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 to represent and handle eigenvalue constraint in optimization

For example consider the problem $\min_X f(X)$ s.t. $\lambda_i(X+A)=\lambda_i(B)$ for $i \in {1,...,N}$ where $A$ and $B$ are full rank N by N matrix, $\lambda_i(X)$ is the i-th eigenvalue of $X$ ...
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101 views

Optimization over vector spaces. Generalized KKT.

I am looking for the extension of the theorem I found in the book by Luenberger called "Optimization by vector space methods." Here is the statement of that theorem from Luenberger: Generalized ...
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23 views

Quick Constrained Optimization Huerstistics

I am wondering if there is a way to find very quick optimization heuristics for the form. $$ f(x) = cx^a $$ $$ s.t. $$ $$ L \le Ax \le B$$ $$ 0 \le x \le \infty $$ I know with only a few variables ...
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20 views

Minimization according to a function

I am working on a simulation project on Matlab/Simulink and I would like to minimize the following function : $min_f \int_{t=0}^{t_1} m(A(t) + \int_{y=0}^{t} B(y) f(y) dy) dt $ With $m : \Bbb{R}^6 ...
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4answers
30 views

Optimization problem for a rectangle with the greatest possible area.

A rectangle is inscribed with its base on the $x$-axis and its upper corners on the parabola $y=7−x^2$. What are the dimensions of such a rectangle with the greatest possible area? Would this be a ...
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52 views

Find a point on the line $-2x + 6y - 2 = 0$ that is closest to the point $(0,4).$

I understand that you would use the distance formula here, but I'm confused as to how you calculate x and y after that. Thanks.
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1answer
17 views

Optimization problem expressing the area of a page in terms of $x$

I have this problem which I have already answered some steps correctly but I got stucked on one of the steps (shown in the picture by the pop-up of answers). How do I get the area of the page if the ...
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2answers
2k views

Optimization of the surface area of a open rectangular box to find the cost of materials

A rectangular storage container with an open top is to have a volume of 10 cubic meters. The length of the box is twice its width. Material for the base costs ten dollars per square meter and for the ...
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45 views

Non-linear system vs minimisation problem

If you have a non-linear system of equations which can be formally written as : \begin{equation} \begin{cases} F_1(\mathbf{x})=0\\ F_2(\mathbf{x})=0\\ \ \ \ \ \vdots\\ F_n(\mathbf{x})=0\\ \end{cases} ...
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3answers
33 views

Optimization word problem for cost effective fence enclosure.

Here's the question: A fence is to be built to enclose a rectangular area of 200 square feet. The fence along three sides is to be made of material that costs 5 dollars per foot, and the material for ...
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1answer
31 views

Optimizing a function with simple equality and inequality constraints

I'm looking to maximize $$f(x_1,..,x_n)=\sum_{i=1}^n\alpha_i \sqrt x_i+\beta $$ subjected to the following constraints $$a_i\le x_i \le b_i \space\forall i \in\{1,...,n\}$$ and ...
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24 views

Application of Implicit Function Theorem for Constrained Optimization

Here's the problem: Consider the subset $S \subset \mathbb R$ defined by $$ x^4+2xy+y^4+yz+z^3 = 2 $$ Show that there exists a $C^1$ function $g: \mathbb R^2 \to \mathbb R$ defined near $(1,1)$ ...
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464 views

Graph theory problem (edge-disjoint matchings)

Find the smallest number $x$ so that if an $n$-vertex simple graph has at least $x$ edges then it contains $k$ pairwise edge-disjoint perfect matchings* ($k$ is a positive integer, $n$ is an even ...
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19 views

Best set of subgraphs of a weighted complete bipartite graph

Consider a weighted complete bipartite graph, i.e. consider the graph $G=(V,E)$, with $V=X \cup Y$, $X \cap Y = \emptyset$ and $E = X \times Y$, and a set of weights $W=\{w_i : i \in E\}$. Now we ...
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3answers
118 views

Prove $\int_0^1 ((g'(x))^2-1)^2dx \geq 1$ for smooth $g$ with $g(0)=g(1)=0$ [closed]

This came up in an optimization problem. How do you prove that $\int_0^1 ((g'(x))^2-1)^2dx \geq 1$ for any $g$ which is twice continuously differentiable on $[0,1]$ and such that $g(0)=g(1)=0$?
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40 views

Find ALL local maxima numerically

Is there an algorithm that given a function and its derivative gives me all local maxima (in an interval)? All optimization algorithms I know of focus on finding one local or one global maximum. I ...
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488 views

Gradient descent with inequality constraints

Suppose we are given a convex function $f(\cdot)$ on $[0,1]$. One wants to solve the following optimization problem: \begin{equation} \begin{aligned} & \text{minimize} && \sum_{i=1}^n ...
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16 views

