# Tagged Questions

**2**

votes

**2**answers

101 views

### Finding the maximum of a multivariate polynomial of degree one

I need to find the global maximum of the function
\begin{align}
f\left(x\right) & = p_1 \max\left(\sum a_{1i} x_{1i}, \sum b_{1i} x_{1i}\right) - \sum c_{1i} x_{1i} \\
&+\ldots \\
&+ p_n ...

**1**

vote

**0**answers

76 views

### Maximizing an integral over a convex region

Let $C$ denote a compact, convex region in the plane containing the origin with unit area, and let $f$ be a probability distribution on $C$. Let $f^\ast$ denote the distribution that maximizes the ...

**0**

votes

**1**answer

76 views

### Avoiding epsilon in mixed integer linear and quadratically constrained programs

I would like to represent the following constraint as MILP constraint where $x \in [a, b]$ with fixed $a, b \in \mathbb{R}$ and $y \in \lbrace 0, 1 \rbrace$.
$(x = 0 \wedge y = 1) \vee (x \neq 0 ...

**1**

vote

**1**answer

352 views

### for what arguments the function reaches maximum?

Hi,
What is the maximum of the following function?:
$f(x_i,w_i)=\frac { \sum w_i}{ \sum \frac {w_i}{x_i} } - \frac{ 1 - \prod \left ( 1 - w_{i}\right )}{ 1 - \prod \left ( 1 - \frac{w_{i}}{ ...

**5**

votes

**2**answers

498 views

### Gandhi's quote formalized [closed]

Hello,
I hope this question is appropriate for Mathoverflow. Gandhi said, "Be the change that you wish to see in the world". I don't understand anything in Game/optimization theory (I don't know ...

**2**

votes

**1**answer

171 views

### A Function with Exactly $k$ Minima in a Bounded Space

Is it possible to have a function with the following properties?
(i) The function maps a bounded $n$-dimensional space $\mathcal{X}$ (say $\left[0,1\right]^n$) onto a bounded interval $\mathcal{Y}$ ...

**1**

vote

**1**answer

128 views

### Nonlinearly constrained optimization (quadratic)

Hi all -- what would be good methods (and/or software packages) to try for solving a problem minimizing a quadratic function $f(x) = \sum_{i=1}^N{(x_i - y_i)^2}$, where some constraints are non-linear ...

**2**

votes

**1**answer

121 views

### Maximizing positive definite quadratic using the eigendecompoisition

Consider the problem:
$\textrm{max}\;\; x^T Q x$
subject to $||x||_\infty \leq 1$, where $Q$ is a positive definite matrix.
I believe this problem is NP-hard (although I have only found hardness ...

**3**

votes

**2**answers

566 views

### Computational complexity of unconstrained convex optimisation

What is known about the relationship between unconstrained convex optimisation and computational complexity? For example, for which optimisation problems and which gradient descent algorithms is one ...

**2**

votes

**3**answers

601 views

### Maximizing the minimum of piecewise linear functions in high dimensional space

I'd like to compute
$\max_{x,t} t$ such that $\forall i$, $t < a_i + \|x - b_i\|_\infty$.
where $a_i,\ldots, a_n \in \mathbb R$ and $b_1,\ldots,b_n \in [0,1]^{21}$ are fixed, $x \in [0,1]^{21}$, ...

**3**

votes

**1**answer

2k views

### Linear program to maximize the minimum absolute value of linear functions ?

I'd like to compute
$\max_{x,t} t$ such that $\forall i$, $t < a_i + |x - b_i|$.
where $a_i,\ldots, a_n$ and $b_1,\ldots,b_n$ are fixed and $x \in [0,1]$.
Can this be solved with a linear ...

**3**

votes

**1**answer

602 views

### maximization of a quadratic function with a quadratic constraint

I have the following quadratic maximization problem
$\max_{\mathbf X} \quad tr(\mathbf A\mathbf X\mathbf B\mathbf X^H)+tr(\mathbf C\mathbf X)+tr(\mathbf C^H\mathbf X^H)$
subject to the quadratic ...

