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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Return of the lost ant 3D

Starting in the center of a sphere of radius 1, draw a path with the shortest possible length that intersects every plane that is tangent to the sphere. This question appeared as a generalization of ...
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1answer
33 views

What's the solution for $\max_{x\in(0,1]}: \{-1-x\}$

What's the solution for the following optimization problem? Is the constraint set convex? $$\max_{x\in(0,1]}:\{-1-x\}$$
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Smooth approximation of maximum using softmax?

Look at the Wiki page for Softmax function (section "Smooth approximation of maximum"): https://en.wikipedia.org/wiki/Softmax_function It is saying that the following is a smooth approximation to the ...
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7answers
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How to prove the sum of squares is minimum?

Given $n$ positive values. Their sum is $k$. $$ x_1 + x_2 + \cdots + x_n = k $$ The sum of their squares is defined as: $$ x_1^2 + x_2^2 + \cdots + x_n^2 $$ I think that the sum of squares is ...
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29 views

Minimise sum of squares [duplicate]

For real numbers $x_{1},..,x_{n}$, minimise $x_{1}^2+..+x_{n}^2$ subsject to the condition $x_{1}+..+x_{n}=2$. This has cropped up in a stats question.
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1answer
629 views

Maximum total distance between points on a sphere

What is the configuration (set of locations) of $n$ points on the surface of a sphere such that the sum of distances is maximum for $n=1,2,3,...$? The sum of distances is measured by summing the ...
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1answer
17 views

Are there known patterns among minimal expressions?

Let $R = F[z_1, z_2, \dots]$ be the finite-degree polynomials in a countable number of variables. Let $\mathcal{E}(R)$ be the set of all expressions of polynomials. Note that there could be an ...
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1answer
22 views

Convex Subset Projection

Suppose that C is a closed convex subset of $\mathbb R^n$ and $x \in \mathbb R^n$. The projection of $\mathbf x$ onto C is the closest point $\mathbf y \in C : \mathbf z = \mathbf y$ minimizes ...
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0answers
20 views

No critical points means convex or conave? [on hold]

If we don't know whether $f(x)=0$ is convex or concave or not, but we know under certain constraint sets there is no critical points of $f(x)$ inside meaning the solution of $df(x)=0$ is outside the ...
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1answer
309 views

Can a dynamic programming problem be transformed into a linear algebra problem?

Here is a simple standard economic problem: Let Robin Crusoe have an endowment $w_0$ of lembas bread. She is immortal and discounts at a rate of $\beta$ per period. Each period (from $t=0$ on) she ...
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0answers
8 views

Formal definition of convexity for multivariate function?

Let $M\in R^{M\times N}$, a function $f: M\rightarrow R$ is called convex on $M$ if $f\big((1-\lambda)X1+\lambda X2, (1-\lambda)Y1+\lambda Y2\big) \leq (1-\lambda)f(X1,Y1) + \lambda f(X2,Y2)$ For ...
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3answers
60 views

How to compute Lipschitz Constant for multivariate function $f(x,y)=1-xy$?

How to compute Lipschitz Constant for multivariate function $f(x,y)=1-xy$? I know the definition for one variable? What is its definition for multivariate functions?
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68 views

is there a closed form solution to this continuous optimization problem?

Consider the function \begin{eqnarray} \max_{t_1,\ldots,t_p \ge 0} V(p) & = & \sum_{i=1}^p [- \alpha t_i - \beta e^{\rho - \delta^{i-1}\theta} \prod_{k=1}^i t_k^{-\delta^{i-k}\Omega}]. ...
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0answers
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How is the upper bound of a minimisation IP determined during branch-and-bound?

When using the branch-and-bound algorithm to solve an Integer Programming (IP) problem, the entire enumeration tree doesn't need to be evaluated and that's where the speed-up is achieved. Suppose the ...
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0answers
30 views

max and min values on symmetric polytope

Let $-N\leq t \leq N$. Let $A$ be regular $(N-1)$-dimensional simplex with vertices $(t,0, \ldots, 0)\ldots (0, 0,\ldots, t)$ and $B$ be regular $(N-1)$-dimensional simplex with vertices $(t-N+1,1, ...
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11 views

How to find fitting parameters of the function?

