Mathematics Stack Exchange
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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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. Find the dimension of a rectangular piece of a lot that can be covered by 500 meters of barbed wire fence

Solve the given problem. Find the dimension of a rectangular piece of a lot that can be covered by 500 meters of barbed wire fence
Austin John Areza
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Minimizing the time to produce $T$ items with machines that run less efficiently over time

I would like to produce $R$ items in the shortest amount of time possible. For the sake of a visual, call these items bottles of carrot juice. Let $t_1$ be the time to purchase and set up a blender to produce units of carrot juice. However, the…
MirrorEcafrus
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How can i optimize this problem?

I am quite confused, I need to do an optimization of a warehouse filling that depends on prices. That is, when it is cheap I fill it and when it is expensive I take it out to sell, two types of restrictions must be met: Have a minimum and maximum…
ADMGYP
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Adding scalar to indicator function

I'm doing some self-study and came across this paper where it appears in Section 4 the authors are adding a scalar to the vector $\boldsymbol 1_{x\geq 0}$. The other two functions called $f$ are $\min_{\boldsymbol x} -\frac12 \boldsymbol x^T \Big (…
Gustav Lilovic
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(Solved) Lipschitz Continuity of Huber Function

I am reading the book Introduction to Nonlinear Optimization by Amir Beck. This problem is the problem 9.2 in his book. Problem: Consider the Huber function $$ H_{\mu} = \begin{cases} \frac{\|x\|^2}{2\mu}, & \|x\| \leq \mu, \\ \|x\| - \frac{\mu}{2},…
Siamese
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Optimal equation for combining 2 parameters to make prediction

I have a binary classification (true or false). I also have two parameters (A and B). I want to combine A and B in to one equation. Then, based on the output of this equation, I want to be able to classify as true or false. So basically come up with…
user1011182
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Maximizing $q_1+q_2$ subject to $x_{11}q_1^b+x_{21}q_2^b\leq k_1$ and $x_{12}q_1^b+x_{22}q_2^b\leq k_2$. How thorough do I need to be in my argument?

This is more about what I have to do than about how to do it. I have the following maximization problem: $$ \begin{array}{ll} \text{maximize}&q_1+q_2\\ \text{subject to}& x_{11}q_1^\beta+x_{21}q_2^\beta\leq k_1\\ &…
Patricio
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positive semi-definite hessian on open set and optimal minimum

I don't know how to prove this theorem (please help): Let $f$ be a twice differentiable function on an open set $U \subset E$ (normed vector space). Let $x$ be a critical point of $f$, we suppose there exists an open set $B \subset U$ such that $x…
array.02
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Is the method given below correct upto some limitations for finding maxima of product of two functions f(t)g(t) under constraints?

Suppose $a= f(t)>=0$ and $b= g(t) >= 0$ . Can we say that maximum value of $f(t)g(t)$ is $\frac{(f(t) + g(t))^2}{4}$ using AM-GM ? So we can just find maximum of function $\frac{[f(t)+g(t)]^2}{4}$ and lets say its equal to $M$ then $M>= f(t)g(t)$ ?…
Orion_Pax
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Operation sharing in mathematical expressions

I have a set of mathematical expressions (only contains sum (+) operation). Here is an example: $$E1= a+b+c$$ $$E2= a+b+d$$ $$E3= a+b+c+e$$ $$E4= a+b+d+f$$ I want to know if there is a method or algorithm to compute these expressions by using the…
Ana.IM
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How to formulate $\min_{f \in P_n(I)} \int_0^1 \lvert f(x)-g(x) \rvert^2 dx$ over $f_n,\ldots,f_0$?

I am working on the following exercise: Let $I = [0,1]$ and let $g: I \rightarrow \mathbb{R}$ be a continuous function. Let further be >$P_n(I)$ be the set of polynomials $f$ with $\deg(f) \le n$ on $I$. Formulate the problem $$\min_{f \in P_n(I)}…
3nondatur
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An optimization of matrix

I have a matrix optimization problem denoted as below: $$\arg\min_{\alpha_i, \beta_i} \lVert \sum_{i=1}^4 (\alpha_i M_i - \beta_i P_i) \rVert^2_2 \\ \text{s.t. } \alpha_i > 0, \beta_i > 0$$ where $M_i,i=1,2,3,4$ and $P_i,i=1,2,3,4$ are constant…
Johnachale
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Circular motion interception problem

I'm trying to calculate the optimal speed (minimizing time) to intercept another object that is moving in the same circular path as me. Here's the catch, there are some constraints: There is a mininum value for the time ($t_{min}$). My speed has a…
kiel814
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Why do only the original $f(x)$ function and the inequality constraint/term change signs when converting the $\min\max$ problem to a $max\min$

To perform constrained maximization, we can construct the generalized Lagrange function of $-f(x)$, which leads to this optimization problem: $$ \min _{x} \max _{\lambda} \max _{\alpha, \alpha \geq 0}-f(x)+\sum_{i} \lambda_{i} g^{(i)}(x)+\sum_{j}…
Revolucion for Monica
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Importance of the KKT Conditions

Does anyone know why the KKT Conditions are considered such a fundamental result in optimization? As far as I understand, the KKT Conditions appear to be a set of conditions that if satisfied - suggest that the "constraints" within the optimization…
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