**2**

votes

**4**answers

334 views

### Efficient algorithm for finding the minima of a piecewise linear function

Consider real numbers $a_i$ and $b_i$ for $i=1\dots n$ and define a function by
$f(x) = \max_i ( a_i + b_i x )$
We desire to find $\min_x f(x)$. Obviously this occurs at an intersection of two ...

**3**

votes

**2**answers

452 views

### Optimizing a quadratic restricted to the sphere

Let $A$ be $p\times p$ symmetric positive definite with distinct eigenvalues and $x_p\in \mathbb{R}^p$ and consider the problem
Minimize $x'Ax + b'x$
Subject to $x'x=1$
Most of the information I've ...

**5**

votes

**0**answers

360 views

### Maximizing the matrix norm

Hi all,
I wish to find a 3x3 rotation (orthogonal) matrix $\mathbf{R}$ such that it maximizes the following matrix norm:
$||\mathbf{A} \mathbf{D}_r \mathbf{D}_b ||_2$
where $\mathbf{A}$ is a known ...

**4**

votes

**1**answer

2k views

### Homogeneous system of polynomial equations

Hi all,
Previously I asked a question that currently has no satisfactory answer Least sum squares given constraints on subcomponents
It comes from an engineering problem. I was thinking to formulate ...

**10**

votes

**2**answers

1k views

### Maximizing the Smallest Eigenvalue of a Diagonally Dominant Matrix

Assume that we have a full-rank diagonally dominant matrix $A$, all the diagonal elements of which are positive, all the non-diagonal elements are negative, and the sum of the absolute values of the ...

**5**

votes

**1**answer

843 views

### Non-negative quadratic maximization

For a given symmetric and positive semidefinite $n \times n$ matrix $A$, we want to solve the problem $$\max_{||x|| = 1, \ x\geq 0} x^T A x.$$
Here, $x\geq 0$ indicates that $x$ must be component-wise ...

**4**

votes

**2**answers

1k views

### Optimal knot placement for fitting piecewise-continuous linear functions to a nonlinear function

I encountered this problem in my research and it is turning out to be a surprisingly difficult one(for me, at least).
Suppose we have a univariate nonlinear function $f(x)$ where $x \in [L,U]$. Our ...

**3**

votes

**3**answers

554 views

### Some questions about Invexity

Recently, I am looking into a non-convex optimization problem whose points satisfying KKT conditions can be obtained. Then the problem becomes how to decide whether the KKT conditions are sufficient ...

**2**

votes

**1**answer

267 views

### Can subgradient infer convexity?

It is known that
If $f:Uâ†’ R$ is a real-valued convex function defined on a convex open set in the Euclidean space $R^n$, a vector v in that space is called a subgradient at a point $x_0$ in $U$ if for ...

**13**

votes

**1**answer

2k views

### A circle packing conjecture

Consider $n$ circles with variable radii $r_1,\ldots, r_n$ that pack inside a fixed circle of unit radius. In other words, all $n$ variable-radius circles are contained in the unit radius circle and ...

**6**

votes

**4**answers

1k views

### Robust black box function minimization with extremely expensive cost function

There is an enormous amount of information about the common applied math problem of minimizing a function.. software packages, hundreds of books, research, etc.
But I still have not found a good ...

**18**

votes

**5**answers

9k views

### Is all non-convex optimization heuristic?

Convex Optimization is a mathematically rigorous and well-studied field. In linear programming a whole host of tractable methods give your global optimums in lightning fast times. Quadratic ...

**1**

vote

**2**answers

213 views

### how to estimate a polyhedron(convex hull) classifier from data sample

Given a set of points $X\in\Re^D$, they have labels $Y\in${$-1,+1$}. I would like to separate the data labeled +1 and the data labeled -1 by a polyhedron.
$min_w \sum_i \xi_i + \frac{1}{2}\|w\|_2^2$
...

**2**

votes

**0**answers

154 views

### modification of singlestart in global optimization

When minimizing a nonconvex function $f : \Omega \rightarrow \mathbb{R}$ that may have multiple minima, there are some very simple strategies to improve the odds of finding the global minimum point. ...