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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Maximum / Minimum Cost of a Box

this is a sample final question for a multivariable calculus course. "A rectangular box has two opposing sides (left and right) made of gold, two (front and back) of silver, and two (top and bottom) ...
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32 views

Probability density function / maximum likelihood for correlating sequence

I have a stream that contains two consecutive identical sequences, each of length $N$. These sequences have a ideal autocorrelation property. So I want to have the probability density function over $...
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Help with a homework problem involving $\textbf{H}$-conjugate vectors

My problem is the following: Let $\textbf{H}$ be a symmetric $n\times n$ matrix. Are the following claims true? Why? a) If the vectors $\textbf{d}_1$ and $\textbf{d}_2$ and vectors $\...
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Proving the shortest function connecting two points is a straight line WITHOUT assuming the Euler-Lagrange equation

For learning purposes, I'm trying to prove that the shortest function passing through the two points $(x_1, y_1)$, $(x_2, y_2)$ is a straight line, without using the Euler-Lagrange equation. My ...
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1answer
58 views

Multivariate function maximum criterion

Be a concave mutivariate function $f(\textbf{x})=\textbf{y}$. I observed the following conjecture: the maximum value of $f$ is achievable when all entries of $\textbf{x}$ are equal. How to prove such ...
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429 views

Maximum vertical distance between the line $y = x + 30$ and the parabola $y = x^2$ for $−5 ≤ x ≤ 6$

What is the maximum vertical distance between the line $y = x + 30$ and the parabola $y = x^2$ for $−5 ≤ x ≤ 6$? This is what I did but didn't work: Set $y_1=x+30$ and $y_2=x^2$, plugged those ...
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43 views

optimization with non smooth constraint

I am trying to maximize the profit of a power plant. I have a constraint which is that the power plant, when operating, has a minimum and maximum capacity. (So a power block either has an output of ...
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26 views

Second Order Necessary Condition for Optimality

Question: [See context below.] What would be the analog of the Thm when $f$ is only defined on, say, a domain $D\subset\mathbb{R}^n$? In that case we can't take a general $h\in\mathbb{R}^n$ ...
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Minimization of a weighted least-squares problem by Lagrange multiplier method

Problem: Let $Y = (y_1, y_2, \dots, y_m) \in \mathbb{R}^{m \times n}$ and $k \in \mathbb{R}^{m}$ satisfy $\sum_{i=1}^{m} k_i =1$ and $k \geq 0$. Show that $x=Yk$ is a minimizer for $h(x) = \dfrac{1}{...
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54 views

Optimization of a function

I need to optimize $$f(x,y,z)= x^2-y+e^{z}$$ with the restriction $$(x-2)^2+(y-3)^2+z^2=1$$ I've tried to substitute the restriction in $f(x,y,z)$ but it seems not to work. And when trying to use the ...
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77 views

Parameter optimization using a regression model.

I am working on an optimization problem. I build a regression model to understand the behavior of a system which depends on two variables which are functions of another two variables. My regression ...
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3k views

Maximum area of a isosceles triangle in a circle with a radius r

As said in the title, I'm looking for the maximum area of a isosceles triangle in a circle with a radius $r$. I've split the isosceles triangle in two, and I solve for the area $A=\frac{bh}{2}$*. I ...
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1answer
51 views

Optimization with Linear constraint $Ax=0$

I confront with this problem: $$\min_{x \in \mathbb{R}^{n}} \dfrac{1}{2} \left\| x- a \right\|_{2}^{2}$$ subject to $$Ax=0.$$ My tactic is to use Lagrange multiplier method that: $$\mathcal{L}(x, \...
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53 views

Selection of the mean of random variables to optimize the expected value of objective function

Here is the objective function to be maximized: $$ E_{v}(\log(1+v^{\mathsf T} \Lambda v) ) $$ where $v$ is a Gaussian distributed random variable vector $v ∼ \mathrm{CN}(M,I)$ with its mean vector $M$...
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54 views

Transform a minimization problem to LP

This is a past examination question. I was asked (Q.1) to find an equivalent linear programming problem of: $$\min_{x \geq 0} \left \|Ax-a \right\|_{1} + \left\|Bx-b \right\|_{\infty}$$ where $A$ ...
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37 views

Will this optimisation give the global maximum?

