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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Significance of multiplying by weight

I have been reading optimization problems in communication area where it is a common practice to maximize rate of users as below objective function: $\hspace{28mm} \text{ Maximize } \sum_k w_k \log ...
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21 views

Is there a binary operator (besides composition) closed under permutations or a notion of a metric space on permutations?

When i say "a binary operator closed under permutations" I mean, given $2$ (finite, same number of elements) permutations $p_1$, $p_2$ , is there an operator "$+$" such that $p_1+p_2=p_3$ ($p_3$ a ...
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26 views

Optimal partitioning of a planar graph

Consider a planar graph, where each node is associated with a weight. I would like to partition the graph such that the sum of the node weights in each group satisfy a minimum requirement. However, I ...
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24 views

finding the shortest distance of a hermitian matrix to a set of hermitian matricies with specific eigenvalues 2-norm

The title is more general, and all that I require is to show an inequality that I already have verified using random matrices in matlab. Let $\lambda_1 \leq ... \leq \lambda$ and $\mu_1 \leq ... \leq ...
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1answer
26 views

Maximizing a convex function under constraints

Consider the following non-convex problem: \begin{equation*} \begin{aligned} & \text{maximize} & & f(X) \\ & \text{subject to} & & f(X)\le b\\ &&& A_kX = c_k, \ ...
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Distributing resource based on Efficiency

I am trying to form an optimization problem where I have $k$ nodes who transmits packets with rate $x_k$. The objective is to maximize the rate. $\hspace{28mm} \text{ Maximize } \sum_k \log x_k$ ...
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1answer
30 views

Taking Log to find MAXIMIZE summation of variables

I have been reading IEEE papers on communication and in several papers the authors formed objective function like: $\text{Maximize } \sum_k \log r_k $ to maximize the total rate of the system of ...
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3answers
25 views

Minimize given LCM

Find the smallest possible value of $n_1+n_2+\cdots+n_k$ such that $LCM(n_1,n_2,\ldots,n_k)=(2^2)(3^3)(5^5)$. Note that $k$ is not fixed. I know the answer should be $k=3$, $n_1=2^2$, $n_2=3^3$, and ...
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304 views

The longest sequence of numbers with a certain divisibility property

Definition - Denizen A sequence $\lbrace a_k \rbrace$ is a denizen if all of it's members are prime numbers, i.e $a_0, a_1, ... a_n \in \mathbb{P} $; and it satisfies the following condition; ...
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Is it possible to generate an arbitrary surface, which maps an image onto a point with any distortion we please?

I'm trying to understand if it's possible to write a function (even a discrete, iterative one) that can generate a surface S, which will take an incoming light field of vectors V, and map it onto a ...
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1answer
59 views

Second-smallest eigenvalue as $\displaystyle \min_x \frac{x^TAx}{x^Tx}$

In Mining Massive Datasets, page 365, the following theorem is stated without proof: Let A be a symmetric matrix. then the second-smallest eigenvalue of A is equal to $\displaystyle \min_{x} ...
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67 views

minimization problem labor and costs

I would appreciate it greatly if someone could provide me with a solution to the problem below: If the contract runs late the business will be penalized \$1000 for each late day. It is estimated that ...
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1answer
13 views

Non-linear Dirichlet problem with FD

I face to the following problem: $$(1+u_x^2)u_{yy} - 2u_xu_yu_{xy} + (1+u_y^2)u_{xx}=0.$$ It should be discretized by finite differences. Does anybody know, how to proceed? Or does anybody know ...
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2answers
28 views

Lagrange's multiplier method find the highest and the lowest point

Plane $x+y+z=12$ intersects the paraboloid $z=x^2+y^2$ find the highest and the lowest point of this cross-section. What should i do here? I need help solely when it comes to transforming this ...
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2answers
51 views

Maximizing area under $y=e^{−{∣x∣}}$

The coordinates of the point $M(x,y)$ on $y=e^{−{∣x∣}}$ so that the area formed by the coordinates axes and the tangent at $M$ is greatest is what? I tried to plot the graph but after that I'm not ...
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10 views

Best basis selection problem with inequalities solution methods

I'm solving an optimization problem in form $\min \sum x$ subject to, $ A x = b$ (1) $ g x \leq d$ (2) $ x \geq 0$ (3) Optimization variable is x. The number of rows of A is considerably ...
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3 views

Show that a function is quasi-concave by finding a transformation of the function

Assume the function u:R2+ -> R given by u(x1, x2) = (x1+1)(x2+1) Show the u is quasi-concave by finding a transformation of u that is concave Show (without using Hessians) that u is not a concave ...
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1answer
35 views

Unconstrained Optimization - Minimum Distance Between Point and Curve

Background: I am writing some software can fit a mathematical curve to data using different regression techniques. I currently have Ordinary Least Squares and Least Absolute Deviations mostly ...
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1answer
41 views

How can I find the critical curves for the following functional

Find the critical curves for the following functional : $$J[y,z]=\int_{0}^{1} \sqrt{1+y'^2+z'^2}$$ such that :$$y^2+z^2=1$$ and $$y(0)=z(1)=1$$ $$y(1)=z(0)=0$$
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Minimizing $\frac{a^TXb}{\|a\|\|b\|}$ given certain constraints.