How to solve constrained ode problem

Currently I'm facing question in which let's say I have 3 coupled eqn. \begin{align} x = f(x', z', y', t) \\ y = f(x', y', z', t) \\ z = f(x', y', z', t) \\ \end{align} There is initial ...
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3answers
30 views

Use a linear approximation to estimate the given number $64.07^{2/3}$

That's all they give you. I tried putting it into the linear approximation equation of: $$ f(a)+f'(a)(x-a) $$ but I get almost the same value as $64.07^{2/3}$, which is around $16.0116$. Just not ...
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2answers
47 views

Use a linear approximation to estimate the number $64.07^{2/3}$

That's all they give you. I tried putting it into the linear approximation equation of: $$ f(a)+f'(a)(x-a) $$ but I get almost the same value as $64.07^{2/3}$, which is around $16.0116$. Just not ...
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0answers
26 views

Good convergence criterion for stochastic optimization?

This is a question that has bothered me quite long, as I have faced it many different optimization and equation solving problems. The basic idea is that one wishes to minimize $F(x)$ and has one ...
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3answers
33 views

Finding the dimensions of a cuboid for minimal surface area

I have no idea how to even start thinking with this problem: Using the theorem for extrema of a function with two variables, find the dimensions of a parallelepiped with rectangular faces and fixed ...
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340 views

Shortest distance between parallel line and plane

I've been doing questions regarding the shortest distance between lines/planes and points , and I've come across a question asking to find the shortest distance between a line and a plane which are ...
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31 views

Optimisation to solve for trigonometric expression?

I have a question that requires the use of optimisation to solve for the following expression: $$\cos ec{(\cos^{-1}{(-\frac{\sqrt{3}}{2})}+\sin^{-1}{(-\frac{\sqrt{3}}{2})})}$$ I'm a bit baffled, as ...
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40 views

Minimize the squared dot product of two specific vectors

Do you think there exists a efficient algorithm(non brute-force) for the following problem. I search the optimal solution for the following problem: Given a vector $u=(u_1, u_2,..., u_k)^T$ with ...
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1answer
52 views

Function going from $0$ to $1$ with minimal concavity

How small can I take $C>0$ such that there exists an $f\in C^2(\mathbb R;\mathbb R)$ satisfying the following properties: $f(x)=0$ for $x\leq 0$ and $f(x)=1$ for $x\geq 1$ $f'(x)\geq 0$ for ...
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43 views

Distance Problem WITHOUT Calculus

The problem is: Suppose you have two points above a horizontal line. You draw a line from from the first point to the line and draw a line from the intersection of the first point and the horizontal ...
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8 views

Minimization of multivariable function which contains floor

Let $f$ be a function such that: $$f(r,h,n) = 4 \pi (r+h)^2 - \big ( \big \lfloor 2 \frac h {r+h} \big \rfloor + n \big ) 2 \pi h (r+h) $$ where $r,h > 0$ and $n \in \mathbb N^*$ as well as ...
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parameterization of a horizontal line

say I need to parameterize the boundary of a set in order to optimize. The equation is f(x,y) = 3 + x − y + xy and the boundary is the set inclosed by the line ...
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Monotonicity and optima of functions

It is said that the logarithm is a monotonically increasing function, hence the logarithm of a function achieves its maximum value at the same points as the function itself. Is there a similar ...
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22 views

How to solve a combinatorial optimization problem

\begin{align} &\max_{C,E,S} \begin{aligned}[t] &\sum_{t=1}^{T}\sum_{k=1}^{K}min\left\{{\mu(\alpha_{k},e_{k}(t)),\gamma s_{k}(t)}\right\} \end{aligned} \notag \\ &\text{s.t} \notag \\ ...
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24 views

How does $\left[\nabla f\left(\mathbf{x}\right)\right]^\mathrm{T} \nabla f\left(\mathbf{x}\right)$ approximate $\mathbf{H}$?

Page 3 of a guide to Levenberg-Marquardt optimization says that $\left[\nabla f\left(\mathbf{x}\right)\right]^\mathrm{T} \nabla f\left(\mathbf{x}\right)$ approximates the Hessian matrix of $f$. I do ...
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49 views

Is standard eigenvalue optimization problem convex

For any arbitrary symmetric matrix A , is the standard eigenvalue problem convex $ \lambda_{max}(A)= \max_{\|x\| \leq1} x^{T}Ax$
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36 views

Reverse spline interpolation

Say I have a number of sets $(x, y)$ for $x \in \{0, 1, \dots, 255\}$. I want to find the least number of points to reproduce the set with a certain accuracy using linear interpolation. What is the ...
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138 views

Apostol's Calculus: optimize the perimeter of isosceles triangle inscribed in circle

As much as I hate to ask for a hint on this, I've gotta admit--I'm stuck. None of my previous attempts to solve this were successful, and I can't think of a fresh way to look at the problem. This is ...
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18 views

What is the intuition behind functional optimization with constraint $x'(t) = \phi(t, x, u)$?

