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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 to optimize $\min\limits_{B}\ \lVert ABC-AC\rVert_F^2+\lambda\lVert B\rVert_{2,1}$, s.t. $B_{ii}=0$

How can I solve the following equation? by which algorithm? $\min\limits_{B}\ \lVert ABC-AC\rVert_F^2+\lambda\lVert B\rVert_{2,1}$ s.t. $B_{ii}=0$ (Diagonal elements of $B$ must be zero) where: $\lVert .\rVert_F^2 \ $ denotes the power 2 of the…
Mohsen
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Minimizing $\mbox{tr} (WX)$ subject to $\mbox{tr} (KX^{-1}) = n$ and $X \succ 0$

Minimize g: g=tr(WX) and X=TT^t Under the constraint tr(KX^(-1) )=n and X=TT^t K,W are symmetric and positive definite matrices T is not singular T∈R^nxn ,K,W ∈R^nxn , n is scalar For which matrix T we have the minimum g?
Apostolos K
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Minimizing the area between an exponential and a polygonal approximation

$\newcommand{\expq}[1]{{e^{#1}}}$ $\newcommand{\integral}[4]{\int_{#2}^{#3} {#4} \,d{#1}}$ $\newcommand{\dd}[2]{\frac{d #1}{d #2}}$ $\newcommand{\loss}{L}$ Suppose we want to draw the function $\expq{-kt}$ between 0 and $x$ by approximating it with…
Neil G
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computation of minimum value of a function

How will we calculate the minimum value ilof the equation with the given condition.
Khamba
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determination of maximum and minimum value of a function

What is the value of the value indicated in the question. Please explain the solution too.
Khamba
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Intesection of manifolds

I have three manifolds $z = 1/2*(x-\sqrt{x^2+4xy-4y^2})$, $y = 1/2*(z-\sqrt{z^2+4zx-4x^2})$, and $x = 1/2*(y-\sqrt{y^2+4yz-4z^2})$. My thinking is that they don't intersect, but I can't prove it. I know they intersect at zero but I want to know if…
www3
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Maximizing the integral

I have an objective function: $$O(x_1, x_2, x_3, x_4) = \int_{T_1}^{T_2} \sqrt{F_1(t)^2 + F_2(t)^2} dt$$ , where $$F_1(t) = x_1a_1(t) + x_2a_2(t) $$ and $$F_2(t) = x_3a_1(t) + x_4a_2(t). $$ $a_1(t)$ and $a_2(t)$ are known functions. I'd like to…
regress
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Help and Check working with rearranging a formula for the subject

I'm currently trying to copy a method for finding value of a parameter after implementing the Levenberg-Marquardt algorithm, by following an academic paper. I am hoping that somebody could check my answer to see if I'm on the right track or have…
user405218
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Why is ROC analysis not used in optimization problems?

In machine learning and applied fields of statistics, receiver operating characterization (ROC) analysis is commonly used to select optimal algorithms/models. However, at a lecture I once attended on mathematical optimization, I remember the…
oisyutat
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Maximize a function including Gamma function

I have the following optimization problem: $$\max_{x}~\frac{A^{x}}{\Gamma(1+x)\Gamma(1-x)}$$ such that $A > 0$ and $x \in (0,1)$. How can we obtain the optimal $x$? My attempt: I wrote $\Gamma(1+x)\Gamma(1-x) = \frac{\pi x}{\sin(\pi x)}$ and equate…
user389066
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cyclic coordinate descent

is it possible to implement cyclic coordinate descent to numerically solve optimization problem with equality constraints like: $$\mathrm{min} \quad f(x)$$ $$a^\top x=b$$ Suppose f(x): $\mathrm{R}^n\to \mathrm{R}$ convex and it is not hard to update…
franjuncis
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What is the optimal way to cut B chocolate bars to share equally between N people, but all people receiveing congruent shape parts

What is the optimal way to cut B chocolate bars to share equally between N people, but all people receiving congruent shape parts? Without the constraint of congruent shape parts, the question was solved here in this group. I've found this article…
Humberto José Bortolossi
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Maximizing a set of data with constraints

Eight athletes 1, 2, 3, 4, 5, 6, 7, and 8 participate as a team in a multidisciplinary competition. The four disciplines involved are labeled A, B, C, and D. The participation constraints for a team are: Each discipline must have four…
penguinick77
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Intuitive Rosenbrock's Valley/Banana Function.

The following is called Rosenbrock's valley or Rosenbrock's Banana Function, $$f(x,y) = (1-x)^2+A(y-x^2)^2, \qquad (x,y) \in \Bbb R^2$$ Can you explain intuitively why is this function so important in the study of Optimization? What is so special…
user366312
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What are regular points? (optimization)

I am taking Optimization and my homework question just asked for the definition of a regular point. I have tried googling it. I read multiple definitions but still am confused. Is a regular point just a point that is in the feasible region? thanks
Tangenttothecurve69
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