I'm solving a problem which I have reduced to maximizing $\frac{a^TXb}{\|a\|\|b\|}$ given that $X = (AA^T)^{-1/2}AB^T(BB^T)^{-1/2}$, $a = (AA^T)^{1/2}a'$ and $b = (BB^T)^{1/2}b'$. In this case, I ...
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4answers
70 views

Find the minimum value of $f(x)=\sum_{i=1}^n |x-a_i|$. [duplicate]

Let $a_1 \lt a_2 \lt \dots \lt a_n$. Find the minimum value of $f(x)=\sum_{i=1}^n |x-a_i|$. My guess is the minimum occurs at the middle point. However, I don't know how to show this since I can't ...
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1answer
20 views

Find diffeomorphism transforming the following areas:

Find diffeomorphism transforming the following: interior of the triangle T with vertices in $(0,0),(0,1),(1,0)$ onto the interior of the circle of radius 1 and centre in $(0,0)$. Obviously i am ...
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1answer
20 views

Max volume of a cuboid given constraint

Find maximum volume of a cuboid for which the sum of three dimensions does not exceed $108$. I think expression to maximize is: $\left( 108-y-z\right)yz$. From partial derivatives I got that max will ...
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14 views

Max volume of a cylinder

Find maximum volume of a cylinder of which the sum of height and the circumference of the base does not exceed 108 cm. How to solve this? Precisely what is the expression that should be minimized? How ...
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25 views

Dynamic Optimization - Transversality Condition for Infinite Horizon Case

When solving dynamic optimization problem such as $$ \max \int_0^\infty f(t,x,x')dt $$ $$ \ s.t. x(t_0)=x_0 $$ we can use the Euler equation to obtain a differential equation to solve. ...
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variable transformation in optimization

I have an optimization problem with two sets of parameters, $x_i \in [0,1]$ and $y_k \in [-\frac{\pi}{2},\frac{\pi}{2}]$ where $i,k \in \{1...n\}$ are indices. One way to solve this problem is using ...
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1answer
13 views

Find a point from the area tha is closest to the other point.

Given: $$A=\{\left(x,y,z \right)\in \mathbb{R}^3 : 2x-3y+z=1 \}$$ Find a point $\left(x,y,z\right)\in A$ that is closest to $\left(3,-2,1 \right)$. I do not know how to solve that problem, i need a ...
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49 views

Optimizing over an infinite set of variables

This may be a very basic question, but it's been a while since I did any optimization. Suppose I have a sequence $(x_i)$, $i=1,2,\ldots$ in the $\ell^2$ space and the following optimization problem: ...
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37 views

Finding the distance from the origin to the surface $xy^2 z^4 = 32$ using the method of Lagrange Multipliers

Problem: Find the distance from the origin to the surface $xy^2z^4 = 32$. Attempt: The Lagrange equation for this problem is $L(x,y,z, \lambda) = x^2 + y^2 + z^2 + \lambda (xy^2 z^4 - 32)$. Setting ...
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how to project optimal parameters on to feasible region

Hi: I'm trying to understand the concept of projection and I created a toy example that might help me to do that. Suppose that I have a non-linear optimization with 3 parameters theta_1, theta_2 and ...
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20 views

Optimal control with non-convex cost function

For a controllable linear system $$ \dot{x}=Ax+Bu $$ with state $x=(x_{1},\ x_{2},\ \cdots,x_{n})^{T}$ and the non-convex cost function \begin{equation} \tau(x)= \int_{t=0}^{T} ...
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Polynomial algorithm for problem in graphs which can also be solved as a linear programming problem.

I have an (undirected) graph $G = (V, E)$. For each vertex $i \in V$ we have a cost associated $v_i$ and for each edge $e \in E$ we have a prize associated $x_e$. My problem is to find $W \subseteq ...
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36 views

Solving Nonlinear System for two variables

I have an optimization like below: $\text{ minimize } \sum_k - w_k\log_2 x_k $ $\text{subject to: } x_k \leq q , k =1,2, \cdots, N .$ $\hspace{20mm} 0 \leq w_k \leq 1$ I can form the Lagrange ...
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40 views

Fenchel Duality in Prof. Bertsekas' lecture

Please see this link, p.39-41 (sufficient to answer my question), before (1.47) for detailed. For convenience, the relevant part is shown as: I am confused in two things: The ...
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70 views

Show that inf $f(x)$ is achieved. Find $\inf f(x)$.

