Optimization is the process of choosing the "best" value among possible values. They are often formulated as questions on the minimization/maximization of functions, with or without constraints.

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6
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2answers
225 views

Constant such that $\max\left(\frac{5}{5-3c},\frac{5b}{5-3d}\right)\geq k\cdot\frac{2+3b}{5-c-2d}$

What is the greatest constant $k>0$ such that $$\max\left(\frac{5}{5-3c},\frac{5b}{5-3d}\right)\geq k\cdot\frac{2+3b}{5-c-2d}$$ for all $0\leq b\leq 1$ and $0\leq c\leq d\leq 1$? The right-hand ...
2
votes
0answers
22 views

Optimization: constructive solution

Consider the following program \begin{align} \max_{x,y \geq 0} f(x,y)\tag{1} \end{align} I wanna construct a solution with $y^* = 0$ and $x^* > 0$. Suppose FOCs satisfy \begin{align} f_x(x^*,y^*) ...
1
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0answers
23 views

Constrained optimization problem using Largange multipliers: ellipsoid collision detection and response

This one is purely for the mathematics so the result is far less important than the method itself. My task is to implement a fast and efficient ellipsoid collision detection and response algorithm. ...
2
votes
1answer
40 views

Value of $F(r)$ to maximize $\frac{\int_r^1xf(x)dx}{2-F(r)}$

Consider a continuous distribution on $(0,1)$ with probability distribution function $f$ and cumulative distribution function $F$. Define $$g(r)=\frac{\int_r^1xf(x)dx}{2-F(r)}$$ and let $r_M\in(0,1)$ ...
0
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0answers
33 views

how to find lower bound function?

In Expectation Maximization algorithms, first step is to find lower bound function of the objective function that is equals to objective function at current estimate. What are the ways to find lower ...
2
votes
2answers
59 views

Finding maximum value of ${n \choose r}$ for given value of n [duplicate]

While I was solving some binomial theorem chapter questions I encountered many questions which asked me me to find maximum value of ${n \choose r}$ for given value of n. Example: Find n for which $...
0
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1answer
167 views

Puzzle question finding Calvin

How to solve this problem. I have reckoned that I need to take as optimization problem finding minimum value for waiting time. Any suggestions? Calvin has to cross several signals when he walks from ...
3
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1answer
29 views

Understanding Energy minimization and poisson equation

Let $M$ be a Riemmanian manifold and $X$ be a vector field thereon. My question is why are these two problems equivalent?: \begin{equation} \operatorname{argmin}_{\phi}\int_M |\nabla \phi - X|^2 \...
2
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0answers
41 views

Optimizing potential

Let $X$ be a vector field on a Riemmanian manfiold $M$. I recently read that solving: \begin{equation} \operatorname{argmin}_{\phi}\int_M (\nabla \phi - X)^2 d\mu, \end{equation} where $\mu$ is ...
-1
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0answers
15 views

Deciding on supermarket daily orders given a fixed amount of money

I have completely no idea even where to start from, and I am sure this question has been discussed already, simply I need some key words and methods that solve this problem so that to start my survey. ...
0
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0answers
90 views

Solutions to recurrence relations

Consider functions $s_{m},c_{m},d_{m}$ defined by the following recurrence relations $$s_{1}=n$$ $$c_{1}=s$$ $$d_{1}=0$$ $$s_{2}=n$$ $$c_{2}=s-n$$ $$d_{2}=d$$ $s,n, d$ are integers. If $c_{m}>...
0
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0answers
18 views

Optimally superimposing two shapes

I have two sets of points, let's say, $Y$ and $X$. Each set has exactly $N$ elements, and each element $element$ is such that $element \in I\!R^{d}$. That is, I have two shapes with $N$ $r-\text{...
0
votes
1answer
56 views

When is minimizing the sum of images of $f$ equivalent to minimizing the sum of independent variables? [closed]

I have to admit I am not good at math, but this is a problem I am having trouble with. What kind of function $f$ can guarantee that $$\arg\min\sum_{i=1}^Kf(x_i) = \arg\min\sum_{i=1}^Kx_i$$ ...
1
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1answer
37 views

Why the local optimum of this problem is always global?

I am reading the paper "Continuous methods for extreme and interior eigenvalue problems" by G.H. Golub and L.-Z. Liao. The papers says for the following problem, (Lemma 2.1 (i), $0>\lambda_i-c\ge \...
4
votes
1answer
111 views

Find closest point, subject to linear inequality constraints

Given a point $p\in \mathcal{R}^2$, I want to compute the closest point $x \in \mathcal{R}^2$, subject to linear inequality constraints $Ax \leq b$. That is, $$\begin{array}{ll} \text{minimize} & ...
-1
votes
1answer
38 views

Finding the global minimum

Let $f~:~\Bbb R^2\to \Bbb R$ be defined as: $$f(x)=\left\|\begin{bmatrix}2&1\\3&1\\4&2\end{bmatrix}\begin{bmatrix}x_1\\x_2\end{bmatrix} - \begin{bmatrix}2\\1\\7\end{bmatrix}\right\|_2^2$$ ...
1
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0answers
16 views

Notation for arguments that maximize functions with priority

Suppose we have some functions $f_1(x), f_2(x), \ldots, f_n(x)$ with $x \in \mathbb{Z}^n$. We can denote the subset $X_1$ of $\mathbb{Z}^n$ that maximizes $f_1(x)$ as: $$X_1 = \underset{x \in \...
1
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1answer
73 views

Can this optimization problem be solved?

