
(iii) Show that $c^T x=b^T \lambda$, i.e., the optimal objective values of the primal and dual problem are the same.
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