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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Theorem: Trust-Region Dogleg method

The following is part of an assignment so please no full answers. I am trying to prove the following theorem. In the Dogleg method we consider $$ p^u = - \frac{g^Tg}{g^TBg}g $$ and $$ p(\tau) = \tau p^u $$ for $0\leq \tau \leq 1$. Now, let $B$ be…
wrong_path
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Help me answer this optimization problem.

What two nonnegative real numbers $a$ and $b$ whose sum is 23 maximize $a^2+b^2$? Minimize $a^2+b^2$?
Chris
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Minimum value of an expression

What would be the minimum value of $$ (1+a)(1+b)(1+c)(1+d) \left({\frac{1}{a}}+{\frac{1}{b}}+{\frac{1}{c}}+{\frac{1}{d}}\right) $$ Given that $a,b,c,d$ are all positive real non zero number THE ANSWER IS ${\frac{4^5}{27}}$
Saakshya Devat
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Minimization of a weighted average

The optimization problem is $$I = \min_{N_i \in \mathbb N}\max_{1 \leq i \leq m} \frac{t_i}{N_i}$$ given the constraint $$\sum_{i=1}^{n} N_i = N $$ $t_i$ are some known constants. Now, intuitively, the $N_i$'s which most closely proportionately…
Dhanvi Sreenivasan
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$f(x,y) = x^2 -xy +y^2 +3x -2y +1$ Hessian Gradiant

Suppose we have a function $f(x,y) = x^2 -xy +y^2 +3x -2y +1$ To find critical points of $f$, we compute its gradient: $\nabla f=(3+2x-y,\ -2 -x +2y)$ The Hessian matrix for function $f$ is: $$ \nabla^2 f = \begin{pmatrix} 2 & -1 \\ -1 & 2…
Paul-Henri GUET
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Maximization of Utility Function

Kabir’s utility is $U(c,d,h) = 2c+5d-d2-2h$, where $d$ is the number of hours per day that he spends driving around, $h$ is the number of hours per day spent driving around by the other people in his home town and $c$ is the amount of money…
Shraddha
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Optimization with multiple constraints

I would like to solve the following optimization problem. I have the following quadratic object function: $$f=\sum\limits_{i=1}^n (x_i-y_i)^2 = min$$ subject to the following linear constraints: $$\sum\limits_{i=1}^n y_i =…
bkosztin
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Alternating Minimization: global convergence

I am trying to show that the alternating minimization algorithm will converge to the global minimum for my application. Since I am not a mathematician, I would appreciate if someone can either improve my arguments or find flaws in those. My AM…
Dushyant Kumar
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What does it mean for a curve to move in the normal direction?

What does it mean for a curve to move in the normal direction? Especially in the context of shape optimization and level set methods. Does it mean that the boundary of a set is considered a curve and when this curve evolves through different shapes,…
mavavilj
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What is the maximum area of a rectangle circumscribing another rectangle?

The problem says that we have to find the maximum area of a rectangle that circumscribes another rectangle of sides $a$ and $b$.
Josemi
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Maximum entropy problem constraints

https://www2.warwick.ac.uk/fac/cross_fac/complexity/study/msc_and_phd/co904/co904online/lecture-3-4.pdf (the without expectation setting) \begin{align*} maximize \ &\sum_{i=1}^K -p_i log(p_i) \\ \ subject \ to \ &\sum_{i=1}^K p_i =1…
learning
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Optimization of found points to line intersections

I have currently a pattern of points. Through an algorithm I obtain quite accurately the points, but I would like to make the points even more accurate. The pattern looks like: x x x x x It may vary since I see it from a camera…
api55
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Simplex Algorithm with state dependent cost function

The cost function to be minimized by the Simplex algorithm is normally defined as $min(x) = F(x)$ subject to linear constraints $Ax \le b$. Can the cost function be a function of two variables $F(x,u)$ which is subject to constraints $Ax \le B$. As…
Programmer1
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Newton's method calculate the residual at fixed fixed slope

In Newton's method, if the residual delta x is iteratively updated ($X \leftarrow X + \Delta x$) at fixed slope, does it converge? In newton's method, the estimation of X should be done iteratively. In each iteration the derivative (slope) of a…
agwad Elt
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Help with optimality?

Being $A$ a symmetric matrix in min $c^Tx$ subject to $Ax \geqslant c$ $x \geqslant 0$ I need to show that, if $x^*$ satisfies $Ax^*=c$ and $x^*>0$, so $x^*$ is an optimal solution. I've tried to show it by the Complementary Slackness Theorem and…
Gwi
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