Mathematics Stack Exchange
Mathematics Stack Exchange

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").

22512 questions
1
vote
1 answer

is $f(S) =(|S|)^2$ submodular?

I am working on an optimization problem for my graduate research. I would like to know whether the square of the cardinality of a set is submodular or not. Thank you in advance for the help.
Jose Cadena
  • 13
1
vote
1 answer

What is a Kuhn Tucker point?

For a Nonlinear Programming problem, what is a Kuhn Tucker Point? Is a local min. necessarily a Kuhn Tucker point?
ThinkConnect
  • 784
1
vote
0 answers

In this context, what is a Kuhn-Tucker point?

I encountered an algorithm, and a step in the algorithm says "Solve the NLP Problem X, to the extent of computing a Kuhn-Tucker problem." I would like to know, is a Kuhn-Tucker point just a point that meets the KKT conditions? If I find a local…
ThinkConnect
  • 784
1
vote
1 answer

Maximize the sum of numbers subject to an upper bound on their sum of squares

I want to find a maximum for $\sum_i^n x_i$ subject to $\sum_i^n x_i^2 \leq c$, where c is some constant. I've done some searching and that's hinted at using the Cauchy-Schwarz inequality, but I don't yet see how to use it here.
Gaussian0617
  • 704
1
vote
0 answers

Linear optimisation of $x \sin x$?

I'm working on a non linear optimisation problem in xpress. To be able to run in in xpress it has to be a linear equation. The equation is $x \sin x$. The restrictions: $x < 2 \pi$ $x > 0$ I thought about using Taylor series to find an…
Jose
  • 11
1
vote
1 answer

What is the role of the $L_2$-norm in optimization?

I'm a little confused about the role of the $L_2$-norm in optimization. Suppose my data is $n$-dimensional data, and I have some input pairs $(x, y)$ and a function $f(x)$ which I want to learn. For my cost function, I sum up, over all training…
Karnivaurus
  • 827
1
vote
3 answers

Can gradient descent be applied to non-convex functions?

I'm just learning about optimization, and having trouble understanding the difference between convex and non-convex optimization. From my understanding, a convex function is one where "the line segment between any two points on the graph of the…
Karnivaurus
  • 827
1
vote
1 answer

Proofs of mathematical optimization theorems

How to show that if x minimizes f over S and x belongs to R, which is a subset of S, then x also minimizes f on R Please help me with this proof. Thank you.
Jackson
  • 101
  • 7
1
vote
1 answer

What is the difference between "optimizing" and "minimising"?

When mathematicians speak of 'optimising' do they always mean 'minimising'? Or does the term refer to a basket of techniques? For example Wolfram MathWorld describes Optimization Theory as, "A branch of mathematics which encompasses many diverse…
Robinson
  • 111
1
vote
0 answers

Discrete optimization of function

I'm having troubles with searching for analytical solution of following problem. Let we work in 3-D space and have the set of points (uniform net at cube's…
Sh.N.
  • 91
  • 5
1
vote
1 answer

Find maximum of quotient of functions

To maximize $f(x)=\frac{g(x)}{h(x)}$ on all of $R$, I think I have found a method. First, find the point at which $g(x)$ is maximized. Call it $a$. Second, find the point at which $h(x)$ is minimized. Call it $b$. Finally, compute $f(a)$ and $f(b)$.…
P Prog uzz Coderam Golflesming
  • 11
1
vote
1 answer

Minimize the volume bounded by a plane

A plane passes through the point $(a, b, c)$. Find its intercepts with the coordinate axes if the volume of solid bounded by the plane and the coordinates planes is to be a minimum. What I have tried: Let $\langle x,y,z \rangle$ be the normal…
user134927
  • 500
1
vote
1 answer

Optimization of shoe manufacturing

I cannot seem to figure out the best way to optimize the shoe manufacturing algorithm in order to minimize the costs in the company I work for. Let me describe the problem a bit. A customer makes the following…
emihir0
  • 113
1
vote
2 answers

Underdetermined Equation Optimization

For the equation: $$5X + Y + Z = 600$$ With constraints: $$92 \le X \le 95$$ $$46 \le Y \le 55$$ I want to find a method that will choose values for $X$ and $Y$ such that $\lvert Z\rvert$ is minimized. Any help would be appreciated!
FishandChips
  • 13
1
vote
1 answer

Penalty Methods for Optimization

Consider the following problem: $$ \text{minimize} \ f(x) \\ \text{subject to} \ g(x) = 0 $$ This is a constrained optimization problem. We want to convert it into an unconstrained optimization problem so that we can use Newton's Method,…
Mark Damien
  • 41
  • 2
1 2 3
…
97 98