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").

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How would you define the following optimization problem?

Axiom: Desires drive people and each person desires to continuously experience minimum injustice and maximum justice with respect to everything. Given the above axiom, how would I define the optimization problem so that it would account each level…
Ramin Melikov
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Understanding a basic optimization problem

In my introduction to optimization course, we are given the following problem as an example: Sorting task: Given real numbers $c_1, c_2, \ldots, c_n \in \mathbb{R}$, we want to find the $k$ smallest numbers. This is the $k$-smallest numbers…
Jon
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Maximum number of interior local optima of a function

Let $f(x) = a_1 + b_1 x + c_1 x^2 + d_1 \sqrt{a_2 + b_2 x + c_2 x^2}$ be a function whose domain is $[0,1]$. It is known that $d_1 < 0$ and $\forall x \in [0,1]$, $a_2 + b_2 x + c_2 x^2 > 0$. What is the maximum number of interior local optima of…
Rittwik Chatterjee
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Minimize Expected Value of Euclidean Norm

Suppose $x = (x_1, x_2, \dots, x_n)^t \in \mathbb{R}^n$. The Probability distribution function of $x$ is $f(x)$. My goal is to minimize the following function, \begin{equation} \underset{a \in \mathbb{R}^n}{\arg\min}…
Robin
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The sum of two optimization problems is the optimization of the sum of the two problem

Suppose I have two independent problems of the form $\max L(f)$ and $\max L'(g)$ for two objective function $L$ and $L'$. We can assume that the space over which we try to find solutions $f,g$ are well defined and the two solutions exist. How can we…
swissy
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Specifying a Restricted Domain that can Make a Function Convex

Define a function $f : S_1 \to \mathbb R$ by: $$ f (\vec x) = (x_1 - 2)^2 - (x_2-1)^2 $$ with domain $$S_1 = \left\{\vec x \in \mathbb R^2 : 0 \le x_1 \le 1, \lvert x_2\rvert \le \frac{x_1}{2}\right\}.$$ Sketch $S_1$. How do we go about sketching…
Reese Cha
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Maximum values of a a function using numerical methods

How can I numerically find the local maximum values of a function? $y=1000 + (100/6)xe^{\sin x}$. Interval $[0 - 100]$. I tried differentiating it but I get stuck at solving the differential being equal to zero. Is there a way to find this?
Lemour
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nonlinear optimization problem,a) solve the problem geometrically.b) verify the optimality of the function by the Kuhn - Tucker conditions

Can you guide me on how to solve this exercise? I need your help please. I don't understand the correct way to do it. being a nonlinear programming problem. can it be solved by simplex method? or the simplex method only applies to linear…
angelabayona
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How to model drop off when having no order on the graph nodes

I'm trying to model a pickup and delivery problem under some constraints. In addition to a binary variable $ x_{i}^{pd}$ which is for pickup, I have another binary variable like $ y_{i}^{pd}$ for drop off. x(i,p,d) = 1 if product p be picked up by…
yaodao vang
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Minimum value of a product $(x+a)(y+b)(z+c)$ for $xyz=d^3$

Show that the product $(x+a)(y+b)(z+c)$, where $a,b,c$ are positive constants, subject to the condition that $xyz = d^3$, where $d$ is a positive constant, has its minimum when $x=$$ad\over\mu$, $y=$$bd\over\mu$,$z=$$cd\over\mu$ and $\mu =…
Natalie
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Lagrange Multipliers to find shortest distance

Use Lagrange multipliers to determine the shortest distance from a point $\,x \in R^n\,$ to a plane $\{y\mid b^Ty = c\}.$ I don't even know where to start!
Natalie
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Minimum point of Constraint set

Let $ c \in R^n $ be non-zero, and consider the problem of minimizing the function $f(x)=c^Tx $ on some constraint set $ S$. Show that a minimum point of this problem cannot lie in the interior of the set $S$.
Natalie
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Simultaneous optimization

Consider a continuous and differentiable function $f(x,y,z)$ with $x,y,z\in \mathbb{R}$. For each $(y,z)$, there exists a unique $x_1(y,z)$ that maximizes $f(x,y,z)$. For each $(x,z)$, there exists a unique $y_1(x,z)$ that maximizes $f(x,y,z)$.…
Paul
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Partial minimization over variable

Suppose that we have a general mathematical program $(P)$ with value \begin{align} \nu(P) := \min_{x,y}\; \{f(x,y) : g(x,y) \leq 0\}. \end{align} Is there a standard way to define functions $\tilde{f}$ and $\tilde{g}$ so that we can write the…
jjjjjj
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Show $\arg\max E_Y[\min(f(x),Y)]\le \arg\max f(x)$

Suppose $f(x)$ is a concave function and $Y$ is a random variable. What is the relationship between $\arg\max E_Y[\min(f(x),Y)]$ and $\arg\max f(x)$. I feel the former is less than or equal to the latter, but do not have a clue to show it.
Justin
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