1
vote
0answers
20 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
26 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
20 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
25 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
0answers
46 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
51 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
18 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
42 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
56 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
49 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
57 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
43 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
53 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
40 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
165 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
51 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
72 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
24 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
24 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
26 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
38 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
82 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
39 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
44 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 ...
2
votes
3answers
96 views

Lagrange Method Problem

I am from engineering background and I am currently studying calculus. I had a question from assignment to be solved from a course on coursera but I could not do it. People have posted solution in the ...
1
vote
2answers
35 views

Positivity of a function in $\mathbb{R}^{n}$

We place ourself in $\mathbb{R}^{n}$. We consider a given increasing function $$ g : \begin{aligned} &\mathbb{R}^{+} \to \mathbb{R} \\ &x \;\;\,\mapsto g(x) \end{aligned}$$ Finally, we ...
2
votes
0answers
32 views

Notation for near optimal solution

Usually, $x^*$ is used to denote the optimal solution to a maximization problem. I need a notation to describe a solution that is not optimal but "good enough." In my case this solution is the first ...
0
votes
2answers
77 views

Finding local max and min of a transcendental function

So.. there's no way in heck i'll be solving the derivative of this algebraically -- what can I do? here's a hint from my professor.. Since the picture may be unclear, the function is $$f(x) = ...
0
votes
0answers
34 views

Bounds of the solution space

I have a continuous function $f(x)=a x^2+b x+c$ that is defined on $]0,X[$. I know that the function values are bounded, $Fu <f(x) <Fl$ for all values of $x$. I want to find the bounds on the ...
2
votes
1answer
75 views

$f$ is convex function iff Hessian matrix is nonnegative-definite.

Let $f: \mathbb{R}^2 \rightarrow \mathbb{R}$, $f \in C^2$. Show that $f$ is convex function iff Hessian matrix is nonnegative-definite. $f(x,y)$ is convex if $f( \lambda x + (1-\lambda )y) \le ...
3
votes
1answer
39 views

Monotonicity of an optimizer

Let us consider an optimization problem over $[0,1]$. That is, we are given two continuous functions $$ f,g:[0,1]^2\to \Bbb R $$ such that $f(x,y)$ is non-decreasing in $x$ and non-increasing in ...
1
vote
1answer
62 views

Intuition and counterexamples for higher-order derivative test

In the higher-order test we keep differentiating a function till we find the n'th derivative (n being even) to be greater than or less than zero thereby identifying it as a minimum or maximum. My two ...
0
votes
1answer
22 views

Quadratic Program over Box Constraints

Consider $f:\mathbb{R}^n \rightarrow \mathbb{R}$ defined as $$ f(x) := x^\top x + c^\top x $$ for some $c \in \mathbb{R}^n$. Define the (compact) "Box" $$X := \{ x \in \mathbb{R}^n \mid x_i \in [ ...
0
votes
0answers
28 views

Show that the Rosenbrock function is strictly convex for a specific region

So we know that the Rosenbrock function is a test function of sorts, but can anyone prove that a specific region is strictly convex? Rosenbrock eqn: $ f(x_{1},x_{2}) = 100(x_{2} - x_{1}^{2})^{2} + ...
1
vote
1answer
59 views

Extermum under constraint of parabula

Find the closet point on $2x^2-4xy+2y^2-x-y=0$ to the line $9x-7y+16=0$. Hint: the distance between $(x_0,y_0)$ to $ax+by+c=0$ is $d = \frac{|ax_0+by_0+c|}{\sqrt{a^2+b^2}}$. For using lagtrange ...
3
votes
5answers
82 views

Local minimum global

Let $f:(a,b)\to\Bbb R$ be continuous. Assume that $f$ has a local minimum at some point $x_0$. Further assume that this is the only point where $f$ has a local extremum. Does it follow that $f$ has a ...
1
vote
1answer
26 views

Semicontinuity of the product of two functions

Let $f(x)$ be a left continuous and non-increasing real-valued function. Can I prove that $f(x)x$ is upper semicontinuous?
2
votes
0answers
38 views

Fast solution to problems involving Lagrange multipliers

Suppose we have a function $f:\mathbb{R^n} \rightarrow \mathbb{R}$ subject to the constraint $g(x_1,...,x_m)=0$ for some natural $m$. We can find the local maxima and minima of $f$ on $g$ by setting: ...
1
vote
1answer
38 views

Extrema on a given set

Could you tell me if my approach to finding extrema on a set is good? Let's take a function $f(x,y,z)=x+y+z$ and a set $N= \{(x, y, z) \in \mathbb{R}^3 : x^2 + y^2 \le z \le 1 \} = \{(x, y, z) \in ...
1
vote
1answer
27 views

Clarification about global extrema

Is everything in my statements $\textbf{(1)}$ and $\textbf{(2)}$ correct? A real valued function $f$ defined on a domain $X$ has a global maximum point at $x^{\bigstar}$ if $f(x^{\bigstar}) ...
2
votes
1answer
136 views

Find $x, y$ such that $\left | \frac ab -\frac xy \right |$ is minimal

Given positive integers $a, b, D$. How to find $x, y \in \mathbb{Z^+}$ such that $$M =\left | \frac ab -\frac xy \right |$$ is minimal and $x + y \le D$? For a solution, I can get it by ...
2
votes
1answer
155 views

$\inf$ and $\sup$ of a set.

Let $n\geq3$ be an arbitrarily fixed integer. Take all the possible finite sequences $(a_{1},...,a_{n})$ of positive numbers. Find the supremum and the infimum of the set of numbers ...
0
votes
1answer
50 views

Maximum of the function of multivariable?

I need to find the maximum of the function given by $z=x^3+xy$ in $A=[0,1]\times[0,1]$. I think I need to use partial derivatives, but I'm not sure exactly how.