0
votes
1answer
24 views

Finding the minimum distance between two lines

I really don't know how to tackle this optimization problem: We consider the two lines $$a(x) = x \begin{pmatrix}1\\2\\3\end{pmatrix}, b(y) = ...
0
votes
2answers
23 views

The maximum volume of Tetrahedron

A optimization problem: Get the maximum volume of a tetrahedron its 4 vertices on the surface of cube whose edge length is 1 . From the geometrical intutition ,we can get : Selecting ...
3
votes
1answer
112 views
+50

Lipschitz continuity of parametric optimizer

Consider the parametric optimal solution $x^{*}: \mathbb{R}^n \rightarrow \mathbb{R}^n$ defined as $$ x^*( y ) := \arg\min_{x \in X } \ \ x^\top x + x^\top A y \\ \quad \qquad \text{subject to: } \ ...
3
votes
0answers
36 views

Finding disjoint intervals from Cantor Set

Consider $C$ the classic Cantor ternary set in $[0,1]$. I am interested in the following problem: Find the largest constant $0<k<1$ such that it is true that any interval $[a,b] \subseteq ...
0
votes
0answers
14 views

Classify the stationary points of the following function

I´m asked to find the stationary points of the equation $$f(x,y)=2+y^{2}-2xy+\frac{81}{y^{2}}-\frac{81}{y}\sqrt{2-x^{2}}$$ I know that we should verify when does $\nabla f = 0$, but the resulting ...
1
vote
0answers
68 views

Find all $(x,y)$ pairs

Find all $x$ , $y$ $\in$ $\mathbb {R^+}$ such that for all $\epsilon>0$, $$x \left(\dfrac{\ln \left(1+\dfrac{1}{x}\right)-2\epsilon}{\ln xy-(1-\epsilon)}\right)\geq \left(\dfrac{\ln ...
0
votes
2answers
84 views

Spivak “min” notation confusion

Spivak uses a notation: min$(1, \frac{\epsilon}{2|a| + 1})$ What does he mean by this notation? especially by "min"??
3
votes
1answer
68 views

Unconstrained optimal control - $J = \int_0^{t_1} (x^2 + ux + \frac{1}{2} u^2) dt$

I've been given the following problem to solve, and I'm having a lot of difficulty in understanding what I can do. The system $\dot x = x + u$, where $u = u(t)$ is not subject to any constraint, ...
0
votes
1answer
37 views

How to find the convex hull of a given set?

$A=\{(0,0),(0,1),(1,0)\}$ $B=\mathbb{Q}^2$ $C=\{(x,\sqrt{x})\in \mathbb{R}^2:x\ge0\}$ I have to find Conv(A), Conv(B) and Conv(C). My attempt Conv(A) is the boundary (correction: obviously it ...
0
votes
2answers
23 views

Quick help on showing a set is bounded

I am working on a constraint optimization problem. I have found the extrema and all I need to do now is to show that the set S that the critical points are defined in is bounded and closed (therefore ...
0
votes
0answers
24 views

Minimizing 1-dim problem containing quadratic and sum of absolute value functions

I stumbled on a problem which I am not sure how to most efficiently solve - I want a solver in code which I need to repeat several times with various constants. Basically I want to minimize a 1-dim ...
1
vote
1answer
15 views

Coercive or not?

I had this problem in the exam. Let $X = [x_1,...,x_d]^T$, $a\space \epsilon$ $\mathbb{R}^d$ and $C$$\epsilon$$\mathbb{R}$. Argue for or against. $f(X) = a^TX + C||X||^2$ is coercive only for $C ...
0
votes
1answer
24 views

Extreme of a function

Let us have the following function on $x\in[0,1]$ $$ y =f(x)= x + a\left(\max(0,b-x)\right) $$ where $a>0$ and $b\in[0,1]$ are known parameters. Could you please find the solution of this $$ ...
1
vote
0answers
24 views