I have the function describing the experimental data - $f(x)$. I also have another function - $g(x, \bar{p})$, which is the theoretical function for the process involved. Here $\bar{p}$ - is the ...
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6 views

Optimizing single element of vector wrt. second order cone constraints

Can anybody put me on the right path to solving the following problem analytically: Given a vector $\bf{x}=(x_1,...,x_n)^T$, how do I find the bounds for a single element subject to second order cone ...
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17 views

Optimize profit given complete market information

Assume there are $N$ market participants (on the order of several hundred), and $M$ items (several thousand) being bought and sold on a market. For each participant/item pair, you know how many units ...
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0answers
19 views

Optimizing over a set of optimization problems

This is my first time asking an optimization question on here, so I am looking forward to see what will happen here. In the lack of a better title, I wrote it as it is. At a high-level, I can perhaps ...
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1answer
17 views

making a function non-linear using a Lagrangian function

How Is this formula a Lagrangian function ? And how can a non-linear element be added to a function using this "Lagrangian function" This is where i got this In order to improve the performance ...
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Maximize $\int_0^1 x^2f(x)~\mathrm dx - \int_0^1 xf(x)^2~\mathrm dx$ among continuous $f:[0,1]\to\Bbb R$

For a function $f$, let $$ a = \int_{0}^{1} x^2f(x) \mathrm{d}x\\ b = \int_{0}^{1} xf^2(x) \mathrm{d}x, $$ where $f$ is a continuous function from $[0,1]$ to $\mathbb{R}$. Then find ...
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3answers
36 views

Ladder Optimization Problem

A fence 4 feet tall runs parallel to a tall building at a distance of 4 feet from the building. What is the length of the shortest ladder that will reach from the ground over the fence to the wall of ...
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how does probabilistic optimization differs with other types of optimization?

probabilistic optimization and other types of optimization are very confusing. how can you differentiate it? i would want to optimize an equation using MATLAB. however, i do not know which or where ...
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Issue with CVX geometric programming

So I'm trying the following geometric optimization problem in CVX and I'm running into this weird issue where I get a higher optimal value if I remove a constraint. Here's the code I have run. The ...
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2answers
41 views

Minimize Function over Convex Subset

Suppose that C is a closed convex subset of $\mathbb R^n$ and $x \in \mathbb R^n$. The projection of $\mathbf x$ onto C is the closest point $\mathbf y \in C : \mathbf z = \mathbf y$ minimizes ...
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0answers
11 views

Golden search method iterations and minimum.

Theoretically, how many iterations should it take to minimize f to be within 〖10〗^(-m) over [a,b] using the Golden Search Method?
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1answer
39 views

Convex Functions and Subsets

Suppose that $f, g: \mathbb R^n \to \mathbb R $ are $C^1$ convex functions. Show that $C = ${$\mathbf x \mid g(\mathbf x) \leq 0$} is a convex subset of $\mathbb R^n$. Show that if $\nabla f(\mathbf ...
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24 views

Lagrangian Method Proof

Suppose $f(\mathbf x)$, $g(\mathbf x)$ are smooth functions where $\mathbf x^*$ is a constrained local minimizer of $f(\mathbf x)$ subject to $g(\mathbf x)=0$. If $\nabla g(\mathbf x^*) \neq 0$ and ...
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3answers
93 views

Find min of $\frac{\left( \sum kx_k \right) \left( \sum x^2_k \right)}{\left( \sum x_k \right)^3}$

With $n \ge 2$ and $x_1,\ x_2,\ \dots,\ x_n > 0$. Find the minimum of: $$ M = \frac{(x_1 + 2 x_2 + ...+ nx_n)( x^2_1 + x^2_2 +...+x^2_n)} {\left( x_1 + x_2 +...+ x_n \right)^3}$$ For specific ...
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23 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 ...
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14 views

maximize the acos function of more than one variable

I want to find the maximum angle which is defined as follows: $\theta = \cos^{-1}(\frac{1 + x_1 + x_2}{\sqrt{1 + x_1^2 + x_2^2}})$ now I should find $x_1$ and $x_2$ values so that $\theta$ has its ...
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2answers
31 views

Max and Min using Lagrange Multipliers

Suppose A is a symmetric matrix. Show that the maximum and minimum of $\mathbf x ^T A \mathbf x$ subject to the constraint $\mathbf x ^T \mathbf x=1$ are the maximum and minimum eigenvalues of A. I ...
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1answer
502 views

hessian matrix not positive definite at a minimum?

I have a function for which I want to find the global minimum. The function is: ...
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39 views

Optimization issue, how to obtain the maximal value?

