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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Product of numbers in any two cells sharing a side is $2$

In a $3\times 3$ square, every cell has a positive number. The product of numbers in any two cells sharing a side is exactly $2$. What is the minimum sum of all the numbers? We may color the cells in ...
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0answers
16 views

Lagrange multiplier method

I am doing some data mining algorithm self learning tutorial. I came up with a problem which I need your help to solve. In order to minimize the resource consumption, a car manufacturer considers how ...
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39 views

What kind of an optimisation problem am I dealing with? [on hold]

I have a connected graph made up of $x$ vertices. Each vertex has a probability $p$. I want to determine the total probability in traversing as many vertices as possible, Edges have a certain cost to ...
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0answers
14 views

Existence and uniqueness of a maximum

Consider $\alpha \in [0,1]$, $\beta>0$, $\delta \geq 0$. Let $1\{...\}$ be the indicator function taking value 1 if the condition inside is satisfied and zero otherwise. Let $$ f(x,y;\alpha, \beta, ...
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22 views

How do you find the minimum distance between two skew planes?

I know how to find the minimum distance between 2 skew lines. Does that help me with skew planes?
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1answer
16 views

How does $\mathcal{A}\cup \mathcal{B}$ indicates that there is at least one augmenting path on $\mathcal{A}$?

I had an exam and there was the following question: $\mathcal{A}$ and $\mathcal{B}$ are matchings in a graph $G$, with $|\mathcal{A}|< |\mathcal{B}|$, study the graph formed with the edges of ...
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1answer
24 views

What is the best choice given a probability and a cost for each choice?

I've been dealing with this problems for a few hours now and think I could use some outside help. The scenario is the following: We are given different choices with each one having a probability of ...
2
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1answer
29 views

Derivatives defined on a discrete state space

Ive been looking at certain economic papers, and optimal control papers. They define a state variable, $\omega$, which follows a discrete time Markov Chain. Then they define a utility function ...
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1answer
15 views

Truncated geometric progression on the complex unit circle - how to minimize the maximum real value

Let $a = \text{e}^{i 2 \pi k}$, and let $n$ be a natural number. Then I have a set defined as follows: $S = \{ \text{Re} (a), \text{Re} ( a^2 ), \ldots, \text{Re} (a^n) \}$ I want to minimize $T = ...
0
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4answers
39 views

If $2000 m^{2}$ of material is used to to construct a box…,then what is the largest possible volume of the box?

If $2000 m^{2}$ of material is used to to construct a rectangular box with a square base and an open top,then what is the largest possible volume of the box? I put an equation for the volume : $V = ...
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3answers
31 views

Optimization Problem multivariable calculus or single variable

Problem is that a right circular cylinder is inscribed in a sphere of radius a .What is height of cylinder when its volume is maximal ? As per suggested by answer i attempted Any hints please ? ...
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26 views

Solving simple decision-making model over multiple periods

Consider the following model. Each period t=0,1,..., an agent makes an effort $x\in R_+$ to solve a problem. The value from solving the problem is $V>0$. The relationship between effort and ...
2
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1answer
27 views

Optimization Word Problem AP Calculus Final

A large window consists of a rectangle with an equilateral triangle resting on its top. If the perimeter of the window is 33 feet, find the dimensions of the rectangle that will maximize the area of ...
2
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1answer
34 views

Description of a constraint for a mixed integer program.

Suppose we have 100 items that are labelled from the set $P = \{A, B, C, D, E\}$. My constraints are as follows: I want to choose exactly seven items. The choice should have at least one item of ...
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0answers
27 views

Why is this conic dual problem infeasible?

The problem is: $$\min \ x_2 : Ax -b = [x_1 \ 2x_2 \ x_1]^T \ge_{L^3} 0$$ where $L^m$ is the Lorentz cone. Which I found to have an optimal solution when $x_2 = 0$. I have shown that the conic ...
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0answers
24 views

Convex optimization with groups

I am relatively new to convex optimization and am looking to solve a resource allocation problem. I understand, that if my utility function is concave the following problem constitutes "an ...
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0answers
10 views

Meta Euler Lagrange

Consider the following generalized standard Euler Lagrange Problem which is to maximize the quantity $$ \int_{x_1}^{x_2} L(x,y,y', y'' ... y^{(n)}) dx $$ It can be solved by first determining a ...
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0answers
20 views

Is the ratio trace problem convex?

