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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Finding an integral's max and min

I've been asked to find the max and min of the following: $F(x)=\displaystyle\int_0^{2x-x^2}\!\cos\dfrac{1}{1+t^2}\mathrm{d}t$ I tried applying the Fundamental Theorem of Calculus (taking the ...
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calculus of variations or optimize over function form

I have a question about optimizing the following quantity over function form . Given unknown function $f(\theta)$ such that $f(\theta)\geqslant 0$ and $\int f(\theta)d\theta\leq \infty$. And ...
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Lagrange Optimization

I would like to ask for the optimization problem: $$\max_{x,y} g(x,y)$$ st. $x+y=1$ Would there be any difference if we formulate the problem as: $$g(x,y) + \lambda(1-x-y)$$ as opposed to ...
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Maximization of Harmonic mean

Suppose x is a vector of size N with positive real elements sorted in decreasing order. Is it possible to find the analytical solution (no iterative solution) to the optimum value of M (1<= M ...
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How to describe $\lbrace \mathbf{x}\in \mathbb{R^n}: |x_j|\le1 $ for$ 1\le j\le n \rbrace $ in terms of $x_j=x_j^+-x_j^-$

How to describe the set $A$=$\lbrace \mathbf{x}\in \mathbb{R^n}: |x_j|\le1 $ for$ 1\le j\le n \rbrace $ in terms of $x_j=x_j^+-x_j^-$ where $x_j^+\ge0$ and $x_j^-\ge0$ The answer says: $B$=$\lbrace ...
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Identify if optimization problem is convex or non-convex?

I have formulated optimization problem for building, where cost concerns with energy consumption and constraints are related to hardware limits and model of building. To solve this formulation, I need ...
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What is a good resource for a more intuitive/flexible understanding of optimization

Take the following example of optimization: $$cost = 10*x + 20*y$$ Where x = cans of soup, y = cans of juice It is easy to see in this scenario what we need to do in order to minimize cost. Just ...
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Calculus optimisation with the speed formula

For a ship travelling at ${x}$ km/h the running cost in £ is ${(x^2 + {13500\over x})}$ per hour. Find the speed that minimises the cost of a 300km journey. The speed formula is ${speed = ...
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A basic minimization problem

Does the following function has a global minimum $$f(x) = \frac{2x +1}{1-e^{-(1-\alpha) x}}$$ where $x$ is a positive real number. for $0< \alpha < 1$
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Optimization with trace and eigenvalues

Let $M \in \mathbb{R}^{n \times n}$ be a symmetric matrix with given eigenvalues $\lambda_1,\lambda_2,\cdots,\lambda_n$ with $\vert\lambda_1\vert > \vert ...
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Optimization: second order condition

This is the condition Where $L(x, \mu\,\lambda)$ is the Lagrangian function in a given point that satisfy the first order condition. Problem $ min (-4x -y)$ $ -x^2 -y^2 +1 <= 0 $ $ y- 1 ...
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Duality and the Positive Lagrange Multiplier

Suppose I have the following optimization problem: \begin{align} \min &f(x) \\ & f_1(x) \leq 0 \\ & \vdots \\ & f_k(x) \leq 0 \\ & g_(x) = 0 \\ ...
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Inapproximability of Combinatorial Optimization Problems

I've been reading the "Inapproximability of Combinatorial Optimization Problems" by Luca Trevisan (see: link). On pages 3-4 it mentions that a polynomial time algorithm for 3SAT would exist if there ...
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51 views

Optimization problem on graph with weights on nodes and edges

I am solving a problem where I have a complete undirected graph with weights on the nodes and on the edges. The weight on the node represents a profit that you obtain if you select that node. The ...
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44 views

Finding minimum of a two variable function

Let $D=\{(x,y)\in\Bbb R^2:1\le x\le1000,1\le y\le1000\}$. Define $$f(x,y)={xy\over2}+{500\over x}+{500\over y}$$ Then the minimum value of $f$ on $D$ is Finding $f_x=\frac y2-{1000\over ...
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Want to factorize one matrix into three, with L1 regularization, which optimization algorithm to choose?

I need to factorize one matrix $R$ into three component: $ R = P^TAQ $, in which I want to apply L1 regularization on $A$ to encourage sparsity, and apply L2 regularization on $P$ and $Q$ to prevent ...
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Optimization Problem Maximize $z= 60x_1+20x_2$

Restate the absolute value constraint as a combination of two linear constraints: I know how to find the optimal solution (std form, canonical form, simplex algorithm ...etc) I don't know how to put ...
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Optimization: determining maximum volume of a tube

I am unsure of how to go about solving this, the context is that there is a rectangular piece of paper with a perimeter of 100 cm that is to be rolled to form a cylindrical tube. The question wants to ...
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51 views

Given $f(x)=4-e^{-cos(x-2)}$, find the maximum value of $f(x)$ in the range $[-2,0]$.

