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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How do I solve the following equations?

I have the following problem: \begin{align} Y &= A X \\ Y &= R \exp \left(j \Phi\right) \text{element-wise}\\ X, R, \Phi &\in \Bbb R \\ A, Y &\in \Bbb C \end{align} I know what A is, ...
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On solving min-max optimization problems in original ways (that is, avoiding the frenzy of differentiation)

As I see from the students I'm tutoring, once faced with a min-max problem, the average student is taken by the frenzy of differentiation. I would like to show that sometimes it is better to use ...
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1answer
28 views

What is a simple proof that something is np complete that does not use np completeness of something else?

What is a simple proof that something is NP complete that does not use NP completeness of something else? Every proof seems to reduce to something else being NP complete.
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Find optimal matrix to maximize the expected expression

I am interested of the following function with $Q$. $$f(Q)=h^TQh-\frac{1}{(h^TQh+1)^2}-g^TQg+\frac{1}{(1+g^TQg)^2}$$ where $h$ and $g$ are both given $N\times 1$ vectors. And $Q$ is $N \times N$ ...
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Find max cone inscribed parallelepiped volume

There is a right cone with H = 20 and R = 12. How to find the best inscribed parallelepiped parameters which would provide its ...
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39 views

Maximising the Area of a Cyclic Quadrilateral

In cyclic quadrilateral $ABCD$, $AB = AD$. If $AC = 6$ and $AB/BD = 3/5$, find the maximum possible value of $[ABCD]$. (Source: SMT 2014) If we let $AB=AD = 3x$ and $BD=5x$, from Ptolemy, we have ...
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The optimization problem of soft margin Support Vector Machine: How to interpret?

I try to understand what exactly we are trying to optimize in the case of Support Vector Machine problem, which supports soft margins. The original problem is posed first as, without soft margins ...
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Show that weak local minimum of a convex function $\mathbb{R}^N\rightarrow \mathbb{R}$ is its weak global minimum.

Show that weak local minimum of a convex function $\mathbb{R}^N\rightarrow \mathbb{R}$ is its weak global minimum. Does the same happen to strong minimums? I know that when $f$ is convex, then we ...
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2answers
37 views

Does being a local minimum imply a positive definite hessian?

If $p\in R^{m}$ is a local minimum of $F:R^{m}\rightarrow R$, then can we conclude that $\dfrac{\partial ^2F}{\partial x \partial x'}[p]$ is positive definite? I guess you guys answers have ...
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Bounded polyhedrons [on hold]

Given a bounded polyhedron $P=P(A,b)$ and with $x$ s.t. $Ax<b$, show: $\exists \ \alpha>0 \ \ \ \ \ \text{ s.t.}\ \ \ \alpha^Tx\leq1, \ \ \ \ \forall x \in P $ How I should proceed to prove ...
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Optimization - find the dimensions of a box as functions of volume - minimal surface area

Had a basic calculus course exam today. This was one of the problems: We have a rectangular box of a given volume V. Present the width, height, and length of the box as functions of V so that the box ...
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27 views

If we have $f(x)=e^x$, then what is the maximum value of $δ$ such that $|f(x)-1|< 0.1$ whenever $|x|<δ$?

If we have $f(x)=e^x$, then what is the maximum value of $δ$ such that $|f(x)-1|< 0.1$ whenever $|x|<δ$? I tried to solve this problem with delta-epsilon definition from the definition, 1 is ...
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1answer
39 views

conditions for $A +B$ to be semi-definite.

Suppose $A$ is a positive definite real matrix, and $B$ is symmetric and real matrix with $B_{ii}>0$. Are there conditions on $\sup_{j}|B_{ij}|$ that can guarantee $A+B$ is semi-definite. ...
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1answer
17 views

Optimization, minimizing volume of an open top box given the volume

The question is: An Anacleto box is a square open box: the bottom is a square, the four sides are equal rectangles, and there isn’t anything on the top. The box should have a volume of 1000 ...
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1answer
29 views

Using Lagrange multipliers to find the extrema of $f(x,y) = e^{2xy}$ subject to $x^2+y^2 = 16$

Find the maximum and minimum values of $f = e^{2xy}$ with respect to $x^2+y^2 = 16$. Using Lagrange multipliers, $\nabla f = \lambda\nabla g$. Therefore, the constraints are the following: ...
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1answer
35 views

Minimizing a function in Mathematica

Edit: I simplified the function using $\textbf{Simplify[...]}$ How can I minimize this function of $x$, where $l$ is a positive constant? $$\frac{1}{2} \sqrt{\frac{x}{l}+\frac{l}{x}+4 x^2-2}$$ ...
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3answers
40 views

What is the closest point on the graph of $x^2-y^2=4$ to the point $(0,1)$?

If there is a point (a,b) on the hyperbola $x^2-y^2=4$ that is closest to the point (0,1), then what is this point? I prefer this problem to be solved with the knowledge of the the maximum and ...
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1answer
34 views

An equation with multiple solutions: finding the maximum of the function of the solutions.

Possibly, this is a bad (stupid) question, but sometimes some discussion helps. I have a fixed point equation (involving $\tanh$). I would like to derive the dependency of some function of the fixed ...
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1answer
30 views

how to find norm of the operator Ax(t) = cos (t)*x(t)?

Please explain me how can I get norm of this operator: $Ax(t)=\cos(t)x(t)$ where $A\colon C[-\pi/2,\pi/2] \to C[-\pi/2,\pi/2]$. Thanks
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Rounding half-integers to approximate product

Given two half-integers $a,b\in\mathbb Z+\frac 1 2$, the integers nearest to each is given by $N_a=\{a-0.5,\ a+0.5\}$ and $N_b=\{b-0.5,\ b+0.5\}$. Is there a general method to find, for the two given ...
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25 views

linear programming problem - how much additional resources should I buy?

I have the following linear optimization problem: Maximize $$\sum_{i=1}^{n}x_{i}B_{i}$$ subject to the constraints $$a_{11}x_1+a_{12}x_2+\cdots+a_{1n}x_n \le l_1$$ $$...$$ ...
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Solving nonlinear system of ODEs

I have the following system of differential equations: $$ \begin{cases} \frac{dx}{dt} = (1 - y) x - 0.4 xu \\ \frac{dy}{dt} = (x - 1)y - 0.2yu \\ \psi_1' = - \frac{dH}{dx} = (-1 + 0.4u)\psi_1 + y ...
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+50

Maximizing a Concave function over a Non convex constraint set

I have been working on a problem which involves maximizing a concave function over a non convex constraint set. The problem is the following $$max. \frac{1}{2}x^TBx\\ s.t.\ x_i(1-x_i)=0,\ \ ...
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1answer
14 views

Constrained nonlinear optimizacion

I have encountered this optimization problem while trying to implement the method proposed in this very interesting paper: http://www.mae.cuhk.edu.hk/~cwang/pubs/JCISERealTimeSkeleton.pdf the ...
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1answer
15 views

improving symbolic generation of objective function for optimization

I am currently using matlab to solve an optimization problem. I am generating the objective function using the symbolic toolbox. I planned use the symbolic toolbox to calculate the gradient and ...
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2answers
30 views

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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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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41 views

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

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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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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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
17 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 ...
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1answer
33 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
19 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 = ...
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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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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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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 ...
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1answer
31 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 ...
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1answer
38 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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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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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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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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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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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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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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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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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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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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31 views

Minimization Problem (Calculus) [closed]

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
40 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 ...