I am working on an optimization problem but I am not sure if the problem can be formulated as an integer programming problem. Assume the cost minimization problem for a set of subscribers and ...
0
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0answers
7 views

Optimizing over a set of orthonormal vectors

I am looking for an algorithm to the following optimization problem. Let $X \in \mathbb{R}^{n\times k}$ be a given matrix and $n > k$. Then, I want to maximize $\mathrm{trace}{(FX)}$ over $F \in \...
0
votes
1answer
50 views

Find a least upper bound for $3^{x+y-4}+(x+y+1)2^{7-x-y}-3(x^2+y^2)$ with some constraint?

This is a question in vietnamese national math exam at the end of 12th grade. Given x,y are real numbers which satisfy the condition: $x+y+1=2(\sqrt{x-2}+\sqrt{y+3})$ Find a $m$ such that: ...
0
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0answers
31 views

Negative bounds on variables in Linear Programming formulation

I am new to optimization theory and encountered the following problem: I am reviewing a formulation for a network problem that is fed into a mathematical solver and I noticed that on the "bounds" ...
0
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2answers
56 views

Is Frank Wolfe a descent algorithm?

A colleague was explaining to me that the Frank-Wolfe algorithm is a descent algorithm (i.e. its objective value decreases monotonically at each iteration). However, when I tried simulating it, my ...
0
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0answers
19 views

Impact of convexity under different changes of variables for different parts of optimization

Let $$ \min_x f(x)$$ such that $$ C(x) \le 0$$ where $C$, and $f$ each are convex under respective changes of variables. How does that impact the optimization? If standard algorithms are sensitive ...
0
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1answer
77 views

Efficient ways of minimizing a complicated objective “function”?

My problem at hand neither has any special structure that gives me closed-form solutions nor can be written by a single expression. Yet, it is still an objective "function," as I can compute a value (...
0
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0answers
9 views

How to find initial values when optimizing functions of ODEs?

I want to reproduce parts of a paper and therefore have to optimize a function $f(A,B)$, where $A$ and $B$ are the solutions of the following system of ODEs: $$\frac{d}{dt}A = c_1F + \nu_{\beta} -...
0
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0answers
38 views

Minimum value will be attained by same z

Let $y=\left(\begin{array}{c}y_1\\y_2\\\vdots\\y_n\end{array}\right)$ be a fixed non zero vector in $R^n$. Let $S \subset R^n \setminus\{0,y\}$ is a finite set. Suppose $$min\big\{\sum_{i=1}^{n}{|y_i-...
2
votes
0answers
98 views

Maximum value of $(1-F(t))t$ for probability distribution

Consider a continuous distribution on $(0,1)$ with cumulative distribution function $F$. For the value of $t\in(0,1)$ that maximizes $$P(t)=(1-F(t))t,$$ what is the lower bound of $P(t)$? For example, ...
0
votes
2answers
29 views

Optimization of the function of two variables

I have two functions $f(x,y)$ and $g(x,y)$. I want to minimize the sum of these functions w.r.t $x,y \in (0,1)$. I know that for fixed values of $x$, $f(.,y)$ is a decreasing function while $g(.,y)$ ...
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1answer
15 views

Sum of convex and decreasing function

I have a sum of decreasing function and a convex function over some domain. Can I say that the sum is also a convex function (i.e. there exists a unique minimum)?
2
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1answer
42 views

Solve the closed form solution for argmax of $ x^Ty - x^T\ln(x)$

Let $y \in \mathbb{R}^n, \ln(x) = \begin{bmatrix} \ln(x_1) \\ \vdots \\ \ln(x_n) \end{bmatrix} \in \mathbb{R}^n$ Show that $$x^* = \text{argmax}_{x \in \mathcal{D}} \quad x^Ty - x^T\ln(x)$$ ...
1
vote
1answer
36 views

Maximize the number of non zero elements of a product of binary matrices.

I want to find two binary matrices $A$ of size $N \times M$ and $B$ of size $M \times N$ such that: $AB=C$ is a strictly lower-triangular matrix ($j \geq i \implies C_{ij}=0$) The number of ...
4
votes
1answer
59 views

Maximizing $\frac{\int_r^1xf(x)dx}{2-F(r)}$

Consider a continuous distribution on $(0,1)$ with probability distribution function $f$ and cumulative distribution function $F$. Define $$g(r)=\frac{\int_r^1xf(x)dx}{2-F(r)}$$ and let $r_M\in(0,1)$ ...
0
votes
1answer
16 views

Proximal operator of the sum of two norms

I'm a little new to this and not sure how to evaluate the proximal operator in this context, assuming a closed form exists. Calculate $\text{prox}_{c,f+g}(v)$, where $f(x) = \frac{1}{2}\|Ax-b\|_2^2$...
0
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0answers
24 views

Find rotation matrix to match points in parallel projection

I am given two sets of 3D points (actually 2D, see below) with corresponding pairs. I am seeking two 3D rotation matrices, such that (only) the X and Y components of the rotated points match best (...
0
votes
1answer
32 views

How to find the analytical solution of this optimization problem?