Given $f(x)=4-e^{-cos(x-2)}$ Find the maximum value of $f(x)$ in the range $[-2,0]$. $\forall a \in \mathbb R$, $e^a>0$ Hence, the maximum of $f(x)$ will occur when $e^{-cos(x-2)}$ is a minimum. ...
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About Dual Simplex Method

I have a question about Dual Simplex Method (for minimization problem). While we are solving the method, when we obtain a non-negative $\bar b$, we stop the algortihm. But in addition to $\bar b ...
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How to show this integer program with irrational data has no optimal solutions.

I want to show the integer program with irrational data max$\{x_1-\sqrt{2}x_2:x_1\leq \sqrt{2}x_2,x_1\geq 1,x\in Z_+^2\}$ has no optimal solution, even though there exist feasible solutions with value ...
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41 views

Do standard gradient descent methods work on complex variables

I am currently whishing to optimize a function numerically $f(z)$ where $z \in \mathbb{C} $ ($f(z) \in \mathbb{R}$) . I am doing this via numerical packages (specifically scipy in python) and I have ...
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Optimization: Minimizing the cost of pipeline over land

I have the question "A Gas Outlet is one one side of a river 120 m wide. It is exactly 300 meters downstream and across the river from a cottage. A gas line is to be constructed to join the outlet to ...
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Deriving optimal time to change

I am working in economics and I am trying to build a model that take into account the fact that indivudal can take a decision once in their life time that changes the value of a parameter R. To be ...
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Maximum of $e^{-x} \sin(x)$, $x \geq 0$

What is the maximum of $e^{-x} \sin(x)$ for $x \geq 0$? Is there a closed-form solution? If not, what is a good approximation $y$ such that $\text{max}_{x\geq 0}e^{-x} \sin(x) \leq y$?
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Comparing a maximization to an integration with economics application

This seemingly simple question has interesting interpretation in economics, but I only state the mathematical problem here. Suppose $B(0)=C(0)=C'(0)=0$, $B'(\cdot)>0,\ B''(\cdot)\leq0,\ ...
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What is the range of $y$ if $x+y+z=4$ and $xy+yz+xz=5$ for $x, y, z \in\mathbb{R}_+$

What is the range of $y$ if $x+y+z=4$ and $xy+yz+xz=5$ for $x, y, z \in\mathbb{R}_+$ How to explain the following method? Let $x=z$ then: $$2x+y=4\quad;\quad 2xy+x^{2}=5$$ $$\implies \left( ...
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Conditions for a totally unimodular coefficient matrix of a Multi-Commodity-Minimum-Cost-Flow-Problem

I'm considering the following Multi-Commodity-minimum-Cost-Flow-Problem: This leads us to a coefficient matrix $A$ with $N$ donates the incidence matrix of a directed graph and $I$ is the ...
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Newton's optimization: why wouldn't the results be good if we take $x_0$ between $(-\frac{1}{\sqrt 3},\frac{1}{\sqrt 3})$ for $f(x)= x^3- x-1\;?$

I was reading about the numerical method of Newton for finding the roots of $f(x)$ in Thomas' Calculus ; the author presented an example Find the x-coordinate of the point where the curve $f(x)= ...
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Is closed convex set with finite number of extreme points convex polyhedron

I have this simple question related to convex set and convex polyhedron. As the content in the title, it's basically my question: Is closed convex set with finite number of extreme points convex ...
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What is the maximum number of boxes that can fit in a rectangular container

I'm looking for an algorithm for the following question: What is the maximum number of boxes with sides a,b,c that can fit in a rectangular container with sides $x$,$y$,$z$. For example, the ...
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Find the smallest possible value of $a^4+b^4+c^4-136abc$

Let $a$, $b$, and $c$ be real numbers such that $a+b+c=-68$ and $ab+bc+ca=1156$. The smallest possible value of $a^4+b^4+c^4-136abc$ is $k$. Find the remainder when $k$ is divided by $1000$. I ...
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Weakly lower semicontinuous functional on a bounded closed and convex set