Extremal condition for series expansion coefficients

I want to maximize a coefficient in a series expansion, so the situation is the following. $f \in C^{\infty}$ and $f: \mathbb{R} \times \mathbb{R} \times [0,2 \pi] \rightarrow \mathbb{C}$. Now, we ...
0
votes
0answers
21 views

A Difficult combinatorial optimization problem

Let $\mathcal{J}$ be a closed, bounded, compact, convex set in $\mathbb{R}^L$. (Notations: vector $\mathbf{x}$ is denoted in bold letters and its $i^{th}$ co-ordinate is denoted as $x_i$. ...
0
votes
0answers
53 views

How to find if an integral is possible to compute: Failing to solve integral for quadratic functional

I am trying to solve the below integral, and no computational method seems to be capable of solving this, nor can I do it by hand. Any ideas? $$\int_{t_0}^{t_1}[a(t)((2\dot{x^*}\dot{\eta} + ...
1
vote
0answers
31 views

Finding the source for minimizer of a functional for all $C^2$-curve $x(t_0)=x_0$ and $x(t_1)=x_1$

I am trying to find where this problem comes from and its corresponding proof for my students, but I cannot find the source anywhere. If anyone can find the source of this, or has any ideas where I ...
0
votes
1answer
29 views

Critical points characterization of real function

Providing a real multi-variable function $f(\bar{x})$ twice differentiable with respect of all its variables. Looking for critical points is equivalent to solve $\nabla f = \vec{0}$. And to ...
3
votes
1answer
148 views

Minimum of the function $f(x)=\frac{1}{1+x^2}+\frac{3}{1+(h-x)^2}$, for $0\leq x \leq h$

Find the minimum of the function $f(x)=\frac{1}{1+x^2}+\frac{3}{1+(h-x)^2}$, for $0\leq x \leq h$ Proof that the solution can be expressed as: 1. There exists a $δ>0$ so that for $0\leq h \leq ...
1
vote
0answers
26 views

Equivalent optimization problems?

I am wondering if the set of optimizers of the problem $$ \min_{x \in X} \ f(x) \quad \text{subject to: } g(x) \leq 0, \ h(x) = 1 $$ is the same of the one of $$ \min_{x \in X} \ f(x) + h(x) \quad ...
0
votes
0answers
33 views

Concave Quadratic Program

Let $X \subset \mathbb{R}^n$ be compact and convex. Consider $$ x^* := \arg\min_{x \in X} x^\top Q x + c^\top x $$ where $Q \prec 0$. I am wondering if there are cases where $x^*$ can be written as ...
1
vote
1answer
21 views

Does existence of global minimum imply coercivity?

It is known that a coercive function over a closed, unbounded set has a global minimum. Is the converse true ? The larger context for this question is the following question: Suppose we are given a ...
0
votes
1answer
26 views

$|p- \dfrac xn|>|q- \dfrac xn|$ $\implies$ $p^x(1-p)^{n-x}<q^x(1-q)^{n-x}$?

If $p,q \in (0,1)$ , and $ n \in \mathbb N$ be given and $x$ be given integer between $0$ and $n$ such that $|p- \dfrac xn|>|q- \dfrac xn|$ , then is it true that ...
0
votes
1answer
69 views

Convex optimization approximation

Consider the optimization problem $\mathcal{P}_0$ $$ \min_{x \in \mathbb{R}^2} \left\| x-p \right\|^2 $$ $$ \text{sub. to: } \ A x \leq b, \ \ x_1^2 + x_2^2 = 1 $$ where $p \in \mathbb{R}^2$ is a ...
2
votes
0answers
57 views

Determining extrema of $f(x,y,z)=(xyz)^{\frac{2}{3}}$ on $x^2+y^2+z^2=1$

Determine where on the sphere $x^2+y^2+z^2=1$ the function $f(x,y,z)=(xyz)^{\frac{2}{3}}$ attains its maximum and minimum. Using Lagrange multipliers one gets the solutions ...
0
votes
0answers
20 views

constrained optimization and differential equation

Consider the following differential equation system (cylindrical coordinate system): $\frac{dP_x}{dz} = P_x C \int\limits_0^{2\pi}\int\limits_0^a \frac{f(r, \theta)}{g(r, \theta, z)} r dr d\theta$ ...
4
votes
1answer
48 views

What exactly are the curves that are a best fit to the Harmonic Cantilever?

