0
votes
1answer
36 views

Maximize arccos-function

I need to find a maximum of the function $$y=\arccos\left(\frac{29+12x\sin(22)+6x\cos(22)+x^2} {\sqrt{x^2+6x\cos(22)-20x\sin(22)+109}\sqrt{x^2+6x\cos(22)-4x\sin(22)+13)}} \right) $$ between x=0 and ...
1
vote
1answer
31 views

Fractional part optimization algorithm

I was trying, out of curiosity, to find an efficient algorithm for the problem below which peaked my interest: Let $r$ be a real number. Find an integer $k > 0$ such that $kr$ is "near" an ...
1
vote
0answers
26 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
3answers
48 views

Find min and maxima

Find local min and maxima of $ \sin(x^3)$ on the interval $]-2,2[$. I take the derivative and get: $$3x^2 \cdot \cos (x^3)$$ I set this equal to zero and get $$x^3 = \cos^{-1}(0)$$ $$ \Rightarrow ...
0
votes
0answers
44 views

Local maximun exercise

Let $u, \psi$ functions defined in $\Omega \subset \mathbb{R}^n$ and $O$ an open subset of $\Omega$. You can think the functions smooth if you want. I want to see that if \begin{equation} ...
1
vote
1answer
20 views

$f:R^2\to R$: Determining the Nature of a Critical Point when the Second Derivative Test Fails

I'm reviewing for a final exam tomorrow. This is an exercise that I am having trouble with: The function: $f(x,y)=x^2-y^4$ I determined that there is one critical point, at $(0,0)$. I determined ...
0
votes
0answers
6 views

Closure in convex analysis

Why do sets $ri(C_1) \bigcap ri(C_2)$ and $C_1 \bigcap C_2 $ have the same closure. I'm reading Dimitri P. Bertsekas's Convex Optimization Theory and this question raises in reading the proof of ...
0
votes
1answer
42 views

How to ensure extreme? — using Extreme Value Theorem

I think it's a simple question. How can I ensure the existence of an extreme (maximum or minimum) using the Extreme Value Theorem / Weierstrass theorem? For example, this multivariate case: $$ ...
4
votes
5answers
71 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 ...
3
votes
1answer
48 views

Does having a zero eigenvalue preclude a matrix from being indefinite?

If a $3\times3$ matrix has a positive eigenvalue, a negative eigenvalue, and a zero eigenvalue, is it then, by definition, indefinite? I think so, since the matrix has both a positive and a negative ...
1
vote
0answers
23 views

Strictly convex self-concordant function

Some definitions: A function $f:R^n\rightarrow R$ is convex[strictly convex] if for every $\lambda\in[0,1]$ [$\lambda\in(0,1)$] and for every $x,y$ [$x\neq y$] in $R^n$ we have $f(\lambda ...
0
votes
1answer
45 views

Strictly positive solution of linear equations

Suppose $A\in\mathbb{R}^{m\times n}$, $b\in\mathbb{R}^m$, and $b\in \mathcal{R}(A)$. Show that there exists an $x$ satisfying $x \succcurlyeq 0$, $Ax = b$ if and only if there exists no ...
-1
votes
5answers
133 views

Find the $\max$ and $\min$ of $f(x,y)= \sin(x) + \sin( y) + \sin(x+y)$, where $(x,y)\in [0, 90^\circ]\times[0, 90^\circ]$ [closed]

It would be nice if someone could explain why is the maximum internal? And for what value you have it? Thank you...
0
votes
3answers
81 views

Local extrema for the function $f(x,y)= x^2+y^2 e^{x^2} + x\sin x$?

I would like to find the stationary points if they exist and so I start by finding the partial derivatives for $x$ and $y$ and equal them to zero and from the second equation I know that $y=0$ but I ...
0
votes
0answers
24 views

Optimization problem - best allocation for

I have the following optimization problem: Suppose that you need to divide traffic bandiwth in a shared medium between $n$ users. We call $x=[x_1,x_2..x_n]$ an allocation. We require that each user ...
4
votes
1answer
267 views

Rigorous proof of the “Lagrange-multiplier theorem”

From Marsden's Elementary Classical Analysis: Theorem 8 Let $f\colon U \subset \Bbb R^n \to \Bbb R$ and $g\colon U\subset \Bbb R^n \to R$ be given $C^1$ functions. Let $x_0\in U$, ...
1
vote
3answers
37 views

Constrained optimisation question

Since $f$ has a local extremum at $x_1$, then surely the LHS of equation (3) always zero? If so, then isn't lambda always simply zero too? But this cannot be, otherwise the last sentence of the ...
1
vote
1answer
100 views

Definition of tangent cone in continuous optimization .

