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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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Are these two problems the same?

Let $T$ be a $n\times J$ matrix with non negative elements, $\mathbf{q,α} \in \Bbb{R}_+^n$ with $α_{i}<1$ for all $i$ and $\mathbf{k}\in\Bbb{R}_+^J$ (with $k_1>0$). Consider the problem $$\begin{matrix}\min_{\{q_i\}} \sum_{i=1}^n…
Patricio
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Optimization with strict inequality

I'm very familiar with using Lagrangeans to solve optimization problems with weak inequalities, but I just realized that I don't know how to solve simple optimization problems with strict inequalities. I have the following problem: Choose p and x to…
leecarvallo
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reformulating $\arg\min |x|^{1.5}$

I've asked this question. Essentially I have the same question for problem $\arg\min |x|^{1.5}$. We can consider next variant $\arg\min t^{1.5}$ subject to $x\le t$ and $-x \le t$. But $t^{1.5}$ is not twice continuously differentiable. Intuitively…
ashim
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Exactly one of the following systems is solvable

$Ax\leq b $ and $x\geq 0 $ $b^{T}y<0$, $A^{T}y\geq 0$ and $y\geq 0$ I can show easily that if (1) is true then (2) is not and converse too. That means both statements can not be true at same time. But I want to show that Exactly one of these is…
user63373
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Is the solution that SQP produces always the best or unique? How can I seek for other solutions?

Is the solution that SQP produces always the best or unique? How can I seek for other solutions? By varying the start point, $x_0$? Also, any idea why the start point may be "absurdly" large compared to the solution. I'm trying e.g.…
mavavilj
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An optimization problem with regard to permutation function

So far I only meet optimization problems solved by searching for an optimal point in $\mathbb{R}^n$. But today I met with an optimization problem that asks me to search within a set of functions. My previous knowledge on optimization such as…
Zhou Heng
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Analytical solution for optimization problem

Suppose we have a vector $p \in \Re ^n$. How can I get analytical solution for that optimization problem: $p = \arg \min {f(x)}_{p \in R} $ if $f(p) = ||p-z||^{2}_{2} + x||p||_{1} $, where $|| \cdot || $ are $L^1$ and $L^2$ norm and $x > 0$.…
Vasya Pravdin
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solve $\underset{d \in \mathbb{R}^n}{min}$ $g^Td$ subject to $d^THd = 1$ where H $\in \mathbb{R}^{n \times n}$ is positive definite and symmetric

I would like to solve the optimization problem $$\underset{d \in \mathbb{R}^n}{min} \ g^Td$$ subject to $$d^THd = 1$$ where H $\in \mathbb{R}^{n \times n}$ is positive definite and symmetric $\textbf{without}$ using Lagrange multipliers. Does…
Lgate8
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Why does NLopt have L-BFGS but not BFGS?

NLopt, the "free/open-source library for nonlinear optimization, providing a common interface for a number of different free optimization routines..." has a L-BFGS routine but seemingly, no BFGS routine. The closest I can see in the list of…
Alex
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Is there any way we can test if a function has local optima or not without searching?

Suppose, we have the following function $f(x)$: Is there any way, without searching, we can test if this function has local optima or not? Can we, say, test any property of this function, or, do some calculation to know this beforehand? Note:…
user366312
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Minimum value of given expression?

You have two lists of $N$ integers $x_{i}$ and $y_{i}$ where $i\in\{ 0,1,\dots,N-1\}$. We know the value of $x_{0}$ and $y_{0}$, remaining $N-1$ values is calculated using formula given below: $$x_i=(x_{i-1}*p_{x}+a_{x})\bmod x_{m}…
user1437473
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Fitting power law with loglog or exponential?

I have $x$- and $y$-data, and I want a power-law fit ($y=ax^b$). I always fit $\log(x)$ and $\log(y)$ by $p_1x+p_2$ (Matlab poly1), but when I fit $x$ and $y$ with $p_1x^{p_2}$, I did not get exactly the same result. Why?! And what is the best way…
mehrdad
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Maximize $\frac{x_1x_2}{y}$

\begin{array}{ll} \text{maximize} & \frac{x_1x_2}{y}\\ \text{subject to} & b_1+b_2\geq x_1+x_2\\&b_2\geq x_2\\&b_1\geq y\\&x_1\geq y\geq x_2\end{array} where $x_i,y,b_i\in \mathbb{R}^+$, $b_1$,$b_2$ are given s.t $b_1\geq b_2$. My attempt: Let…
Lee
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Converting a quadratic constraint into a second-order cone constraint

This is straight from the book: Optimization Methods in Finance. I'm trying to gain understanding of how the author derived the cone constraints from the the following quadratic constraint: $$x^TQx + 2p^T x + γ ≤ 0$$ Assuming Matrix $Q$ is positive…
user3547551
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Can you prove this equality?

In their book An Introduction to Optimization, on the chapter on gradient algorithms, to prepare for discussing convergence properties of the descent methods, authors Chong and Zak have following: $f(x) = \frac{1}{2}x^TQx-b^Tx$, where $Q$ is…
O. Altun
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