Let's start with a few references to get an idea: Daniel Goldwater: Harmonic Cantilever Book Stacking Problem Block-stacking problem Harmonic Series and Bricks Interesting related issues: Maximum ...
1
vote
0answers
58 views

Proving boundedness of a function .

Consider the function \begin{eqnarray} f(x_1,x_2,\cdots, x_n) = \frac{\sum_{i}^{n}a_ix_i}{\sum_{i}^{n}b_ix_i}, \end{eqnarray} over the set $S = \{x := (x_1,x_2,\cdots, x_n):-1 \leq x_i \leq 1,\; ...
0
votes
1answer
57 views

Proving boundedness of a function (part 1).

Consider the function \begin{eqnarray} f(x_1,x_2,\cdots, x_n) = \frac{\sum_{i}^{n}a_ix_i}{\sum_{i}^{n}b_ix_i}, \end{eqnarray} over the set $S = \{x := (x_1,x_2,\cdots, x_n):-1 \leq x_i \leq 1,\; ...
0
votes
2answers
25 views

Minima point is a solution point

Consider $$f:\left[0, \dfrac{\pi}2\right] \to \mathbb R$$ defined as $$f(x)=\sup\{x^2,\cos x\}.$$ It is easy to show that $f$ has an absolute minimum point at $x_o \in I$ , but how to show that $\cos ...
1
vote
2answers
36 views

Finding extremal values on a set

Let $f(x,y)=(x-1)^2+y^2+xy$. Find the maximal and minimal values of $f$ on the set $M=\{(x,y):|x|+|y|\leq4\}$. Attempt: By taking partial derivatives and solving the homogenous algebraic system we ...
1
vote
1answer
59 views

New variable in a convex optimization problem

Consider the convex optimization program $$ \min_{x \in X } x^\top P x + p^\top x \quad \text{ sub. to: } Ax = b $$ where $X \subset \mathbb{R}^n$ is compact, $P \succ 0$, $A \in \mathbb{R}^{m \times ...
1
vote
1answer
44 views

Elegant way to solve this extreme value problem

I want to show that $$ \sup_{(x,y)\in \mathbb{R}^2 \setminus \lbrace (0,0) \rbrace} \frac{(ax+by)^2}{x^2+y^2} =a^2+b^2 $$ where $a,b \in \mathbb{R}$ are fixed (this problem appears when one tries to ...
1
vote
2answers
65 views

Extreme value problem, maximize ratio of volume to surface area

For a cylindrical can, how to choose the ratio of the height to the radius such that the ratio of the volume to the surface area gets maximized? The volume ist $V = \pi r^2 h$ and the surface ...
0
votes
1answer
41 views

Maximization of Function with two restrictions.

Maximize $$f(x,y,z)=xy+z^2,$$ while $2x-y=0$ and $x+z=0$. Lagrange doesnt seem to work.
12
votes
2answers
173 views

$|f(x)|\leq \sqrt{\frac{\pi}{3}\int_0^\pi f'^2}$

Let $f\in C^1([0,\pi],\mathbb R)$ such that $\displaystyle\int_0^\pi f(t) dt=0$ Prove that $\forall x\in [0,\pi],\displaystyle|f(x)|\leq \sqrt{\frac{\pi}{3}\int_0^\pi f'^2(t)dt}$ Failed ...
2
votes
0answers
55 views

How to find out the closed form of a function from its parametric form?

In general suppose that we have a parametric curve given by: $$ x = \phi(t) \\ y = \psi(t) $$ Then if $\phi^{-1}$ exists it is easy to get $y$ as a function of $x$ in closed form: $$ y = ...
1
vote
2answers
79 views

Local minimum implies local convexity?