Let $$\Sigma = \{x\in R^3: x_1x_2 +x_1x_3 +x_2x_3=1 \}$$ and $$f(x) = x_1^2 + x_2^2 + \frac{9}{2} x_3^2$$ a) Show that $\Sigma$ is a smooth surface in $R^3$. b) Show that $\inf_{x\in\Sigma}$ f(x) ...
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9 views

Scaling issue with Gradient descent methods

As is the common knowledge that gradient methods are affected by scaling issue of the variables. For example, If minimizing a function of say 2 variables $x_1$, $x_2$. Both variables have different ...
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30 views

Finding KKT conditions for nonlinear optimization problem.

I have an optimization like below: $\text{ minimize } \sum_k - \log_2 x_k $ $\text{subject to: } x_k \leq q , k =1,2, \cdots, N .$ I can form the Lagrange of the problem as below: $L(x, ...
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22 views

Some problems in finding conjugate function

Ask the following fundamental problems: How to derive the conjugate function of $g(y)$ if given "$\underset{y \geq 0}{\text{sup}}\{g(y)-y^Tx\}$"? My attempt is as following: \begin{align*} ...
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18 views

Maximizing the area of a transformed rectangle within some bounds

I need to solve this problem for a program I'm writing, but I'm struggling a bit with the maths behind it. Given a rectangle $R_{max\_layout}$ and a $3\times 3$ transformation matrix $M_{transform}$, ...
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52 views

Conjugacy relation in the primal and dual problem

The following is my derivation in the Conjugacy relation in the primal and dual problem. I am shaky in it; so hope for some advices. Consider the following problem, $f(x),g(x)$ are convex ...
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Minimum vertex cover of two edge disjoint perfect graphs

How well can the minimum vertex cover of the union of a perfect graph and bipartite graph (the two graphs are edge disjoint but not vertex disjoint) be approximated?
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Solving linear constrained optimization problem

So I have the following constrained optimization problem to optimize a circuit (electrical engineering) that I am working on: Minimize the following expression (power dissipation): $$I_{B1}(V - C_1) ...
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2answers
35 views

The sum of the lengths of the hypotenuse and another side of a right angled triangle is given.

The sum of the lengths of the hypotenuse and another side of a right angled triangle is given.The area of the triangle will be maximum if the angle between them is: ...
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Find the critical curves for the following functional with subsidiary conditions

Find the critical curves for the following functional : $$J[y,z]=\int_{0}^{1}\left(y'^2+z'^2-xyz'-yz\right)dx$$ with subsidiary conditions : $$\int_{0}^{1}\left(y'^2-xy'-z'^2\right)dx=2$$ ...
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Solving Nonlinear system with logarithmic objective function

I have my objective function as : $\hspace{25mm} \text{Minimize} \sum_k- \alpha_k \log_2 W_k$ $\hspace{25mm} \text{subject to}: 0\leq W_k \leq q', 0 \leq \alpha_k \leq 1 $ $\hspace{25mm} k ...
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21 views

Winning the relay race for your team

Relay race, members of a team of three take turns running from the point P to a point on the circle; To A for the first, B for the second, and C for the third, starting and returning to point P, ...
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26 views

Minimum of a function given by integral and inequality type constraint

I need your help with the following problem I want to minimize $$2a + \int_0^1 tx(t) \, dt \to \min$$ s.t. $$1 - a - \int_t^1 x(s) \, ds \leq 0\text{ a.e. }t \in (0,1)$$ $$x(t) \geq 0 \text{ a.e. ...
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20 views

Integer (Binary?) Optimisation of a problem

got a question regarding maximal optimisation of a problem. Refer to the table below: $$\begin{array}{c|c|c|} & \text{Area A} & \text{Area B} & \text{Area C} & \text{Area D} \\ ...
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1answer
50 views

How can I solve a nonlinear optimization problem where constraint contains exponential term?

I have the optimization problem as below: $\hspace{10mm} \text{Maximize} \sum_{k} \alpha_k {R}_k $ $ \hspace{10mm}\text{subjcet to:} $ $ \hspace{10mm}\exp \left[ - (2^{{R}_k } -1) \left( ...
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17 views

Optimization problem: solving one implies solving the reverse?

I am looking to solve an optimazation problem $Maximize_{x} [A(x)]$ s.t. $B(x)\geq B_0$, where $B_0$ is a constant. If I solve this problem (i.e, finding the optimal $x^*$ that optmize while ...