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Questions tagged [optimization]

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.

In mathematics, computer science, economics, or management science, mathematical optimization (alternatively, optimization or mathematical programming) is the selection of a best element (with regard to some criteria) from some set of available alternatives.

An optimization problem can be represented in the following way: given a function $f:A\to\mathbb{R}$ from some set $A$ to the real numbers, we want to find an element $x_0\in A$ such that $f(x_0)\le f(x)$ for all $x \in A$ ("minimization") or such that $f(x_0)\ge f(x)$ for all $x \in A$ ("maximization").

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Maximize $(ab+cd)^2$

$a^2 + b^2 - \frac{ab}{2} = c^2 + d^2 + \frac{cd}{2} = 256$ $ac + bd = 240$ where a, b ,c ,d are positive reals, maximize $(ab +cd)^2$ looking at the equations and restrictions, I think Cauchy can be applied $(a^2+c^2)(b^2+d^2)> (ab + cd)^2$ so we…
SuperMage1
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solution of linear equation

I have the following optimization problem: \begin{equation} \begin{aligned} \underset{x}{\text{min}} {\sum_{i=1}^{k}} w_{i}x \end{aligned} \end{equation} Is the solution x = 0? Here $w_{i}$ is given. Both $w_{i}$ and x are scalars.
qwerty
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Area Minimization of Circle

The radius of the circle having minimum area which touches the curve y=4-x² and the lines, y=|x| is? I tried using the normal to the curve and satisfying it with the centre of the circle but it just gives me a complex 4th degree equation which I'm…
Sameer Thakur
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Hessian vs. Bordered Hessian

I am confused as to whether to use the standard hessian or bordered hessian for the following problem. $f(x,y,z) = (x+1)^2+(y+1)^2+z^2 \text{ subject to } x+y=(x-y)^2 \text{ and } z-x-y=1$ We are told that $(0,0,1)$ satisfies the first order…
atd31
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Minimizing $f(x,y)$ subject to $0 \leq x \leq 1, \ 0 \leq y \leq 1$

I have to solve the following problem, and I don't know how to proceed: $$\min \{(x-a)^2 + (y-b)^2+xy\}$$ subject to $0 \leq x \leq 1, \ 0 \leq y \leq 1$. I have tried to use Lagrangian multipliers but, I am not sure how to put the conditions $0…
frank99
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Why is Optimization being taught under different titles in different universities?

I am searching for some pdf lecture notes on Optimization: Google search I have found some pdf lecture notes. But, they have various titles like: Convex Optimization Optimization Methods Optimization-I Numerical Optimization Engineering…
user366312
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How does the Gradient Projection Method work?

I'm trying to get this method working. I cannot understand the computations included in the answers. I know what $B$ and $N$ are and how they are calculated. Can't see where the basis comes from and if it's actually given. I don't even see where…
Hypergeometry
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How do you prove the following maximization task?

If $x + y = a$, then the maximum value of $x * y$ is when $x=y=a/2$. ($x, y$ are positive numbers)
user2609410
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Relationship between optimality and feasibility

Say I have two optimization problems A and B. If an optimal solution of A is feasible for b and vice versa: Q 1) Is every optimal solution for A an optimal solution of B (and vice versa)? Q 2) If the answer to Q1 is false, then is there at least one…
rahs
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Minimizing square of a norm

I have a question and I am fully aware it is "trivial". I wonder the following: If we minimize $\|x\|^{2}$ at the same time also $\|x\|$ becomes minimal. This is trivial insofar as it is very basic intuition. What I have realized though is that I…
MiauPiau
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What is $C^2$ in this theorem?

In their book "An introduction to Optimization", 4th Edition, Chong and Zak has following text. What does the $C^2$ mean? Theorem 6.2 Second-Order Necessary Condition (SONC). Let $\Omega \subset R^n$, $f \in C^2$ a function on $\Omega$, $x^*$ a…
O. Altun
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What is the difference between "Design Function" and "Objective Function"?

What is the difference between "Design Function" and "Objective Function" in the study of Optimization?
user366312
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Consider the elastic-net optimization problem

Consider the elastic-net optimization problem: $$\min_{\theta}||y-X\theta||_2^2+\lambda[\alpha||\theta||_2^2+(1-\alpha)||\theta||_1].$$ I was told that I could turn this into a Lasso problem, using an augmented version of matrix $X$ and vector…
Layan
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Is it okay?? Any lambda is equal to zero

In optimization problem using Lagrange multiplier.. $f(x, y, z) = x + 2z$, subject to $g_1(x, y, z) = x + y + z = 1$, $g_2(x; y; z) = 2x + z = 2$ Find $x, y, x, \lambda_1, \lambda_2$ ............................................. I couldn't find the…
woogawooga
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Global minimum of $g(x)=\sum_{v=1}^k c_v\|x-x_v\|_2^2, \ \ c_v>0 \in \mathbb{R};\ x,x_v \in \mathbb{R}^n$

$g(x)=\sum_{v=1}^k c_v\|x-x_v\|_2^2, \ \ c_v>0 \in \mathbb{R};\ x,x_v \in \mathbb{R}^n$. I need to find the minimum of $g$ and reason why it is a global minimum. I tried the following: $g(x)=\sum_{v=1}^k c_v\|x-x_v\|_2^2=\sum_{v=1}^kc_v…
user597304
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