Consider a real function $f$, and suppose it has a local minimum at $a\in \mathbb R$. It typically looks like What hypotheses can be added to $f$ so that there is some $\epsilon >0$ such ...
0
votes
0answers
37 views

show that M isn't close map

the line search map $M:En\times E_n \rightarrow E_n$ defined below is frequently encountered in nonlinear programming algorithm.the vector $y∈ M(x,d)$ if it solves the following problem where $f:E_n ...
3
votes
1answer
25 views

Find extrema of $f_a(x)=\vert x-a\vert^2$ on $\overline{B_1(0)}$

Let $\overline{B_1(0)}\subseteq\mathbb R^3$ be the closed unit-sphere and $a\in\mathbb R^3$. Find all extrema of the function $f_a(x)=\vert x-a\vert^2$ on $\overline{B_1(0)}$ depending on $a$. ...
1
vote
1answer
34 views

Optimization of a convex target function with inequality constraints

I want to solve the following optimization problem: \begin{equation} \begin{split} \text{maximize} &\;\;\; \ln x_1+\ln x_2+\ln x_3+\ln x_4 \\ \text{s.t} &\;\;\; x_4\le4 \\ ...
0
votes
2answers
33 views

How to find this maximum

We have $$n\in\mathbb{N}\quad k=1,...,n$$ we want to find $$\max_k{\cos(\frac{k\pi}{n+1})}$$ As we don't have a continuous application , we have a set of $n$ points we cannot do the typical ...
0
votes
0answers
26 views

Dual convex pairs

I am currently trying to understand a certain proof. The author uses the term dual convex pair for a pair $(\phi,\psi)$ of convex functions defined on subsets $X,Y$ of $\mathbb R^n$ satisfying: $$ ...
0
votes
1answer
66 views

Given a polynomial of degree 5, get minimum and maximum without using derivatives

Given a quintic polynomial (in my case, $x^5+2x^4+16x-32$), I am supposed to get its maximum and minimum value for the interval $I=[-2;2]$ without using the derivative of the corresponing polynomial ...
0
votes
1answer
29 views

Image of an unbounded set in $\mathbb R^2$ under the function $f(x,y)=x^3+4y^2-4xy$

Given the function $f(x,y)=x^3+4y^2-4xy$ to be evaluated over the set $E={(x,y) \in R^2: 0\leq y \leq 3x/4}$, I'm asked to determinate $F(E)$. I've noticed that the function in continuous, and the ...
0
votes
0answers
44 views

Minimization of product function subject to constraints

I want to minimize the following function: $\prod_{i=1}^{n}{x_i}$ Subject to the following constraints: $\sum_{i=1}^{n}{x_i}=1.1+(n-1)(0.1)$ and $0.1 \leq x_i \leq 1.1$ How should I go about it? ...
1
vote
1answer
38 views

Quadratic programs: is the projection onto constraints optimal?

Consider the Quadratic Program $$ x^* := \arg \min_{ x \in X } \ \{ x^\top x + c^\top x \} \ \text{ sub. to: } Ax=b $$ where $X \subset \mathbb{R}^n $ is a non-empty, convex, bounded polyhedron. ...
0
votes
1answer
106 views

Minimum of sum of increasing and decreasing function

Suppose we have a function $f(x)$ defined for integer $x$ in some bounded interval, which is positive and increasing $$f(x+1)\geq f(x)\\ f(x)>0$$ , and a function g(x) which is positive and ...
0
votes
2answers
47 views

arc wise connected set

I am having confusion in understanding what is arc wise connected set.The definition is a set $S$ is arc wise connected if for any pair of point a,b we can define a continuous function $f$ from ...
0
votes
0answers
52 views

Mean value theorem mindset

So I am to learn to use the mean value theorem to prove these types of problems that I will list. I would really like for someone to provide some visual/intuitive information on how I can imagine the ...