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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Optimization of Integral

$F(x,y)=\int_a^{b} \int_c^{d} G(x,y,s,t)dsdt$. I wish to find $x$ and $y$ (subject to some constraints) that minimize $F$. $F$ is not expressible in closed form. Is the only way to solve this problem as follows: "Replace different $x$ and $y$ values…
Ken Dunn
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optimization, local/global maximizers/minimizers

I'm a bit confused on this question, I've had a go at it and any help would be much appreciated. Determine all the optimal points of the following function, and indicate whether the optimal solutions are global or local: $$f(x) = \left\{ …
user54511
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Optimize configuration/connection of groups for Shortest Path Length

I am trying to find the most optimal configuration of cells/connections in a battery pack of a certain size/shape in order to minimize the voltage drop for each cell in the pack. I have a grid (triangle shaped) with n batteries (aka…
Maarten -Monica for president
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What is the relationship between minimax problems and bilevel optimization problems?

Is there (ever) a way to phrase a minimax problem as a bilevel optimization problem, or vice versa? Can one be seen as a limiting case of another?
capybaralet
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Algorithm to find optimal cuts of pipe

I have varying lengths of pipe in inventory. When a customer requests various lengths I want to find the optimal way of cutting what I have in inventory. I need to make a program that does this. This optimization needs to consider that longer…
Mike
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Relationship between two minimization programs

Let $\mathbf{A}$ be a ${n\times J}$ matrix with $A_{ij}\geq0$ (and $A_{ij}>0$ for most $ij$, there cannot be any rows or columns that consist only of $0$s), $Q=\left\{\mathbf{q}\mid \mathbf{q}\in\Bbb R^n_+\land\sum\limits_{i=1}^n q_i=1\right\}$ and…
Patricio
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Simplex method: Utter, extreme confusion

We want to maximize $ z = 30x_1 + 20x_2$ with $$2x_1 + x_2 \leq 140$$ $$x_1 + 2x_2 \leq 160$$ $$x_1 + x_2 \leq 90$$ $$ x_1, x_2 \geq 0$$ So my book says the first step is writing these to equalities, we do thse by introducing the slack variables…
Ylyk Coitus
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optimization problem to find the maximum of sum of sigmoids

I am having a problem of maximizing a sum of sigmoid functions over different time instants with some constraints. Considering the standard sigmoid function $f(x)=\frac{1}{1+e^{-\alpha x}}$ and it's derivative $f'(x)=f(x)[1-f(x)]$ In my case, it is…
mattia
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Rewrite a max-min Problem by partitioning the domain

Consider the following max-min problem: $$\sup_{x\in D_{X}}\inf_{y\in D_Y} f(x, y),$$ where $f:D_{X}\times D_{Y}\to \mathbb{R}$. Assume that $D_{Y}$ can be partitioned into $k$ disjoint union of sets $D_{Y, 1}, D_{Y, 2}, \dots, D_{Y, k}$, i.e.,…
Chasel Weng
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Find $(x_1,\dots,x_n) \in (\mathbb{R}_+^*)^n$ to minimize $\sum_{k=1}^{n}x_k\prod_{i=1}^{k}{(1 + x_i)}$ such that $\sum_{k=1}^{n}{x_k} = 1$

I want to find $(x_1,\dots,x_n) \in (\mathbb{R}_+^*)^n$ to minimize $$ \sum_{k=1}^{n}{x_k \prod_{i=1}^{k}{(1 + x_i)}} $$ with the following constraint $\displaystyle\sum_{k=1}^{n}{x_k} = 1$.
booster
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Preconditioning for an LBFGS

I am working on a high dimensional (N ~ 1000-60000) optimization problem which is currently solved with an LBFGS algorithm. I have experimented with different diagonal preconditioners as I know that the gradients in some dimensions are oders of…
Mark
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Rewriting a function of two variables as the quotient of another function

If I have a function f(x,y) which is increasing in x but decreasing in y, could I in general, rewrite that function as g(x/y)? For context, I have an objective function F(x,y)*H(x,y) which is increasing in x for F and decreasing for H, and…
Fića
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What's $argmax$ in multiobjective optimization? Is it component-wise or something else?

What's $argmax$ in multiobjective optimization? Is it component-wise or something else? Such as in the weighting method: $$\sum_{i=1}^n w_i p_i$$ then what is $$argmax_p\sum_{i=1}^n w_i p_i$$ Is it the $p$ that produces the largest sum? Or that has…
mavavilj
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Gradient Descent vs Coordinate Descent

Let us assume that we are trying to solve $\min_x f(x)$, $f$ differentiable. Computing a full gradient $\nabla f$ is more expensive that computing the gradient on a single coordinate $\nabla_i f$. Is there any general theory that says that, say, for…
OptStudent
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Maximizing the ratio of inner products

The problem is stated as follows. \begin{equation*} \underset{\|\mathbf{x}\|_2 \le \gamma}{\max}\quad\frac{1+|\mathbf{y}_1^H\mathbf{x}|^2}{1+|\mathbf{y}_2^H\mathbf{x}|^2}, \end{equation*} where $\gamma > 0$ and $\mathbf{y}_1$, $\mathbf{y}_2 \in…
Alex
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