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

Minimization problems that do not depend on data

I'm trying to solve the following minimization problem: $$\underset{\lambda\in[a,b]}{\min}\|A(\lambda)X-X\|_2$$ for given $a
ar_ma_cs
  • 33
1
vote
1 answer

Is the minimizer of condition number of a PSD matrix $A$ the same as the minimizer of $\lambda_{max}(A) - \lambda_{min}(A)$?

Given a positive semidefinite matrix $A\in\mathcal{C}\subset\mathbb{S}^n_+$, where $\mathcal{C}$ is a convex set. Define the following minimizers: $$ \bar{A} = \arg\min_{A\in\mathcal{C}} \frac{\lambda_{max}(A)}{\lambda_{min}(A)}, \quad \tilde{A}…
Shih-Chi Liao
  • 132
  • 7
1
vote
0 answers

Optimizing vector distances to a certain distribution with Gaussian kernel function

Suppose we have a set of vectors on a unit circle $u_i$. And we use the Gaussian kernel to measure the distance among them, which could be described as: $$ G(u,v)=e^{-||u-v||_2^2} $$ One already proved theory is that when we maximize the sum of…
Haochen Li
  • 11
1
vote
0 answers

Is a bi-Level optimization problem is same as an optimization on two variables?

Is a bi-Level optimization problem is same as a constrained optimization on two variables? Does it make sense to compare these two types of problems? Given a two-variable constrained optimization problem, can we consider it a bi-Level optimization…
Myshkin
  • 35,974
  • 27
  • 154
  • 332
1
vote
2 answers

Closed form minimization of this function

I'm trying to find a value of $x$ in terms of positive integers $n,b$ that minimizes this function: $$ \frac{b\,(n+2^x)}{x} \quad (x>0) $$ So far, I've tried differentiaton, but that doesn't lend itself to a clean solution. Any thoughts?
user348382
  • 447
1
vote
0 answers

Minimum of a multivariable function

Conjecture 1: Let $f : \mathbb{R}^n\to \mathbb{R}$ be a continuously differentiable function and $a, b\in\mathbb{R}^n$ such that $a_i \leq b_i$ for each $i\in \{1, 2, \ldots, n\}$. Define $v = \nabla f(a)$. If $f(a) \leq f(b)$ and $v_k < 0$ for…
arxtch
  • 11
  • 2
1
vote
1 answer

Minimization of the sum of absolute values

Here is a sum of absolute values $$V(x)=\sum_{i=1}^n|x_i - x_{i-1}|$$ and the sum of square errors is $$U(x)=\sum_{i=1}^n(x_i - a_i)^2$$ My problem is to minimize $V(x)$ while keeping $U(x)$ as small as possible, so I might formulate a new objective…
avocado
  • 1,209
1
vote
0 answers

Quadratic programming with modified x

In the objective function of quadratic programming, the vector x and its transposed are existing. In the constraints also vector x exists. If there is a way that both in the cost function and in the constraints x be a function of x? Not x itself?
Simin Hesami
  • 11
1
vote
1 answer

optimization problem to minimize two funtions

Here is an minimization problem involving $2$ variables, $x$ and $y$. And there are $2$ functions, $g(x,y)$ and $h(x,y)$. The goal of this minimization problem is to find out the $(x,y)$ which minimizes both $g(x,y)$ and $h(x,y)$ at the same time. I…
avocado
  • 1,209
1
vote
0 answers

Why Lagrange Multipliers does not work for this problem?

The problem statement is find the maximum and minimum of $f(x,y) = \frac 3 2 x^2y^2+5x + 18y + 25$ under the constraints $ -5 \le 2x - y \le 5 $ and $ 0 \le x + 2y \le 2.5$. I graphed the restricted domain here. For unconstrained extrema (aka local…
john
  • 860
1
vote
1 answer

What is the name of this Knapsack-like problem?

I am looking for the name of the following problem: Imagine having a function $f(t)$, which basically resembles the maximum weight from the Knapsack problem for each t (although for what I try to do I just want to be close to f(t) don't have to stay…
mth_nub
  • 11
1
vote
1 answer

From constrained to unconstrained problem

I have a doubt regarding a constrained optimisation problem. Suppose my original constrained minimisation problem is $$ \min_x f(g(x), x) \quad \text{ s.t. }\quad g(x)=3 $$ I would like to know if this equivalent to solving the unconstrained…
Star
  • 222
1
vote
0 answers

Generalized Simulated Annealing Algorithm dividing by zero when choosing qv = 1 and qa = 1?

I am currently trying to understand the Generalized Simulated Annealing (GSA) optimization algorithm proposed by Tsalis and Stariolo in their paper. In the GSA algorithm, they try to generalize the "Classical Simulated Annealing" (CSA) and the "Fast…
CitrusToffee
  • 11
  • 2
1
vote
1 answer

Equivalence of maximizing ratio between two functions with shifted

Suppose, I need to maximize the function $\frac{a(t)}{b(t)}$ over some continuous domain of $t$, and I know the maximum exists. WLOG, the maximum is obtained at $t_1$. Now I'm wondering whether if $\frac{a(t)+C_1}{b(t) + C_2}$, where $C_1, C_2 > 0$,…
peng yu
  • 1,271
1
vote
1 answer

Solving linear equation with max min function

For a particular problem, I got some equations of the form $a = \min(1-b,0.9)$ $b = \max(\min(1-a,0.7)$, $\min(c, 0.8))$ $c = \max(\min(1-a,0.9)$, $\min(1-c, 0.9)) a,b,c \in [0,1]$ The system has a unique solution $a = 0.23333, b = 0.766666, c =…
asraful islam
  • 11
1 2 3
…
97 98