Let $J$ be a sequentially weakly lower semicontinuous functional on $C$ with values on the real line. Moreover let $C$ be a bounded, closed and convex subset of a Hilbert space $H$. Is it true that ...
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Using Lagrange multipliers to identify the Extremes of function $f(x, y)=x-y$, under condition $g(x,y)=x^2 + y^2 - 4=0$

I'm studying in preparation for a Mathematical Analysis II examination and I'm solving past exam exercises. If it's any indicator of difficulty, the exercise is Exercise 3 of 4, part $b$ and graded ...
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Existence of non-negative solution to a diagonally dominant tridiagonal system

Let $D \in \mathbb{R}^{n \times n}$. having only non-negative entries, strictly diagonally dominant (both row-wise and column-wise), tridiagonal. Show that $$\exists\; x \in \mathbb{R}^n \quad ...
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Maximise volume given inequality constraint on its dimensions without using Lagrange, KKT or Linear Programming

The problem (from Calculus for Business, Economics, Life Sciences and Social Sciences 12e): I found this and that, but they use Lagrange/KKT. What I tried: Girth $= 2w + 2h$ Maximise ...
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Maximal distance of a segment

Let a path enclosed by lines as illustrated in this figure Fig. knowing that the widths of the two paths are $\ell$ and $\ell^{\prime}$ respectively. What the maximum distance $x^{\star}$ to be able ...
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KKT conditions for nonlinear problem

I need to state the KKT conditions for the following problem: Minimise $x_1^2 + 2x_2^2$ subject to $(x_1-1)^2 + x_2^2 \le 1$ and $x_2 = 1$. I have that these conditions are: $f(x^*) \le 0$ ...
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Can lagrange multiplier(Kuhn tucker multipliers?) change in corner solution?

If we want to maximize $f(x)$ subject to two constraints, one which says that $x< c$ $c>0$, and another that says that $x\geq 0 $. Assume there are no problems with either $x=0, x>0$ or $\mu ...
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Find min $ax+by+cz$ subject to $0 \le y \le 1, 0\le z \le 1$ and $\max(0,y+z-1) \le x \le \min(y,z)$

I am seeking an elegant way to solve the following problem. Let $a,b,c$ be constant real numbers. Find min $ax+by+cz$ subject to $0 \le y \le 1, 0\le z \le 1$ and $\max(0,y+z-1) \le x \le \min(y,z)$. ...
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Feasible set and level sets

Consider the following problem: Minimise $x_1^2 + 2x_2^2$ subject to $(x_1)^2 + x_2^2 \le 1$ and $x_2 = 1$. Sketch the feasible set and the level sets of the objective function, and determine an ...
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Optimization on a grid

I worked a lot on defining the problem so I will be grateful to get input if i'm not clear enouth and I will fix the question. We have a grid made out of uniform points on $[x,y],$ $x,y\in[0,1],$ ...
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what is the maximum value of $x(x+y)^3$ given that $x^2+y^2/d=1$?

Without losing generality, we can assume $x,y\geq 0$ and then use $x$ to replace $y$. This is complicated. Instead I use $x=\sin\theta$, $y=\sqrt{d}\cos\theta$, and then I only need to get the ...
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Solving SVM classifier with two weight vectors

I am trying to implement a paper that basically proposes the following way to train two classifiers on some data with two types of labels. I do not know how to tweak existing solvers for SVM to do the ...
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Design a circuit for a function

I am so confused on this problem. We are given a function $f$ and told to design a circuit that has four inputs labeled $b_3,...,b_0$, and an output $f$, where $f = 1$ if the 4-bit input pattern is a ...
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Multivariable optimization for time to build a ship in a game, and maybe some possible application in “everyday” life

I precise first that english is not my monther tongue and I may will not be as clear as I would like, just ask me question if you need, thank you. I am playing a game (Galaxy Empire) for a while, ...
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Envelope theorem for Conditional value at risk

Let $X$ be a Gaussian random variable and suppose $f(p,X)$ is a strictly increasing and continuous function in $p \in \mathbb R$. Conditional value at risk is defined in the following way ...
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MInimum value of the sum of three numbers

if product of three numbers is 1, how do you find the minimum value of the sum of those three numbers? i tried to find the possible values of the numbers that would give a product of one but I'm not ...
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Minimising logistic loss function to find optimal matrix

Please take a look at this paper on classifying triples (re link prediction): http://arxiv.org/pdf/1510.04935v2.pdf The question is about how to solve equation 2 using stochastic gradient descent. It ...
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Optimising volume of a truncated cone

Given a slant height h and radius r1 how can I find a truncated cone with largest volume?. Is there any calculus involved in ...