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
2 answers

Local minimum of $\sqrt[x]{n\over v-x+1}$

I'm trying to find the local minimum of $\sqrt[x]{n\over v-x+1}$ with respect to $x$. The restrictions on $x$ are that it must be $\le v$ and $\ge 1$. Also, $v$ and $n$ are fixed, and $v
Avi
  • 1,780
  • 11
  • 21
1
vote
1 answer

How to interpret a discontinuity in 2D Pareto Frontier?

I've solved a bi-objective optimization problem by means of NOMAD solver from OPTI Toolbox and as a result I've obtained a Pareto frontier: How to interpret the visible "gap" in the Pareto frontier?
mc2
  • 181
1
vote
0 answers

Find unknown such that four dependent quantities have the same value.

I have $12$ unknown $a_i, b_i, c_i, i=1,\ldots,4$, that should satisfy equations $$ \sum_{i=1}^4n_ia_i=a,\quad\sum_{i=1}^4n_ib_i=b,\quad\sum_{i=1}^4n_ic_i=c, $$ where $n_i,\,i=1,\ldots,4$ and $a,b,c$ are positive and known constants. Some other…
Vincenzo Tibullo
  • 10,955
1
vote
0 answers

Unconstrained optimization problem (lasso with modification)

I am looking to solve the following unconstrained optimization problem: $$\arg \min_U \|b-A(UY^*)\|_F^2+\lambda\|U\|_1$$ where $\|.\|_F$ is frobenius norm. I know that the solution without the $\lambda\|U\|_1$ term, will be the least square…
Parisa
  • 11
1
vote
1 answer

The equality of gradient between different calculations?

Suppose there is a problem $$\min\limits_v\max\limits_x E(v,x).$$ $E$ is a concave function w.r.t. $x$. But w.r.t. $v$, $E$ is a convex function plus a concave function. I can get $x^*=\arg\max\limits_x E(v,x)=\phi(v)$. Since $E$ is a convex…
Heather
  • 163
1
vote
2 answers

MaxMin: how much does the min "see"?

Consider the following quantity: $$ \max_{a \in \{-1,1\}}\left( \min_{b \in \{-1,0,1\}} ab\right).$$ Since the min is inside, we apply it first, but what value $b$ will be chosen? If the minimum "knew" that there was a maximum waiting for it…
Mark Perlman
  • 725
1
vote
1 answer

Need help with optimization concepts.

In some optimization problems with inequality constraints some of the aforementioned constraints can be x>=0 , y>=0 and so on. I think these constraints are called non negativity constraints; they just state that some of the variables of the…
Werther
  • 85
1
vote
1 answer

Need help with the graph of a function

In the optimization problem max: $$6x+2xy-2x^2-2y^2$$ subject to $x+2y\le2$ and $-x+y^2\le1$ I need to draw the graph of the feasible region in order to determine if the problem has global solutions, but I don't know what the graph of $-x+y^2=1$…
Werther
  • 85
1
vote
1 answer

Help me out with this optimization problem

This excercise has been taken from an exam. In the following problem: opt:x+y^2-2 subject to y^2<=x and x<=2-y and y>=0 I've found the green area to be the feasible region. (Sorry for the poorly drawn graph). I determined, by Weierstrass' theorem,…
Werther
  • 85
1
vote
2 answers

Minimizing $f(x,y,z)=\dfrac{|x|+|y|+|z|}{xyz}$ on a sphere

I need to find the minimum of the function: $$f(x,y,z)=\dfrac{|x|+|y|+|z|}{xyz}$$ with the condition: $$x^2+y^2+z^2=r^2$$ Using numerical methods it's quite easy to solve the problem. How can I proceed to solve it analytically? Thanks.
Riccardo.Alestra
  • 10,546
1
vote
2 answers

Clarification on optimization problem

While reading a combinatorics paper about packing densities in compositions, I encountered the following optimization problem. Maximize $$f(\alpha_1, \dots, \alpha_5) = \sum_{1 \le i < j < k \le 5} \alpha_i \alpha_j \alpha_k$$ with the…
Daniel R
  • 3,199
1
vote
3 answers

Max $f(x,y,z) = \min\{x, 5y+2z\}$ subject to $x+15y+7z=44$

Max $f(x,y,z) = \min\{x, 5y+2z\}$ subject to $x+15y+7z=44$ As well, $x,y,z \geq 0$ I have guessed that the extrema point will be a point such that $x=5y+2z$ and tried solving for the curve of intersection of $z=(x-5y)/2$ and the constraint…
user11135
  • 773
1
vote
0 answers

Optimization problem, how to obtain the optimal F?

The objective is as follows: $\min_{\mathbf{F}} a Tr(\mathbf{F} \mathbf{F}^H) - \mathbf{b}\mathbf{F}^H \mathbf{C} \mathbf{F} \mathbf{d}$ $s.t.\ \ \ Tr(\mathbf{F} \mathbf{F}^H)
Michael Fan Zhang
  • 121
1
vote
0 answers

Algorithm for redistributing wealth

This seems like a trivial problem, but dumb me thinked forever without proving any solution as the optimal. So the problem is, given $n$ people, each with wealth $w_1,\dots ,w_n$, with $w_i$ being any real number, how to move money among the $n$…
ithisa
  • 2,763
1
vote
1 answer

Constrained maximization in L dimensions

The problem is like $max_{\mathbf{x} } \:u(x_1, x_2, ..., x_L) = - \sum_{i =1}^{L} \mid x_i - a_i\mid$, $ s.t. \sum_i^L x_i \leq C$, for each $i$, $a_i > 0$ is a scalar; $C$ is a constant that is strictly greater than $0$; $\mathbf{x} = (x_1,…
user123818
  • 69
1 2 3
…
97 98