I have an optimization problem of the form $$\begin{align} \text{maximize}\quad&\sum_{i=1}^{k}\sum_{j=1}^{n}w_{ij}x_{ij}\\\text{s.t.}\quad \quad\quad\,\,& \sum_{i=1}^{k}x_{ij}\leq 1,\;\forall ...
3
votes
0answers
81 views

Optimal cyclic permutations (Formulate as standard problem)

How can we find cyclic permutations $\prod_i$ to be applied to each of corresponding $i$'th rows of a square matrix $X$ of size $n \times n$ such that a given sum of pairwise costs $\sum_{ij}C\left[\...
1
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2answers
36 views

Maximizing the sum of the squares of numbers whose sum is constant

I wonder how one goes about to find the maximum of $\sum v_i^2$, the $v_i$'s being positive integers whose sum $\sum_i v_i$ is fixed.
1
vote
1answer
55 views

Invertibility of bordered Hessian

I have an optimization problem: $max_{x \in C} f(x)$ s.t. $Ax=b$, where $x \in R^n$ and $b \in R^m$, $m \le n$, adn $C$ compact. I know that $f$ is strictly quasi-concave, and that $A$ has rank $m$ (...
0
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0answers
17 views

Momentum Potential Term in Optimization Problem for Implicit Euler Solver

I'm trying to understand the explanation of the implicit Euler solver (Section 3.1) set forth in this paper: Projective Dynamics: Fusing Constraint Projections for Fast Simulation For the purposes of ...
0
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0answers
17 views

Solving $I^* = \arg\min_{I'} \left( \|\phi_\ell(I) - \phi_\ell(I')\|_2^2 + R(I') \right)$ with gradient descent

I am trying to create the results from this a paper that is trying to understand the types of features a convolutional neural network is learning to recognize. I don't think understanding ...
0
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1answer
35 views

Is sup of max, same as max of sup?

Let $\sigma_1, \sigma_2 \dots \sigma_n$ be functions of $\omega \in \mathbb{R}_+$. Is $\sup_{\omega}(\max_{i=1:n} (\sigma_i))$ same as $\max_{i=1:n}( \sup_{\omega}(\sigma_i))$? Could you also please ...
0
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1answer
14 views

In Constrained Optimization, Restrict Domain to Open Set $A\subset\mathbb{R^N}$?

In constrained optimization and context of economics (e.g. utility function with quantity of goods as arguments subject to wealth), why do textbooks always restrict domain of the objective function ...
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0answers
18 views

alignment of two sets of vectors

I have a maximisation problem to do with aligning two ordered sets of 3D unit vectors. I want to apply the same rotation to all the vectors in one set so that they are in closest alignment with those ...
3
votes
1answer
68 views

Team grouping troubles

Imagine there are 12 teams, numbered 1 through 12. There are 10 games those teams can compete in, with two teams needed per game. There are 10 rounds, and it is important that after the 10 rounds are ...
4
votes
2answers
55 views

Maximizing product of five-digits numbers

From a French 2016 puzzle and math contest, where no calculator is allowed Using each of the digits $0,1,2,3,4,5,6,7,8,9$ exactly once, find two five-digit integers such that their product is ...
0
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0answers
11 views

Direction of a gradient at maximizer on the boundary

Let $u \in C(\bar{B})$ where $B=B_1(0) \subset \mathbb{R}^n$ is the unit ball. Assume $u$ attains its maximum at $x_0 \in \partial{B}$ and $\nabla u(x_0) \neq 0$. What can we say about the direction ...
0
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0answers
22 views

Calculate the division between total height by base's diameter.

One silo for grains storage was built in a form of a cylinder (floor and walls) with a hemispherical roof. The silo is design to have a certain volume $V$. Calculate the division between total height ...
0
votes
1answer
35 views

Minima of symmetric polynomials subject to two symmetric constraints

The homogeneous symmetric polynomial of degree $k$ in $n$ variables is $$ f_k(x_1,x_2,\dots,x_n) = \sum_{i_1<i_2<\cdots<i_k}x_{i_1}x_{i_2}\cdots x_{i_k}. $$ Consider the following ...
0
votes
1answer
19 views

Why does Frobenius norm make BFGS scale-invariant?

On slide 11 here it is claimed that the weighted Frobenius norm leads to a scale-invariant optimization method. Similar claims about this norm can be found throughout the literature see 1,2,3. In ...
3
votes
1answer
72 views

Maximization of a determinant

I'd like to compute $$ \DeclareMathOperator*{\argmax}{arg\,max} A^*=\argmax_{\substack{A\in\mathbb{R}^{d\times k}\\A^T A=I}} \det(A^T \Lambda A) $$ where $k\leq d$, $\Lambda=\operatorname{diag}(\...