MY-ASSIGNMENTEXPERT™可以为您提供manoa.hawaii MATH414 Optimization Theory最优化的代写代考和辅导服务！
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）
(Mehrotra’s Predictor-Corrector Algorithm)
Given a point $(x, \lambda, s) \in \mathcal{F}{P D}^o$, show that the $\Delta x$-component of the primal-dual affine-scaling step in Mehrotra’s predictor-corrector algorithm, obtained by setting $\sigma=0$, satisfies $$ \Delta x=-D P{\operatorname{Ker}(A D)}(D c), D:=S^{-1 / 2} X^{1 / 2},
$$
where $P_{\operatorname{Ker}(A D)}$ denotes the projection onto $\operatorname{Ker}(A D)$.
4 Points
(Tangent and normal cone to the feasible set)
Let the feasible set $\mathcal{F}$ be given by
$$
\mathcal{F}=\left{x \in \mathbb{R}^n \mid h(x)=0\right}
$$
where $h: \mathbb{R}^n \rightarrow \mathbb{R}$ is smooth.
Prove that for $x \in \mathcal{F}$ with $\nabla h(x) \neq 0$ the tangent and the normal cone to $\mathcal{F}$ at $x$ are given by
$$
T_{\mathcal{F}}(x)=\operatorname{Ker}\left(\nabla h(x)^T\right) \quad, \quad N_{\mathcal{F}}(x)=\operatorname{Im}\left(\nabla h(x)^T\right),
$$
where Ker and Im refer to the kernel and range, respectively.
4 points
(Limiting directions and the tangent cone)
Given $x^* \in \mathcal{F}:=\left{x \in \mathbb{R}^n \mid h_i(x)=0,1 \leq i \leq m, g_i(x) \leq 0,1 \leq i \leq p\right}$, the set
$$
F_1:=\left{\alpha d \mid \alpha \geq 0, \begin{array}{l}
\nabla h_i\left(x^\right)^T d=0 \ \nabla g_i\left(x^\right)^T d \leq 0
\end{array}, \quad \begin{array}{c}
1 \leq i \leq m, \
i \in \mathcal{I}_{a c}\left(x^\right) \end{array}\right} $$ is a cone. Prove: If LICQ is satisfied, $F_1$ is the tangent cone to the feasible set $\mathcal{F}$ at $x^$.
4 points
Quadratic Programming Problem
Consider the QP
$$
\begin{aligned}
& \operatorname{minimize} \quad 3 x_1^2+2 x_1 x_2+x_1 x_3+2.5 x_2^2+2 x_2 x_3+2 x_3^2-8 x_1-3 x_2-3 x_3 \
& \text { over } \quad x=\left(x_1, x_2\right)^T \in \mathbb{R}^2 \
& \text { subject to } \quad x_1+x_3=3 \quad, \quad x_2+x_3=0 .
\end{aligned}
$$
Identify the matrices $B \in \mathbb{R}^{3 \times 3}, A \in \mathbb{R}^{2 \times 3}$ and the vectors $b \in \mathbb{R}^3, c \in \mathbb{R}^2$ of the KKT system and compute its solution $x^, \lambda^$. Moreover, determine the null-space basis matrix $Z$.
4 points
MY-ASSIGNMENTEXPERT™可以为您提供MANOA.HAWAII MATH414 OPTIMIZATION THEORY最优化的代写代考和辅导服务！