I have a ratio trace problem described as follows: $\arg\max_{w} trace((w^tAw)*inv(w^tBw))$, where A and B are full rank matrices. This problem can be solved via generalized eigenvalue problem. ...
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1answer
26 views

what is the convex hull of such a matrix cone?

A matrix cone is in the following form: $M: = \begin{pmatrix} 1 \\ x\end{pmatrix}\begin{pmatrix}1 & x^T\end{pmatrix}$ where $x\in F$ , let $F = \{x: x\in [l,u]^n\}$ How to express the convex ...
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64 views

Find a function that maximizes $\int_{0}^{1}f(x)\,\rm dx$ with given constraints

Find a function $f(x)$ that maximizes the following integral $$\max\int_{0}^{1}f(x)\,\rm dx\quad \text{s.t.}\quad \frac{d}{dx}ln(f(x))<0$$ $f(x)$ also continues, $f:[0,1]\rightarrow R$ and we ...
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2answers
50 views

How to find $\theta$ at which $d$ is the maximum possible?

I have an equation: $$d=\dfrac{v\cos \theta}{g}\left(v \sin \theta + \sqrt{v^{2} \sin^{2}\theta + 2gh} \right),\ g≈9.81 \dfrac {m}{s^{2}}$$ How to find $\theta$ at which $d$ is the maximum possible? ...
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21 views

Generating primal solution from dual solution of a LP

How to get the primal solution from a dual solution in general? For example, let the primal problem is $$ \text {maximize } 2r_1+2r_2-2c_1-2c_2 $$ where $$ r_1-c_1\leq1\\ r_1-c_2\leq1\\ ...
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7 views

Optimization of nested models

Hi everyone, I'm programming in Matlab and I have the following optimization problem in calibrating several nested model specifications. Summary: I have two models ($1$ and $2$, $1$ is nested in ...
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1answer
30 views

Minimization Problem (Calculus) [on hold]

Not quite sure how to go about solving this one: A post $1 \,\mathrm{m}$ high is $6 \,\mathrm{m}$ from another post that is $2 \,\mathrm{m}$ high. A line to be run from the top of one post to a ...
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2answers
35 views

Maximize the difference of two linear expressions

Given two $1\times N$ complex vectors h and g. I want to find a $N\times 1$ complex vector w(normalized to unit norm $ \Vert w \Vert^2=1$), which maximizes the following expression: $$w_0=\arg\max_w ...
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1answer
45 views

Find smallest $x$ such that $a^x \equiv b \bmod p$

Problem: How do we find smallest $x$ such that $a^x \equiv b \bmod p$, where $p$ is a prime and $1 \le b,a \le p$ and $a$, $b$, and $p$ are given and fixed. If there is no such $x$, how do we check ...
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2answers
26 views

Minimization problem, both terms in function positive

I have the following problem: Using the simplex method, minimize $z = 10x + 3y$ given the following conditions: $$2x + y \le 12$$ $$4x + y \ge 12$$ $$2x + y \ge 8$$ I've been told that minimizing ...
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1answer
12 views

MINLP optimization with matlab reaching different solutions every run

I have written a program for optimizing a set of generators. I have hourly price and cost data and need to figure out when a generator should run or just stay off. I describe the problem in more ...
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2answers
24 views

Calculus optimization cone dimensions [duplicate]

How can I determine the dimensions of a cone with surface area 1 and maximal volume?
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13 views

Is this function upper hemi-continuous?

I want to determine whether a function $f(x)$ always attains a maximum value for some $x\in X\subset\mathbb{R}$, where $X$ is a compact set and $x\ge 0$. Thus, I need to check whether $f(x)$ is upper ...
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1answer
58 views

Minimization of Variational - Total Variation (TV) Deblurring

Under the Linear Blurring Model - $ f = H \ast u $. I'm trying to calculate the Euler Lagrange of with respect to $ u $ of the functional: $$ E \left( u \right) = {\left\| f - H \ast u ...
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1answer
13 views

matlab MINLP optimization with ga

I have written a program for optimizing a set of generators. I have hourly price and cost data and need to figure out when a generator should run or just stay off. There are additional constraints but ...
0
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1answer
20 views

integer linear programing in matlab with the symbolic toolbox

I am writing a program to optimize a set of generators. I have hourly data and but dont want to necessarily optimize the whole time series. For a similar problem in the past I used the symbolic ...
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1answer
16 views

discrete nonlinear convex optimization relaxation over a dense set

Be a discrete nonlinear convex optimization problem $P$ \begin{align} \underset{x\in \mathrm{C}^n}{\mathrm{min}} \ \ \ f(x) \\ Ax=b \\ c \leq x \leq d \end{align} $C$ is a dense in $F$. Is solving ...
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0answers
32 views

Generalized Eigenvector Problem

I am reviewing a paper in which the solution to $$x =\max_\bf{v}\frac{\alpha \bf{v}^\dagger \bf{h}\bf{h}^\dagger \bf{v}}{\beta+\gamma\bf{v}^\dagger \bf{D} ...
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1answer
24 views

What is the demand function p(x)?

