1
vote
0answers
48 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
24 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 it's 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 ...
1
vote
0answers
11 views

Extremal for variable end point problem. Optimization

I feel like this is not correct, could anyone verify? $\int_0^T (\dot{x}^2 + 2x\dot{x} + x^2) \mathrm{d} t,\;\; x(0)=1 \text{ and for } T\gt 0, x(T) \text{ lies on a given curve } x = c(t) = 3$ ...
0
votes
1answer
23 views

Unique extrema of sum of monotonically increasing and decreasing functions on an interval

If I have two functions, f and g, defined on the interval [0, 1] with both f and g non negative (i.e. f(x), g(x) >= 0) f(x) is monotonically increasing, while g(x) is monotonically decreasing. and ...
0
votes
0answers
10 views

Quasiconcave condition for a power function

Let $f(x, y)= (ax^2+by^2)^n$ where $a, b, n$ are positive, $x, y\in \mathbb{R}$. What is the condition of $n$ so that $f(x, y)$ is a quasiconcave, and concave function? My idea is only calculate ...
0
votes
1answer
14 views

Finding the extremal curve satisfying a variable endpoint

Below is a question I am trying to solve, and my attempt. $\int_0^T \frac{\dot{x}^2}{t^3} \mathrm{d} t$, where $x(0)=1 $ and $x(T)$ lies on the curve Transversal condition: $$f-(\dot{c} ...
0
votes
1answer
19 views

Finding extremal of a fixed end point problem. Optimisation

I want to find the extremal of the fix-end point problem $\int_1^2 \frac{\dot{x}^2}{t^3}$ with $x(1)=2,x(2)=17$ First I check the euler-lagrange equation is equal to $0$. We have: ...
1
vote
0answers
12 views

Local minima: Sufficient conditions. Comparison of Calculus verses Calculus of Variations

My lecturer has written: Let $y=x^*+\epsilon \eta$ where $x^*,\eta,y\in \mathbb{R}^2$ $0\leq f(y) - f(x^*) = \epsilon V_1 + \epsilon^2 V_2 + O(\epsilon^3)$ $V_1 = \nabla f(x^*)\eta$ $V_2 = ...
1
vote
0answers
19 views

Constrained optimization minima and maxima and non-degeneracy answer check

Find the critical points of $$\def\1{x_1}\def\2{x_2}\def\3{x_3}\def\f{f(\1,\2,\3)}\def\l{\lambda}$$ $$\f=\1\2+\2\3+\3\1$$ subject to constraint $\1+\2+\3=1$ First I will construct the Lagrangian: $$L ...
1
vote
0answers
21 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 ...
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 ...
1
vote
0answers
23 views

Hint for KKT Optimization problem

Can anyone help me with the following optimization problem please? I have to find the $\max f(c,y_1^1,\cdots,y_{N-1}^1,\cdots,y_1^M,\cdots,y_{N-1}^M)=c$ subject to the constraints ...
2
votes
2answers
52 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
30 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
37 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
61 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
34 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
80 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
15 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
43 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
25 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
214 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
59 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
38 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
39 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
37 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
9 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
83 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
72 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
56 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
28 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
423 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
214 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
126 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
59 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
805 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
135 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
68 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
170 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
116 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 $$\max_{A \in S} \det A = \det(I_n)$$ ...