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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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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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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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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Fitting a grid inside a circle, is the solution always symmetric?

Let us construct a grid consisting of rectangles of height $h$ and width $w$. When we place a circle of radius $r$ over this grid, there is a certain amount of rectangles that completely lie within ...
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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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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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How to do an optimization using asymmetrical loss functions (LINEX) for time series. [closed]

ong time Lurker, first time asking. For a research paper, i'm required to optimize some parameters of a certain function using an asymmetrical loss function, specifically LINEX and compare it to 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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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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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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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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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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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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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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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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Classify the stationary points of the following function

I´m asked to find the stationary points of the equation $$f(x,y)=2+y^{2}-2xy+\frac{81}{y^{2}}-\frac{81}{y}\sqrt{2-x^{2}}$$ I know that we should verify when does $\nabla f = 0$, but the resulting ...
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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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What is a closed chain (or circuit) that is used in solving a transportation problem (a special type of linear programming problem)?

What is a closed chain (or circuit) that is used in solving a transportation problem (a special type of linear programming problem)? I'm having some problems with it. Please clarify it. I have posted ...
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24 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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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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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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Why in the Uncapacitated Facility Location problem does each client not order all their goods from one location?

Here is the problem: A set of potential depot locations N = {1, . . . , n} has been identified. A set of clients M = {1, . . . , m} is known, each of which buys products that could be delivered from ...
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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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Reference request: Time-optimal trajectories

I am looking for some lecture notes or a textbook for time-optimal trajectories. Any help is greatly appreciated. I am having plenty of trouble with understanding switching POQ curves etc.
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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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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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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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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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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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Importance of checking corners in the interior point method?

I was asked to analytically find the global minimum of $$ f(x_1,x_2) = 4x_1^2-x_1x_2+4x_2^2-6x_2 $$ on the triangle with corners in (0,0), (0,1) and (1,1). My answer was marked wrong, and I am ...
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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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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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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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Maxima of a recurrence

The following recurrence has a maxima around $k = \lceil \log_d{n}\rceil$, where $n > 0$, $d > 3$: $$b(n,k) = {b(n-1,k-1)\over {d^{k-1}}} + (1-{1\over {d^{k}}})b(n-1,k)$$, where $0 \le k \le ...
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How do you determine the values of alpha, beta and rho in ant colony optimization? [closed]

I am implementing a software application that leverages the ant colony optimization algorithm and I'm having some trouble figuring out what the values of alpha (pheromone weight), beta (heuristic ...
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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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Optimization across markets - How can I solve?

I am unsure how to solve problems involving several markets and optimizing the price across all my markets. Note: I am looking to be pointed in a specific direction of study, not a solution to the ...
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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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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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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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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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How to find a function which maximizes a stochastic process containing sum?

Let $X=\lbrace X_t : t\geq 0\rbrace$ denote a Lévy process with initial value $X_0=0$. Let the process be sampled equally in time ($t_n-t_{n-1}=const.$). I am looking for the ...
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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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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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Ways to deal with Boolean constraint in optimization

For the optimization of $\text{min}_\alpha Q(\alpha)$, such that $\alpha_i \in \{0,1\}$, what will be popular way to deal with the Boolean constraint? Is there any methods to approximate the Boolean ...
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28 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 ...
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Weierstrass conditions, what does strong mean, and are both conditions required?

I have the Weierstrass condition: In order that the extremal $\bf{C^*}: x = x^*(t)$ give a strong local minimum to $\bf{J[x]}$ it is sufficient that: # ...
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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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Check my solution for optimization problem

A piece of wire 40 units long is to be cut into two pieces. One piece will be bent to form a circle; the other will be bent to form a square. Find the minimum and maximum values of the area. I found ...