My book optimises a type of functions using the lagrange method. From calculus I remember that we had to check the boundary when using lagrange, because it only gave local max, but it is not mentioned ...
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112 views

Find the Maximum and Minimum of the Given Function on the Given Plane Region

I've been good with most of the max/min finding in different regions, but this one's really messing with me. Can anyone lend a hand? Thanks. z = 2xy Region is the circular disk $x^2 + y^2 =< 1 $
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Polar cones' property [duplicate]

I am trying to prove: $A \subseteq B \implies B^\circ \subseteq A^\circ$ where $A^\circ$ is polar cone of $A$ ($A$ convex cone) and $B^\circ$ is polar cone of $B$ ($B$ convex cone)
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82 views

Slice an ellipsoid into equally thick slices for maximal surface

After seeing a colleague slicing a nearly ellipsoid piece of ginger for his cup of tea into almost equally thick slices to get more surface area (so the tea would suck out the ginger taste better), i ...
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4answers
60 views

The minimum value of $\frac{a(x+a)^2}{\sqrt{x^2-a^2}}$

The problem is to find the minimum of $A$, which I attempted and got a different answer than my book: $$A=\frac{a(x+a)^2}{\sqrt{x^2-a^2}}$$ where $a$ is a constant $A'=\frac{(x^2-a^2)^{\frac{1}{2}}(...
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40 views

Solving for f(t) in presence of f'(t)

Here's the situation: I have a function $$e(t) = \frac{a~d(t)}{b + d(t)}$$ with first derivative $$e'(t) = \frac{a~b~d'(t)}{[b+d(t)]^2}$$ where $a$ and $b$ are constants. For a given constant $K$ I ...
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36 views

Maximize area of a corral

See problem 7 and figure 9 in picture What I've done so far: Not sure if $P=2l+2w$ or just $l+2w$ (dashed line makes me think the latter) $600=\pi r+l+2w$ $600=\pi r+2r+2w$ $w=\frac{600-\pi r-...
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93 views

Shannon Entropy Continuity Constraint

I have the following problem: I want to find the probability density $p$ which maximizes the Shannon entropy \begin{equation} S := - \int_{x_b}^{x_c} dx ~ p(x) \log (p(x)) \end{equation} under the ...
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41 views

What is this sort of optimisation called?

I am reading a book in mathematical finance. There is something about constrained optimisation. They have specialised it for the financial market, but I am wondering what the general name for this ...
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208 views

What does 'the level set is bounded' exactly want to tell?

'The level set is bounded.' occurs in many theorems and other places. I think I can understand the definition of 'level set' but I don't know what does 'it's bounded' want to tell me exactly in ...
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20 views

Binary Linear Programm: Check for feasability and multiple solutions

Assuming, I have binary integer program, e.g. given by: $ \arg\min_x \quad 0\\ \text{such that}\quad A_\text{eq} x = b_\text{eq}, x_i \in \{0,1\} $ Where also $[A_\text{eq}]_{ij} \in \{0,1\} $ and $...
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85 views

How to show these two problems have equivalent solutions

I have two problems, where $A$ is positive definite: $$\inf \{ x^TAx + b^Tx : 1-x^Tx \ge 0,\ x \in \mathbb R^n\} \ (1)$$ and $$ max_\lambda \ q(\lambda) = -0.25b^T(A+\lambda I)^{-1}b - \lambda : \...
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174 views

Derivative of a trace w.r.t matrix within log of matrix sums

I'm trying to solve an optimization (sub)problem and am running into trouble with a tricky derivative. I'd like to find the matrix $C \in \mathbb{R}^{n\times d}_+$ which minimizes the following ...
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How can I show that these two problems have the same optimal solution?

How can I show that these two problems have the same optimal solution: $$\inf \{ x^TAx + b^Tx : 1-x^Tx \ge 0,\ x \in \mathbb R^n\}$$ $$\inf \{ x^TAx + b^Tx : 1-x^Tx = 0,\ x \in \mathbb R^n\}$$ when ...
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Discretization of a convolution integral for constrained optimization problem

I'm working on a constrained optimization problem in which an unknown forcing function, $u(\eta)$, is in the integrand of a convolution integral. To find an optimal shape for $u(\eta)$, the integral ...
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45 views

Property of monotone operator (Positive definite)

I would like to prove this statement: "$F$ is monotone if and only if $\nabla F$ is positive semidefinte." I only know $F$ is monotone with respect to $\Omega$ if and only if $$(u-v)^{T}(F(u)-F(v))...
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295 views

Given a fixed volume for a solid cylinder, is it possible to find the minimum or maximum curved surface area?

Given a fixed volume for a solid cylinder, is it possible to find the minimum or maximum curved surface area $ 2 \pi r h $ of this cylinder?
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Prove $\lambda=\min_{i = 1,\ldots, n}\max_{0 \le k \le n-1}\left(\frac {p_i(n)-p_i(k)}{n-k}\right)$

Prove the minimum directed cycle mean cost satisfies: $\lambda = \min_{i = 1,\ldots, n} \max_{0 \le k \le n-1} \left(\frac {p_i(n) - p_i(k)} {n-k}\right)$ using the Bellman-Ford algorithm. Let $\...
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82 views

Given an optimal solution to the LP, show how it can be used to construct a directed cycle with minimal directed cycle mean cost.