Looking at the definition of tangent cone in continuous optimization : If $M$ is a open subset of $\mathbb R^n$ $x \in M$, The tangent cone of $M$ at $x$ is defined by $$\mathbb T (M, x) = \big\{d ...
1
vote
0answers
37 views

Prove that $\{(x, y): x\in ri (dom f), y >f(x)\}\subset ri (epi f)$

$f: R^n \to (-\infty, \infty]$ is a convex function. Prove that $ri (epi f)=\{(x, y): x\in ri (dom f), y >f(x)\}$ I have used definition of convex functions to prove that the right-side is ...
0
votes
3answers
110 views

Calculus of variations, what is a functional

I'm reading a bit about the calculus of variations, and I've encountered this bit: Suppose the given function $F(.,.,.)$ is twice continuously differentiable with respect to all of its arguments. ...
0
votes
1answer
43 views

How is min max f(x,y) defined when solving a dual problem?

I am trying to solve a dual problem. And it is said that min max f() is always smaller or equal to max min f(). For example, $\max_{y \in Y} \min_{x \in X} f(x,y)$ is always smaller or equal to $ ...
1
vote
1answer
20 views

Limit of maximizer not equal to maximizer of limit

I am looking for functions $f_n,f$ defined on a subset of $\mathbb{R}$ with unique maximizers $\alpha_n, \alpha$, such that $f_n$ converges to $f$ pointwise, but the $\alpha_n$ do not converge to ...
1
vote
1answer
308 views

Deducing that a matrix is indefinite using only its leading principal minors

$A$ is indefinite iff $A$ fits none of the above criteria. Equivalently, $A$ has both positive and negative eigenvalues. Also equivalently, $x^TAx$ is positive for at least one $x$ and ...
3
votes
1answer
116 views

The 2nd total derivative (Hessian) of a composite function -Version 1

Let $f\in C^2(\mathbb R^n,\mathbb R)$ and $Df:A\subset\mathbb R^n\to L(\mathbb R^n,\mathbb R)$ so that $Df_x:\mathbb R^n\to\mathbb R$ is $f$'s total derivative at $x\in\mathbb R^n$. ...
1
vote
2answers
63 views

Why does determining the nature of local extrema for $\mathbb R \to\mathbb R$ functions require twice continuous-differentiability?

In the text Elementary Classical Analysis, why does Marsden specify the condition "twice continuously differentiable" here? Isn't mere twice-differentiability sufficient for the purpose indicated? ...
7
votes
1answer
144 views

$\max\min$ problem

Suppose there is a real valued function $f(x,y)$ where $x,y$ are vector variables $x\in\mathbb{R}^n$ $y\in\mathbb{R}^m$. Moreover $f$ is strictly/concave in $x$ and strictly/convex in $y$. $f$ is also ...
3
votes
2answers
96 views

Maximize the determinant

Over the class $S$ of symmetric $n$ by $n$ matrices such that the diagonal entries are +1 and off diagonals are between $-1$ and $+1$ (inclusive/exclusive), is $$\max_{A \in S} \det A = \det(I_n)$$ ...
0
votes
0answers
34 views

Relating a Taylor-expansion to a maximization problem

Suppose a continuous and twice-differentiable function $f_a(x): [\underline{x}(a), \overline{x}(a)] \rightarrow \mathbb{R}$ has a Taylor-expansion around $x^*$ such that $f_a(x) = \text{const} - r a ...
5
votes
1answer
50 views

Fundamental Optimization question consisting of two parts.

A) Find all extrema of $$f(x)=\sum_{k=1}^{n} x_{k}^{2} $$ subject to the constraint $\sum_{k=1}^{n}\vert x_k\vert^p=1$ B) prove that $$\frac{1}{n^{(2-p)/(2p)}}(\sum \vert x_k\vert^p)^{(1/p)}\le (\sum ...
0
votes
0answers
52 views

Points positioned on a surface with maximum distance

Given a spherical shell with area A. I want to arrange n points on this surface in such a way, that the distance between those n points is maximal. Do you know how to do this?(Can we say something ...
0
votes
0answers
24 views

From optimization problems to explicit solver function

I have the general constrained optimization problem: $$\min_{x\in \mathbb{X}} V(x;\beta)$$ where: $\beta \in \mathbb{R}^m$ is a parameter. $V(x_1,x_2,v,w;\beta)$ is typically the bilinear $x_1^T A ...
0
votes
1answer
93 views

Condition or Proof: Minimizer of one function is maximizing another function

I have two real functions $f(X),g(X)$ where the argument $X$ is a real matrix. The solution $X^*$ for the problem of minimizing $f$ is ending up maximizing $g$ as well. I am looking for a way to prove ...
2
votes
1answer
349 views

Lasso - constraint form equivalent to penalty form

We know that there are two definitions to describe lasso. Regression with constraint definition: $$\min\limits_{\beta} \|y-X\beta\|^2, \sum\limits_{p}|\beta_p|\leq t, \exists t $$ Regression with ...
0
votes
1answer
102 views