The marginal revenue of a certain commodity is $R^1(x)=-3x^2+4x+32$ where $x$ is the level of production in thousands. Assume $R(0)=0$ Find $R(x)$. What is the demand function of $p(x)$? I took the ...
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1answer
50 views

How can I determine the center of this circumference?

I have the following question: if I have an irregular symmetric polygon, how can I determinate the circumference with the least area that contains this polygon? I believe (in case that the polygon ...
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0answers
28 views

Notion of complex optima

Consider the function: $$y = \frac{1}{3}x^3 + x$$ Suppose we wanted to determine its local optima, but instead of looking at local optima with domain $R$ we instead consider domain $C$ and range ...
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Proof of nearest integer equality

Let $N(n)$ be the nearest-integer function undefined on half-integers. There are many valid ways to define $N(n)$, I like to choose $N(n) =\arg \min_{z \in \mathbb{Z}} |n-z|$. Consider the function ...
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0answers
26 views

Finding saddle points with some optimization

Consider a function that has multiple saddle points e.g. $$(x+y)(xy+xy^2)$$ We know for functions of two variables we can use the 2nd derivative test to locate saddle points $$Saddle ...
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1answer
15 views

KKT for not convex problems

In my optimization course we learned something about KKT for not konvex problems: $$min \; f(x)$$ $$s.t. \; c(x)=0$$ $$d(x)\geq 0$$ $$f(x): \mathbb{R}^n\rightarrow \mathbb{R}$$ $$c(x): ...
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0answers
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Optimal value of decision variable leads to inconsistency

$\epsilon$ is a random variable with support in $(0.8,0.95)$ and pdf $f(\epsilon)$. The following equation arises out of a business problem: $ENP=800*A*E(\epsilon)+ 9000 - ...
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0answers
10 views

How to express these in AMPL. [closed]

max $(\prod_{t=1}^Tx_t)^{1/T}$ which is also the geometric mean of vector ${x}$, i.e., geo_mean($x$). How can I express this in AMPL?
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Minimization using logarithmic barrier function

I'm thinking of the quadratic problem(QP) \begin{align} &\underset{x\in \mathrm{R}^n}{\mathrm{Minimize}}\ \ \ \frac{1}{2}x^\top{}Qx + f^\top{}x\\ &\mathrm{subject\ to}\ \ \ \ a_ix \leq b_i\ ...
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2answers
45 views

Minimizing the Frobenius Norm

I would like to minimize the following expression with respect to matrix X: $$\left \| A-BX \right \|_{F}$$ where A and B matrices are given and all the matrices have positive integer elements. Any ...
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0answers
24 views

Minimization problem involving a set of prime numbers and modular arithmetics

I'm a student working for curiosity on a general minimization problem where I suppose that there is no efficient algorithm for solving it. I'd like to ask for your valuable advice. Let $P$ be a set ...
0
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0answers
27 views

How does this optimization problem satisfy Karush-Kuhn-Tucker Conditions?

I am following Andrew Ng's course notes on Support Vector Machines at: http://cs229.stanford.edu/notes/cs229-notes3.pdf There is something in these notes which I do not understand. SVM's basic ...
0
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1answer
73 views

A binary min-max optimization problem

I encountered a very special optimization problem for a practical application. We have a variable $$\mathbf{s}=(s_1,s_2,s_3, s_4)^T$$, where $s_i$ can only take $1$ or $-1$, and we also have a ...
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0answers
22 views

What is $F_P$ and $E(P)$?

I'm reading Handbook of Graph Theory: At this section, he speaks about $F_P$ and $E(P)$. It's not really clear what they are. I guess there is enough context for someone to answer me but if ...
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32 views

closed-form solution for this constrained optimization

I want to find a closed-form solution for the vector $w=\left[\begin{array}{c} c\\-b \end{array}\right]$where $c$ and $b$ are column vectors, such that the following MSE is minimized: $\begin{align} ...