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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The maximum volume of Tetrahedron

A optimization problem: Get the maximum volume of a tetrahedron its 4 vertices on the surface of cube whose edge length is 1 . From the geometrical intutition ,we can get : Selecting ...
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Orthogonal Vectors in a 2D Lattice with minimum area

I came across an interesting problem in my research (not a mathematician). Here it goes: Suppose, there is a 2D lattice $\Lambda$ in the X-Y plane with basis vectors $\vec{a}$ and $\vec{b}$, which ...
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optimization of nasty expression with nice symmetry between expressions

Consider the function $\ f(x,y,z,\rho_a,\rho_b)=$ $ \log \left(1+ (x+ \rho_ay)^2 + \frac{(z+ \rho_by)^2}{1+(x+ \rho_by)^2} \right)+ \log\left(1+ (x+ \rho_by)^2 + \frac{(z+ \rho_ay)^2}{1+(x+ ...
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95 views

gradient descent - cost reduces and then increases

I am optimizing a function using Gradient Descent. The learning rate is fixed. First for few iterations the cost decreases after that it starts increases. What is the reason for this?
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64 views

Taking the dual of this non-standard linear program

I am just beginning to learn linear programming have a question about taking the dual of a non-standard LP specifically the one below: $\min M\\ 2x_1 + 3x_2 + 4x_3 \leq M \\ 2x_1 - x_2 + x_3 \leq M\\ ...
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517 views

Why does SVD provide the least squares solution to $Ax=b$?

I am studying the Singular Value Decomposition and its properties. It is widely used in order to solve equations of the form $Ax=b$. I have seen the following: When we have the equation system $Ax=b$, ...
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60 views

Please explain to optimization of maximizing revenue.

I'm finding a hard time understanding optimization. The equation I'm having a hard time figuring out is Maximizing revenue. Suppose the quantity demanded per week of a certain dress to the unit ...
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54 views

optimization calculus, cannot find restriction, I am stuck!

A capsule formed by a cylinder and two half spheres on the top and the bottom have a minimal volume of $\pi / 12$. What is the height and radius of the capsule? The volume is $$\pi r^2 (4 \times \pi/3 ...
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43 views

Maximum of $A^tB^{1-t}$

What is the maximum of $A^tB^{1-t}$ for positive $A$ and $B$ and for $t \in [0,1]$. I think the maximum occurs at end points either $t=0$ or $t=1$, right?
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225 views

Optimization of a rectangular container

A rectangular sheet of tinplate is $2k$ cm by $k$ cm. Four squares, each with sides $x$ cm, are cut from its corners. The remainder is bent into the shape of an open rectangular container. Find the ...
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22 views

Distributing years in a Leap Year system.

First let's take our Leap Year system: a Leap Year is a year of 366 days, as opposed to normal years of 365 days. It occurs every 4 years, except if the year is divisible by 100, and not divisible by ...
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When result of max of min problem is equal to min of max problem

Let's assume there are two functions $f(x)$ and $g(x)$. I want to know when the optimal $x$ of max of min of $f(x)$ and $g(x)$ is not equal to optimal $x$ of min of max of $\frac{1}{f(x)}$ and ...
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98 views

Minimum volume cone.

What would be the radius and the altitude of a right circular cone that circumscribes a sphere with a radius 8 cm if the volume of the cone is to be minimized? Here is my rough sketch; My idea is ...
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47 views

Independent variables in optimization

I'm not sure whether I'm asking very obvious/stupid question, but essentially I'm looking for references. I am looking for the notion of independence in the context of optimization problems (I am ...
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55 views

Is the set of all projection matrices a convex set?

The set $\phi=\{P| P^2=P\}$ contains all projection matrix. Is this set $\phi$ convex?
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372 views

Convergence of Steepest Descent: Proving Orthogonality of Exact Line Search Steps

For the following assume that $f(x) = 0.5x^TQx - b^Tx$, where Q is symmetric, positive definite $n$ x $n$ matrix, and $b$ belong to $R^n$. Assume that $x^*$ is the unique local minimizer of $f(x)$ and ...
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What is the minimum value of $(\tan^2)(A/2)+(\tan^2)(B/2)+(\tan^2)(C/2)$, where $A$, $B$ and $C$ are angles of a triangle