A hard multivariate optimization problem in $n-1$ variables

For $n>1$, I want to find the smallest value, and corresponding $x_i$ values, of $f(x_2,\dots,x_n) = \prod_{k=2}^n (x_k+1)^k$ subject to the constraints $x_j > 0$ for all $j$ and $\prod_{k=2}^n ...
2
votes
1answer
93 views

lagrange multipliers fails

I am looking for a certain counter example. Assume a $C^1$ function $f$ is to be optimized with respect to a $C^1$ constraint $g=0$, and we have at a point $(x,y)$, the existence of a lagrange ...
0
votes
0answers
100 views

Simpson's rule characteristics

I just wanted to ask a quick question in regards to simpson's rule for integration. I have been reading up on the trapezoidal rule, and have found the notations and have an understanding such that: ...
2
votes
3answers
213 views

A robust convex optimization problem

Consider a function $ f: \mathbb{R}^n \times \mathbb{R}^m \rightarrow \mathbb{R} $ such that $\forall x \in \mathbb{R}^n$ the map $f(x,\cdot)$ is convex, and $\forall y\in \mathbb{R}^m$ the map ...
1
vote
1answer
89 views

Question on the perimeter of any quadrilateral

Is it true that the perimeter of any convex quadrilateral inside a unit circle is no more than $4\sqrt{2}$?
0
votes
0answers
65 views

Calculus of variations-fields and weierstraß excess function.

if i have a lagrangian $$L (t,x(t),y(t),\dot{x}(t),\dot{y}(t))$$ that depends on two functions and one parameter. Then I will get two Euler-Lagrange equations as a test for extrema. Let us assume ...
1
vote
4answers
147 views

Maximize $f(x) = x^3-3x$ subject to constraints

I would like to understand more about how to maximise functions of one variable subject to constraints. How can you find the maximum value of $f(x) = x^3 - 3x$ subject to $x^4+36 \leq 13x^2$? The ...
1
vote
1answer
73 views

Taylor expansion with integral?

I have looked at a version of a Taylor expansion that has an integral- for the first time. Is this the same as the usual version of a Taylor expansion without integrals? Also, do the $\alpha's$ have ...
2
votes
1answer
97 views

Finding the max. of an integral

I have a question which asks: Let $g\in C[-1,1]$ and the usual inner product $\langle f,g\rangle = \int_{-1}^{1} f(x)g(x)dx$. Find the max value of $\int_{-1}^{1}g(x)x^3dx$ where $g$ is subject to ...
6
votes
1answer
555 views

Why are additional constraint and penalty term equivalent in ridge regression?

Tikhonov regularization (or ridge regression) adds a constraint that $\|\beta\|^2$, the $L^2$-norm of the parameter vector, is not greater than a given value (say $c$). Equivalently, it may solve ...
0
votes
2answers
31 views

Change $\inf$ order

Is it true that if $f: X \rightarrow \mathbb{R}$ and $g : X \times Y \rightarrow \mathbb{R}$ then $$ \inf_{x \in X} \left( f(x) - \inf_{y \in Y} g(x,y) \right) = \inf_{x \in X} \sup_{y \in Y} \left( ...
3
votes
1answer
42 views

$ \log_{\frac 32x_{1}}\left(\frac{1}{2}-\frac{1}{36x_{2}^{2}}\right)+\cdots+ \log_{\frac 32x_{n}}\left(\frac{1}{2}-\frac{1}{36x_{1}^{2}}\right).$

Let $x_{1}$, $x_{2}$, $\ldots$, $x_{n}$ be $n$ real numbers in $\left(\frac{1}{4},\frac{2}{3}\right)$. Find the minimal value of the expression: $ \log_{\frac ...
1
vote
3answers
138 views

Min-max theorem

If $f$ is a polynomial. Prove that there is a $y\in \mathbb{R}$ which $|f(y)|\le |f(x)|,$ for every $x\in \mathbb{R}$: I said since $f$ is a polynomial it's continuous in $\mathbb{R}$ so we can use ...
3
votes
1answer
134 views

For any curve of unit length there exists a closed rectangle with area at most $A$ that covers the curve.

Find the minimum possible value of $A$ such that for any curve of unit length there exists a closed rectangle with area at most $A$ that covers the curve.
10
votes
3answers
285 views

Find the maximum and minimum of $\sum_{i=1}^{n-1}x_ix_{i+1}$ subject to $\sum_{i=1}^nx_i^2=1$.

Find the maximum and minimum of $$ \sum_{i=1}^{n-1}x_ix_{i+1} $$ subject to $$ \sum_{i=1}^nx_i^2=1 $$ for all $n\in\mathbb{N}-\{1,0\}$.
1
vote
1answer
767 views

Show this function is convex.

Could someone point me in the right direction for proving the following? Given that $f:\mathbb{R}^n\rightarrow \mathbb{R}^m$ is an affine map given by $f(x)=A\mathbf{x}+\mathbf{b}$, ...
2
votes
3answers
270 views

Epigraph of a function.

I hope you can give me some suggestions on convex functions. the function $f:(0,\infty)\rightarrow \mathbb{R}$ given by $f(x)=\dfrac{1}{x}$ is convex and continuous, but its epigraph is closed in ...