Let $\mathcal G = (\mathcal V, \mathcal A)$ be directed graph with associated edge costs $c_{i,j}$ that has at least one directed cycle. Define the directed cycle mean cost to be $\frac {\{\text {sum ...
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446 views

Local optimality of a KKT point.

Consider the problem \begin{equation} \min_x f(x)~~~{\rm s.t.}~~~ g_i(x)\leq 0,~~i=1,\dots,I, \end{equation} where $x$ is the optimization parameter vector, $f(x)$ is the objective function and $g_i(...
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44 views

How to get the peak value of this logarithmic equation?

Is there a way to get the peak point of the following equation? $$ (a_1-a_2 x)\ln\left(1+\frac{b_1 x}{b_2 x+b_3}\right),$$ where $a_1,a_2,b_1,b_2,b_3$ are all positive constant values and $x$ is also ...
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Optimizing a set of rules to better predict the outcome of events

I'm trying to better predict the top three finishers of the next 1000 800m mens freestyle swimming race. I've got a set of rules to rate the swimmers: 1) Add 5 points if the swimmer won his last ...
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29 views

One solution of a diophantine system

How to find one solution of $Ax = b$, where $A$ is a $(m, n)$ matrix and $x$ a vector of size $(n, 1)$. $A$, $x$ and $b$ are matrices of integers entries. How to check whether is a solution exists?
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53 views

Find those values 'a' which belongs to the Convex Hull

Find those values of 'a' for which (1,a,1) belongs to the convex hull of $$\{(0,0,0), (1,1,2),(2,4,-6), (1,3,8)\}$$ Give me hints as much as you can, I would like to understand the mindset rather ...
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88 views

A interesting max min problem

Let $\mathcal{S}\subset\mathbb{R}^2$ be a bounded, closed, compact, convex set which contains origin in its interior. Define \begin{align} c_1^{\star}=\min_{{(x_1,0)\in\mathcal{S}}}~&x_1 \end{...
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Prove an artificial variable that leaves the basis will never return.

This is in the context of the Big M Method in the simplex algorithm in linear programming. Prove an artificial variable that leaves the basis will never return. I have no idea how to start this. ...
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Prove optimal solution to dual is not unique if optimal solution to the primal is degenerate and unique.

How do I prove an optimal solution to dual is not unique if an optimal solution to the primal is degenerate and unique? What I tried: Let the primal be $$\max z=cx$$ subject to $$Ax \...
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167 views

Conditions for unique solution of a maximization problem?

Let $S\subseteq \mathbb{R}^2$, $d:=(d_1,d_2) \in S$, and $s:=(s_1,s_2)$ a generic point of $S$. Assume that there exists $s \in S$ such that $s_1>d_1$ and $s_2 >d_2$. Consider the following ...
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473 views

Can calculus optimization problems be turned into linear programming problems?

I found a Linear Programming textbook somewhere, and I skimmed through the first few pages. While I am not nearly enough ready to go through it, the things it dealt with seemed very much like calculus ...
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24 views

Are there two notions of flow?

I'm reading Jungnickel's Graphs, Networks and Algorithms. He defines the flow as a mapping $f:E\to \mathbb{R}_0^+$, which seems to mean the value of the flow of each edge, but in here: When he ...
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51 views

Maximizing $\log(|A|)-\text{Tr}(AB)$ for pd and symmetric $A$ and $B$

Let $A$ and $B$ be two symmetric and positive definite matrices of the same size. Then the function $$ f(A)\equiv\log(\det(A))-\text{Tr}(AB) $$ is maximized uniquely by $A=B^{-1}$. This is ...
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163 views

Is there Lipschitz property for subdifferential?

I'm trying to bound the quantity $\langle \nabla \Psi(x),\bar{x}-x \rangle$ above, with the bound depending on $\|x-\bar{x}\|$ and perhaps also of $\|x-y\|$ for fixed (but not varying) points $y$. ...
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54 views

Finding the Dual of a primal LP

Suppose that we have the following primal LP: $\min z=c^Tu+d^Tv \\ \mbox{s.t.}\ \ \ \ \ \ \ \ \ u+Av=b, u\geq 0, v\geq 0$ I want to find the dual problem of this LP but I am slightly confused as ...
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26 views

How to get the updating rules? after derivation

In the picture i brushed yellow, it dose make no sense to me to get formulas(2) and (3). If anyone could point out or give some references? Thanks a lot!
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49 views

Finding L^1 centers of sets of probability distributions

Let $\mathcal{P}^n = \{ x \in \mathbb{R}^n : x \geq 0, \sum x = 1\}$. Suppose I have $p_1, \ldots, p_m \in \mathcal{P}^n$. I want to find an $L^1$ center for these points. i.e. $q \in \mathcal{P}^n$ ...