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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Maximizing sum of sign functions

I want to solve the following optimization problem: $$ \max_{{\bf u}} \sum_{i=1}^N \left(a_i \textrm{ sign}({\bf u \ . x_i}) \right), $$ where ${\bf u}$ and ${\bf x_i}$ are $p\times1$ vectors, $a_i \in \{+1, -1, 0\}$ are known constants, and…
MMKK
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How to minimize the number of functions to be projected on

I have a set of functions $f_i,\, i=1,2,\ldots,n$ defined on an interval $[a,b]$, and a function $F$ also defined on $[a,b]$. I would like to project F on a subset of functions $f_i$ so that the number of $f_i$ used to represent $F$ is minimum, but…
Karla
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Book which has a proof of NP-completness for Partition Problem

I need the name of a book which has a proof of NP-completness for Partition Problem. I need to cite it in a my bachelor thesis.
user3680510
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How to find the extrema of a function?

I'm having problems finding the extrema of the function $h(x, y) = 2x sin(y) + y^2−x^2$. There is supposed to be one saddle point but I can't seem to get that. I tried taking $f_{xx}f_{yy} - f_{xy}^2$, but I can't seem to end up with 0. Anyone knows…
thbcm
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No free lunch theorem

How does the No free lunch theorem apply in linear programming? Given a linear Programm. Calculate the optimal solution. Then you can calculate with the simplex method the solution in finite steps. My understand of this theorem is now averaged over…
user3613886
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Trouble with formulation of objective function (constraint optimization)

I am new to optimization and I will try to state my question as clear as I can. I need to solve this constraint optimization problem. I want to find real vectors $\mathbf{f}$ and $\mathbf{g}$ that minimize $e^2$ $$ e^2 =…
NAASI
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IF a cone is inscribed in a larger cone,then what will be the radius of the small cone if it has the maximum volume?

If a smaller cone is inscribed in a larger cone as shown, then what will be the radius of the smaller cone if it has the maximum volume? Attempt I know that the volume of a cone =$\dfrac{1}{3}\pi r^2h$ , and the maximum volume can found by…
Maher
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How to solve an optimization problem where the size of the solution is part of the objective

I want to find the smallest vector $\vec p$ such that some constraints are satisfied, so something like: $$\hat p = \underset{\vec p}{\arg \min} \; |\vec p| \\ s.t. \; F(x_i, \vec p) \leq \epsilon_i \; \forall i $$ Is there a general approach for…
rhombidodecahedron
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Transform a minimization problem to LP

This is a past examination question. I was asked (Q.1) to find an equivalent linear programming problem of: $$\min_{x \geq 0} \left \|Ax-a \right\|_{1} + \left\|Bx-b \right\|_{\infty}$$ where $A$ is an $m \times n$ matrix, $a$ is in…
nam
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Maximum remainder

I got this programming challenge - and can solve it by brute force. But I would love to solve it using a more mathematical approach. /* Square Remainder Challenge * * Here's a challenge for the math lovers. Solving this challenge can be…
stulster
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Projected Gradient for Bounded constraint Problem

For the one-side bounded constraint problem, $\min_x f(x)$ s.t $x\geq0$, I know the projected gradient is given as $\nabla^p f(x) = \nabla f(x)$ if $x>0$ $\nabla^p f(x) = min(0,\nabla f(x))$ if $x=0$ I wanted to know the projected…
Learner
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Optimization Question Regarding proving Maximum Area of Window

A window has the shape of a rectangle of height $h$ surmounted by a semi-circle of radius $r$. The area of the window is given by $A = 2rh + \frac{1}{2}\pi r^{2}$ NOT drawn to scale. If the perimeter of the window is constant, prove that the area is…
Aspiring Mathlete
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Find the optimal Q to minimize $\text{tr} (Q^TQ)$

Min $\text{tr} (Q^TQ)$ subject to $A^TQ+QA=C$ all the matrix is n*n matrix, the problem is to find Q such that $\text{tr} (Q^TQ)$ gets minimal value I try to view the $A^TQ+QA=C$ as the linear transformation of Q,define the linear operator L,and…
fengdi
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Asymptotic properties for the Lasso

I try to figure out a proof of Keith Knight and Wenjiang Fu regarding the asymptotic property of Lasso-type estimators (http://www-personal.umich.edu/~jizhu/jizhu/KnightFu-AoS00.pdf), especially for theorem 2. The Lasso-estimator…
Theodor Ignuf
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Optimization of a cone

Find the largest possible volume of a cylinder inscribed in a cone with a height of 8 cm and radius of 4 cm. Do I use the equation for the volume of a cylinder or cone?
Amanda
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