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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45 views

Finding the max of $x^2/(2 + x^2)$

I tried to find the maximum value of the following function using the first derivative and equate it to zero : The function : $y=x^2 /(2+x^2)$ The first derivative: $4x/(x^2 +2)^2 =0 \implies x=0$ ...
-2
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0answers
22 views

It is an optimization problem, how can i set it up? [on hold]

Your car runs out of gas on a road running north/south at a point 3 miles from the highway (which runs east/west). The nearest gas station is 6 miles east down the highway from the point your road ...
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1answer
23 views

Algorithm to distribute numbers so that average of sets is closest to total average

Given a collection of N non-unique decimal numbers d1, d2, d3 ... dN whose average is μT. Assume we distribute d1, d2, d3 ... dN across K sets with sizes s1, s2 ... sK and therefore: $$\sum_{i=1}^K ...
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0answers
12 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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0answers
21 views

Maximize profit with dynamic programming

I have 3 tables… $$\begin{array}{rrr} \text{quantity} & \text{expense} & \text{profit}\\ \hline 1 & 2312 & 3212 \end{array}$$ All equal structure but with different values and ...
2
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1answer
11 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 ...
1
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3answers
45 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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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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3answers
19 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 ...
1
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1answer
21 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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0answers
14 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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0answers
17 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 ...
1
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0answers
37 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 ...
0
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3answers
27 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 ...
1
vote
2answers
45 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 ...
0
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0answers
14 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 ...
2
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3answers
113 views

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

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 $g(0)=g(1)=0$?
0
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0answers
14 views

Mean-variance portfolio optimization MATLAB [on hold]

I am doing just a simple mean-variance optimization in Matlab. I know my question may be stupid/easy to solve, but I am a beginner user so your help would be very appreciated... I successfully get ...
1
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0answers
20 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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0answers
18 views

Handle (boolean/integer) decision-variables in heuristic optimization algorithms [on hold]

Decision variables (at least in my problem) tend to change the problem in a rather drastic way. The different states control like on-and-off parameters how the objective function/or constraints are ...
0
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3answers
31 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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2answers
28 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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1answer
35 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 ...
1
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1answer
46 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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0answers
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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0answers
42 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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0answers
10 views

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 ...
2
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1answer
15 views

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 \\ ...
0
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1answer
22 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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1answer
44 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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17 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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0answers
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. $E(x_i(t_1)x_j(t_2)^*)=0 \space \space \forall t_1,t_2, \space i\neq j$ ...
2
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1answer
22 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) = ...
0
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0answers
19 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 ...
0
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1answer
20 views

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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0answers
16 views

Longest chord in the intersection area of n circles

Given n circles that intersect, there is a shape in the space where the intersect. Given that, what is the longest chord, more generally longest line, that can be drawn in this space? For n=1, this is ...
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0answers
10 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 ...
0
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0answers
25 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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0answers
27 views

Find max volume of a cone [on hold]

I have problem with following task: You have two cones "glued" together (base to base), so that the figure is created by a orthogonal triangle: a + b + c = 1 Also, you can rotate the triangle around ...
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2answers
30 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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1answer
60 views

Inequalities with $e$, no calculus (Challenge) [on hold]

So, here's a little bit of fun I came up with. (If this isn't the best place for a challenge that I already know how to solve, let me know... I hope it's OK with you guys.) So, when we try to sum ...
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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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0answers
38 views

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
82 views
+100

The minimum of $I_{n,k}=\int_0^{2\pi}\sqrt{3+2\cos(nx)+2\cos(kx)+2\cos(nx+kx)}dx$ is attained for $k=n$

I have the following conjecture: ``For each given $n\in\mathbb{N},\ n\ge 2$ the minimum of the sequence of integrals $I_{n,k}=\int_0^{2\pi}\sqrt{3+2\cos(nx)+2\cos(kx)+2\cos(nx+kx)}dx,\ k=1,2,\dots,n$ ...
2
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1answer
31 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 ...
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0answers
22 views

Setting up the Bellman equations for dynamic programming

I have the following question I want to understand. The owner of a chain of three grocery stores has purchased five crates of fresh strawberries. The estimated probability distribution of ...
0
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1answer
41 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 ...
0
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1answer
21 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 ...
1
vote
2answers
53 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 ...