Suppose we want to find the extremums of the $J(x,u)$ subject to some constraints: $\begin{cases} J(x,u) = \int_{t_0}^{t_1} L(t,x,u) dt + \Psi & \to \text{extr} \\ x' = \phi(t,x,u) \\ J_i(y,u) ...
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15 views

a problem of linear optimization

I struggle with the following problem: Given equation: $y = Hx$ where -> $x$ is a complex random process of $N$X1 dimentions. $E(x_i(t_1)x_j(t_2)^*)=0 \space \space \forall t_1,t_2, \space i\neq j$ ...
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59 views

To Calculate Maximum Volume by finding out Maximum value

Snacks will be provided in a box with a lid (made by removing squares from each corner of a rectangular piece of card and then folding up the sides) http://www.imageshack.com/a/img661/6094/fZUQXg.jpg ...
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1answer
25 views

Critical points of a function $f(x) = x\sqrt{x-a}$

Find the critical points of a function $f(x) = x\sqrt{x-a}$. A function $f(x)$ is said to have critical points at points $c$ such that $f^\prime(c)$ is $0$ or undefined. For a function $f(x) = ...
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20 views

How can I solve $\inf_{x \in X} x^T(Ax + b) + \sum_{i=1}^n x_i \log(x_i)$ in a closed form depending on projection?

I'm trying to solve the minimization problem $$\inf_{x \in X} x^T(Ax + b) + \sum_{i=1}^n x_i \log(x_i)$$ where $b \in \mathbb{R}^n$, $X \subset \mathbb{R}^n$ is a convex set. And $A$ is a symmetric ...
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Undefined case in Lagrangian method

I am trying to finding the minimum distance between the point(1,1,0) and points on the sphere $$x^2+y^2+z^2-2x-4y=4$$ An easy way to do this is to graphical intuition and get the distance, since the ...
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11 views

relation within Gauss-Newton method for minimization

If we study model fit on a nonlinear regression model $Y_i=f(z_i,\theta)+\epsilon_i$, $i=1,...,n$, and in the Gauss-Newton method, the update on the parameter $\theta$ from step $t$ to $t+1$ is to ...
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55 views

How to solve a matrix equation for a scalar?

Given matrices $Q, P \succeq 0$, a vector $q$, a real number $\gamma$. How can one solve the equation $ q^T (Q+\lambda P)^{-T}P(Q+\lambda P)^{-1} q = \gamma$ for the scalar $\lambda$ in an efficient ...
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27 views

Can I do gradient ascent this way for a non-differentiable function?

I have a probability distribution $P(x)$ where $x$ is a N-dimensional vector with constraints $sum(x)=1$. This distribution $P(x)$ does not have a closed form. $P(x)$ is a function where I query the ...
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2answers
40 views

Can I optimize area of cylinder with no givens?

I have a problem which should be very easy (as the rest of them are on this worksheet) but this one has me stumped. The question reads: A metal can is in the form of a cylinder. It has a bottom ...
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How to solve for CE?

I have a function $f(CE) = 2 \cdot \sqrt{4^2+(3-CE)^2} + \sqrt{4^2+CE^2 }$ And it's derivative $f^′ (CE)= \dfrac{−6 + 2 CE}{\sqrt{25−6CE+CE^2}}+\dfrac{CE}{\sqrt{16+CE^2 }}$ And then I'm trying to ...
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1answer
43 views

Is $\mathbf{y}^*$ a local minimizer of $f(\mathbf{h}(\mathbf{y}))$?

Let $f(\mathbf{x})$ be a twice differentiable function, where $\mathbf{x} \in \mathbb{R}^n$. Let $\mathbf{x}^*$ be a local minimizer of $f(\mathbf{x})$. Consider a differentiable and invertible ...
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1answer
30 views

Method for maximizing simple functions

I am wondering if there is a general method or approach to maximizing ( or minimizing) multivariable functions. For example, consider $f(x,y)=49+4x-x^2-2y^2$ over $\mathbb R^{2}$ Now, it could be ...
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1answer
23 views

Does it make sense to use optimization algorithms(Like ACO) in weighing average to find weighs

I am going to use a DEM fusion method using simple weighing average,I am going to use 2 inputs to create my fusion function W1X1+W2X2/(w1+W2)=result this is a ...
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1answer
33 views

Shortest path in the plane under derivative constraint

A colleague posed a toy problem to me today that degenerates to finding the curve y(x) of shortest length than connects two points in the plane (WLOG: y(0) = 0, y(a) = b), such that y'(0) = 0. This ...