$ max f(\beta)=\frac{\beta}{1+\beta}\cdot \left(1- \frac{\binom{N+B}{B}\cdot\beta^B} {\sum_{i=0}^B {\binom{N+i}{i} \cdot \beta^i}} \right)$ where $\beta\in[0,\infty)$, $N$ and $B$ are identified ...
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1answer
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Maximizing the trace

Say i have the following maximization. $ max_R$ trace $(RZ): R^TR = I_n$ where $R$ is an $n$ x $n$ orthogonal transformational vector. Also, the SVD of $Z = USV^T$. I'm trying to find the optimal ...
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1answer
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Why is that a risk averse consumer buys the optimum insurance when there is actuarially fair insurance?

I've asked the same question at the Quantitative Finance StackExchange. Consider the following example: "As a risk-averse consumer, you would want to choose a value of x so as to maximize expected ...
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3answers
41 views

Minimizing sum of squared distances from point to spheres

Given some spheres with known radius and known origin in three dimensional space, I want to find the point P that lies "closest" to all these spheres. The meassure of closeness, I guess, will be the ...
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0answers
17 views

Regularized least squares

In Image Restoration, a true image f (in vector form)can be related to degraded data y through a linear model of the form $$y = Hf + n$$ where H is 2d blurring matrix and n denotes noise vector and ...
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1answer
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Find max and min of $F=ax^2+2bxy+cy^2$ when $x^2+y^2=1$

Find the Maximum and Minimum of $$F=ax^2+2bxy+cy^2$$ when $$x^2+y^2=1$$ The variables a,b, and c are just real numbers. I have attempted using partial differentiation in order to solve for the ...
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0answers
21 views

minimizing sum of squares

Say i have the following optimization problem. min $\sum\limits_{i=1}^m \parallel r - (y_i - Rx_i) \parallel_2^2$. where we are optimizing over $r \in R^n$ and also $R \in R^{n, n}$ is given. Also, ...
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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 ...
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Linear Programming, Optimal Solutions

I posted the whole question to give some context, but my problem lies with (iv). I think you're meant to use a formula for the generalization of the optimal solution, but I'm not really sure what ...
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Inequality optimization, KKT condition.

So we have the problem: maximize $x^2+y^2$ subject to $x^2-y \leq3$ and $y\leq 1$. And I sorted out the KKT conditions for the problem (is here where the problem is?): $2x=\lambda _12x$, ...
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Using math for interior decorating with lamps

When I was in college, I owned three lamps and had a dark apartment. I kept trying to position them in different areas of the room, but it was still dark. Then I decided to model the problem with ...
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A function to maximize a variable and minimize the other partially.

So I need a function that has two input variables $x$ and $y$. Let' say that $x$ is a variable representing a score. And $y$ is a variable representing steps taken to reach that score. I need a ...
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1answer
197 views

Prove or disprove this argument

Let $L>0$ and let $\Omega$ be the set of all integrable functions from $[0,L]$ to $]0,+\infty[$. For all $\varphi, \psi \in \Omega$ define $\left \langle \varphi,\psi \right ...
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1answer
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Understanding optimization on non-compact region

Say we have $f(x,y) = x^2 e^{-x^2 - y^2}$ and we want to optimize it over $\mathbb{R}^2$. The minimum value is $0$ since $f(x,y) \geqslant 0$; the question is whether a maximum value exists or not. ...
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Linear programming to find minimal additive and multiplicative factors

Consider samples $\{x_i,y_i\}$ with $x_i\in\mathbb{R}^N$ and $y_i=\pm1$ and additional $z\in\mathbb{R}^N$. Can one use linear programming to find the minimal $m>0$ and minimal $\epsilon>0$ (e.g. ...
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2answers
169 views

Google Code Jam's Cookie Clicker Program…

Today, the Google Code Jam's cookie clicker problem was something like this. Problem In this problem, you start with 0 cookies. You gain cookies at a rate of 2 cookies per second, by ...
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45 views

The minimum of $\dfrac{x(1+y)+y(1+z)+z(1+x)}{\sqrt{xyz}}?$

Can we use $AM\geq GM$ inequality to find the minimum of $\dfrac{x(1+y)+y(1+z)+z(1+x)}{\sqrt{xyz}}?$ I can find out that minimum is $6$, but can we use $AM\geq GM$ to show this?