What is the minimum value of $(\tan^2)(A/2)+(\tan^2)(B/2)+(\tan^2)(C/2)$, where $A$, $B$ and $C$ are angles of a triangle? I know that the sum of the angles is $\pi$, but I am unable to find the ...
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Finding disjoint intervals from Cantor Set

Consider $C$ the classic Cantor ternary set in $[0,1]$. I am interested in the following problem: Find the largest constant $0<k<1$ such that it is true that any interval $[a,b] \subseteq ...
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125 views

Transversality conditions in optimal control with non-linear final pay-off

I have a doubt regarding transversality condition in the case of a non linear final pay-off. For instance, I need to solve with the Pontryagin maximum principle the following optimization problem ...
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44 views

Describing the minimizers of this function

Consider the following continuous function over $x$, with $a,b>0$: $$f(x)=\begin{cases} ax-\sqrt{x} & \text{for }x\leq b^{2}\\ ab\sqrt{x}-b & \text{for }x>b^{2} \end{cases}$$Note that ...
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97 views

Finding the MLE estimates of a beta, binomial hierarchical model

Consider $M$ observations ($x_i$, $n_i$) where $x_i$ is a realisation from $X_i \sim \mbox{Binomial}(n_i,p_i)$ and $p_i$ is a realisation from $P_i \sim Beta(\alpha, \beta)$. I would like to find the ...
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236 views

Proof of Simplex Method, Adjacent CPF Solutions

I was looking at justification as to why the simplex method runs and the basic arguments seem to rely on the follow: i)The optimal solution occurs at some vertex of the feasible region (CPF points) ...
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28 views

How can I find the maximum value of $x_6-x_1$ subject to the two constraints $\sum_{j=1}^{6} x_j^2$ and $\sum_{j=1}^{6} x_j = 0$

I currently have six variables $x_1, x_2, x_3, x_4, x_5, x_6$. I am trying to determine how large I can make the difference $x_6-x_1$ while satisfying the constraints: $\sum_{j=1}^{6} x_j^2 \leq 1$ ...
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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 ...
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78 views

Find all $(x,y)$ pairs

Find all $x$ , $y$ $\in$ $\mathbb {R^+}$ such that for all $\epsilon>0$, $$x \left(\dfrac{\ln \left(1+\dfrac{1}{x}\right)-2\epsilon}{\ln xy-(1-\epsilon)}\right)\geq \left(\dfrac{\ln ...
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28 views

Problems of choice with immediate effect under conditions of uncertainty

A farm, in order to commercialize a product, may select between two intermediaries, which offer the following conditions: A) A fixed cost of 2000 dollars for any level of production; B) A variable ...
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115 views

A simple dual problem in economics: profit v.s. cost

The setup is simple but a bit lengthy. Please bear with me. Suppose that I have a production function $F(K,L)$ that is: constant return to scale; increasing in each factor: $F_K>0$, $F_L>0$ ...
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99 views

Optimal Placement of Points inside a Set

Say I have to place N points in $\mathbf{R}^2$ inside a circle of radius $R$. I want to position them so as to maximize the sum of nearest distances i.e. solve the following problem \begin{align} ...
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125 views

Improving the Edit Distance Algorithm

I applied an Edit Distance Algorithm for similarity between two strings over the lowercase latin alphabet, where the first string has length $m$ and the second length $n$. However I want to improve ...
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102 views

Meal Platters Optimization Problem

Mark has to buy hamburgers, hot dogs, and pig's feet for an event. The restaurant he is purchasing from offers two Platter options. Platter A comes with 4 hamburgers, 3 hot dogs, and 2 pig's feet. ...
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45 views

Conversion of a general linear program into a standard linear program

I am trying to teach myself the basics of optimization of linear programmes, for example the following question: How do I tackle such a question?
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25 views

Maxima and Minima, If s = 60, what should the side of the cut out be…

A square piece of steel, s cm on a side, is to made into an equipment chassis by cutting equal squares out of the corners, folding up the sides , and welding the seam to form a pan. $A)$ If $s=60$, ...
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Find a function where the mode is the minimum

Let $a_i\in\Bbb R$ some collection of data points where $0\le i\le n$. Define the function $$f(x)=\sum_{i=0}^n(x-a_i)^2$$ It is clear that the minimum value of $f$ occurs when $x$ is the mean of ...
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106 views

