2
votes
2answers
51 views

Find an equation for a moving rod

The two endpoints of a 1-metre long rod have an initial position at $(0,0),(0,1).$ The rod slides continuously to the position $(1,0),(0,0)$ sweeping out a region in the positive quadrant. Determine ...
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 ...
0
votes
0answers
27 views

Proof of sufficient condition of existence of Lagrange multipliers

Consider the optimization problem $$ (P) \quad \inf\{ f(x) : g_i(x) \le 0, i = 1,\ldots, m, x \in \mathbb R^m \} $$ where $f, g_i : \mathbb R^n \to (-\infty, \infty]$ are convex and $0 \ne ...
0
votes
1answer
32 views

Sufficient condition on convex function such that $f(x) > -\infty$ for all $x$.

Let $f : \mathbb R^n \to [-\infty, \infty]$ convex and let $f(\overline x) > -\infty$ for $\overline x \in \mbox{int}(\mbox{dom}(f)$. Show that $f(x) > -\infty$ for all $x \in \mathbb R$. ...
1
vote
2answers
51 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
0answers
33 views

Finding $p$ of the binomial cdf…

Please bear with me, I'm only a biologist ^.^: I have a need of solving this cdf so as I can plug in known values $Pr, n, k$, and get an answer for $p$. $$f(k;n,p) = Pr(X\le k) = \sum_{i = ...
4
votes
1answer
75 views

Important topics in Matrix Analysis

I'm doing a course in Matrix analysis, and I'm supposed to prepare a presentation about any topic in Matrix theory. We already covered the book "Matrix Analysis" by Horn, so preferably I need a topic ...
1
vote
0answers
14 views

Analytic solution for the maxima of a bivariate

I found the maximum of the function $f(x) = \frac{x e^{-x}}{1+1/k-e^{-x}}$ by reducing the first order necessary condition to $ke^{-x}+(1+k)(x-1)=0$, and from there the solution obtained with a ...
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: $$ ...
4
votes
1answer
42 views

Find the points in the graph (my solution) - high school.

my math problem is on Swedish so i'll try my best to translate it so you can understand. I'd appreciate it if someone could point out if I did anything wrong and if there is anything that I should add ...
0
votes
0answers
23 views

What’s the relation between the constraint parameter and penalty parameter in lasso regression?

Lasso regression adds a constraint that ∥β∥1, the L1-norm of the parameter vector, is not greater than a given value (say c). Equivalently, it may solve an unconstrained minimization of the ...
3
votes
2answers
212 views

Two halls 6 and 9 meters perpendicularly intersect. Optimization

Two halls 6 and 9 meters perpendicularly intersect. Find the length of the longest straight bar to be passed horizontally from one aisle to another by a corner without deformation. and this is my ...
2
votes
1answer
58 views

Existence of global minimum

Could someone help me with this problem? Let $C$, $D$ convex and closed sets such that the intersection is empty. I want to show that the function $f: \mathbb{R^n} \to \mathbb{R}$ defined by $f(x) = ...
0
votes
1answer
37 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
38 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 ...
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
3answers
50 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 ...
1
vote
1answer
35 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
7 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
48 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: $$ ...
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 ...
3
votes
1answer
71 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 ...
2
votes
0answers
31 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
53 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 ...
0
votes
2answers
103 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
26 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 ...
5
votes
1answer
393 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
40 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
204 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
43 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
2answers
125 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
56 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
23 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
702 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
133 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
66 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
167 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
113 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
35 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
57 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
1answer
98 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
441 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
106 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
105 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
109 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
226 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
91 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 $4\sqrt{2}$?
1
vote
4answers
176 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
75 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 ...