MY-ASSIGNMENTEXPERT™可以为您提供sydney MATH3971 Convex Optimization凸优化课程的代写代考和辅导服务！
这是悉尼大学凸优化课程的代写成功案例。
MATH3971课程简介
The questions how to maximise your gain (or to minimise the cost) and how to determine the optimal strategy/policy are fundamental for an engineer, an economist, a doctor designing a cancer therapy, or a government planning some social policies. Many problems in mechanics, physics, neuroscience and biology can be formulated as optimistion problems. Therefore, optimisation theory is an indispensable tool for an applied mathematician. Optimisation theory has many diverse applications and requires a wide range of tools but there are only a few ideas underpinning all this diversity of methods and applications. This course will focus on two of them. We will learn how the concept of convexity and the concept of dynamic programming provide a unified approach to a large number of seemingly unrelated problems. By completing this unit you will learn how to formulate optimisation problems that arise in science, economics and engineering and to use the concepts of convexity and the dynamic programming principle to solve straight forward examples of such problems. You will also learn about important classes of optimisation problems arising in finance, economics, engineering and insurance.
Prerequisites
At the completion of this unit, you should be able to:
- LO1. analyse static optimisation problems with constraints
- LO2. formulate deterministic and stochastic dynamic optimisation problems, that arise in scientific and engineering applications, as mathematical problems
- LO3. understand the importance of convexity for optimisation problems, and use convexity to determine whether a solution to a given problem exists and is unique
- LO4. check if a certain controlled system is controllable, observable, stabilisable
- LO5. apply the Maximum Principle in order to solve real world control problems
- LO6. formulate the Hamilton-Jacobi-Bellman equation for solution of dynamic optimisation problems and solve them in special cases
- LO7. explain the derivations of key theoretical results and discuss the role of mathematical assumptions in these derivations
- LO8. use game theory to formulate optimisation problems with many competing players
- LO9. identify important solvable classes of optimisation problems arising in finance, economics, engineering and insurance and provide solutions.
MATH3971 Convex Optimization HELP（EXAM HELP， ONLINE TUTOR）
Exercise 1 Which of the following sets are convex?
1) A rectangle, i.e., a set of the form $\left{x \in \mathbb{R}^n \mid \alpha_i \leq x_i \leq \beta_i, i=1, \ldots, n\right}$.
2) The hyperbolic set
$$
\left{x \in \mathbb{R}_{+}^2 \mid x_1 x_2 \geq 1\right}
$$
3) The set of points closer to a given point than a given set, i.e.,
$$
\left{x \mid\left|x-x_0\right|_2 \leq|x-y|_2 \text { for all } y \in S\right}
$$
where $S \subseteq \mathbb{R}^n$.
4) The set of points closer to one set than another, i.e.,
$$
{x \mid \operatorname{dist}(x, S) \leq \operatorname{dist}(x, T)}
$$
where $S, T \subseteq \mathbb{R}^n$, and $\operatorname{dist}(x, S)=\inf \left{|x-z|_2 \mid z \in S\right}$.
5) The set
$$
\left{x \mid x+S_2 \subseteq S_1\right}
$$
where $S_1, S_2 \subseteq \mathbb{R}^n$ with $S_1$ convex.
Exercise 2 For each of the following functions determine whether it is convex or concave or not.
Optional: Determine if they are quasiconvex or quasiconcave.
1) $f\left(x_1, x_2\right)=x_1 x_2$ on $\mathbb{R}{++}^2$. 2) $f\left(x_1, x_2\right)=1 /\left(x_1 x_2\right)$ on $\mathbb{R}{++}^2$.
3) $f\left(x_1, x_2\right)=x_1 / x_2$ on $\mathbb{R}{++}^2$. 4) $f\left(x_1, x_2\right)=x_1^\alpha x_2^{1-\alpha}$, where $0 \leq \alpha \leq 1$, on $\mathbb{R}{++}^2$
Exercise 3 Show that following functions are convex
1) $f(X)=\operatorname{Tr}\left(X^{-1}\right)$ on $\operatorname{dom} f=\mathbf{S}{++}^n$ 2) $f(X, y)=y^T X^{-1} y$ on $\operatorname{dom} f=\mathbf{S}{++}^n \times \mathbb{R}^n$ Hint: express it as a supremum
3) $f(X)=\sum_{i=1}^n \sigma_i(X)$ on $\operatorname{dom} f=\mathbf{S}^n$, where $\sigma_1(X), \ldots, \sigma_n(X)$ are singular values of a matrix $X \in \mathbb{R}^{n \times n}$. Hint: express it as a supremum
Exercise 4 We define the monotone nonnegative cone as
$$
K_{\mathrm{m}+}=\left{x \in \mathbb{R}^n \mid x_1 \geq x_2 \geq \cdots \geq x_n \geq 0\right}
$$
i.e., all nonnegative vectors with components sorted in nonincreasing order.
- Show that $K_{\mathrm{m}}+$ is a proper cone.
- Find the dual cone $K_{\mathrm{m}+}^*$.
Exercise 5 Derive the conjugates of the following functions.
1) Max function. $f(x)=\max {i=1, \ldots, n} x_i$ on $\mathbb{R}^n$. 2) Sum of largest elements. $f(x)=\sum{i=1}^r x_{[i]}$ on $\mathbb{R}^n$.
3) Piecewise-linear function on $\mathbb{R}$. $f(x)=\max _{i=1, \ldots, m}\left(a_i x+b_i\right)$ on $\mathbb{R}$. You can assume that the $a_i$ are sorted in increasing order, i.e., $a_1 \leq \cdots \leq a_m$, and that none of the functions $a_i x+b_i$ is redundant, i.e., for each $k$ there is at least one $x$ with $f(x)=a_k x+b_k$.
MY-ASSIGNMENTEXPERT™可以为您提供SYDNEY MATH3971 CONVEX OPTIMIZATION凸优化课程的代写代考和辅导服务！