Functions of 2 variables and applications to economics

Given the production function $Q := \sqrt K + L^2$, determine the optimal level of production and the relative demand of the two inputs capital $K$ and work $L$. The cost of a unit of capital ...
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Help finding value of N that minimizes a sum

Suppose we have the following inequality: $\sum\limits_{k=N+1}^{1000}\binom{1000}{k}(\frac{1}{2})^{k}(\frac{1}{2})^{1000-k} = \frac{1}{2^{1000}}\sum\limits_{k=N+1}^{1000}\binom{1000}{k} < ...
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Time-optimal control - Coupled system of equations, control to the origin

I want to find the time-optimal control to the origin $\underline 0$ for the following: $\dot{x}_1=-3x_1 + x_2$ and $\dot{x}_2 = x_1 - 3x_2 + u$, $|u|\leq 1$ How do I go about doing this. I ...
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35 views

Differentiable L-1 Regularization

In machine learning we are often faced with optimization problems where we want to minimize some energy function using L1 regularization over some of the parameters, e.g.: $$ E(a,w) = [\text{sum of ...
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Are the Taylor polynomials of a function the results of a minimization problem?

Here is an example to better explain my question. Consider the function $f(x) = \cos(x)$. I want to approximate it in the set $[-\pi; \pi]$ using a polynomial $g(x) = a + bx + cx^2$ of order $2$. ...
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Least sum of distances

Problem: Let $A, B, C, D$ be points in a $3$-dimensional space. Find the point $X$ that minimizes the sum of the distances $AX+ BX + CX + DX$. Context: During a course, I was assigned a ...
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102 views

Unconstrained optimal control - $J = \int_0^{t_1} (x^2 + ux + \frac{1}{2} u^2) dt$

I've been given the following problem to solve, and I'm having a lot of difficulty in understanding what I can do. The system $\dot x = x + u$, where $u = u(t)$ is not subject to any constraint, ...
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135 views

Maximizing a Strictly Convex Quadratic Function Over a Convex Set

I need to solve a special case of non-convex QCQP (Quadratically Constrained Quadratic Programming) with the general form: $$ \begin{align} & \max {x^T}{A_o}x \\ & \text{s.t.}\left\{ ...
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Optimization, point on parabola closest to another point

The problem is as follows: Find the point on the parabola $2x=y^{2}$ closest to $(1,0$). I was highly surprised because I ended up with the correct answer doing something completely different than ...
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When can I solve in closed form this curve fitting problem?

I have $n$ real values $x_1,x_2,\ldots,x_n$ and $n$ real values $y_1,y_2,\ldots,y_n$; then I have a function $f(x,\boldsymbol\theta)$ from $\mathbb{R}$ to $\mathbb{R}$ and depending on $m$ parameters ...
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74 views

Monotonically decreasing function for multiplication product?

I have a set of numbers $S = [100,999]$ for which I want the maximum product $p$ such that $p = a \times b$ for all $a,b \in S$ also fulfilling some condition $C$. I would like $p$ to be monotonically ...
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62 views

Optimization problem in the standard form

Let $x\rightarrow x^{T}c$ be an objective function of an optimization problem in the standard form, for which the optimal solution doesn't exist. Does then exist an optimal solution to $x\rightarrow ...
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Convert a problem o minimizing a function to linear programming problem in standard form

I have to 1) convert a problem o minimizing a function to linear programming problem in standard form. It is something new to me. Can somebody explain it to me? $$\min(\mathbb{R}^2\ni(x,y)\rightarrow ...
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111 views

Argument to “linearize” an objective function

I have this optimization problem on the variables $\lambda_\ell^+, \lambda_\ell^-$ such that $ \lambda_\ell^+ \geq \lambda_\ell^-$ with $\ell=1,\ldots,n$ , and fixed $P\in [1/(n+1),1]$ \begin{align} ...
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Find the minimum possible order at a restaurant for a party of n people

I want to find an efficient algorithm for determining the minimum possible order total for a party of n people at a restaurant, assuming that the items in the order are unique, and they will each ...
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450 views

Maximise the volume of an open triangular prism

An open container is to be constructed out of 200 square centimeters of cardboard. The two end pieces are equilateral triangles. The open top is a horizontal rectangle. Find the lengths of the sides ...
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55 views

Second order derivation optimization

Recently I am thinking about a problem that might be easy to answer but for me is a big challenge. Assume you have a function $f(x)$ that is second order derivative. So I am looking for a way to ...