**0**

votes

**1**answer

160 views

### (probably simple) optimization question

Suppose you have a concave function defined over a non-polyhedral convex cone and you are interested in the infimum. What would be standard approaches to tackle the question? (The cone is actually PSD ...

**1**

vote

**2**answers

108 views

### LP/QP with not-so-constant linear constaints

I have an otherwise standard LP or PSD QP problem as below:
$\min\limits_x {c}' x$ subject to $Ax\leq b$
or
$\min\limits_x \frac{1}{2}{x}' Qx + {c}' x$ subject to $Ax\leq b$
the only exception ...

**2**

votes

**1**answer

175 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

474 views

### Solutions to Heat Equations with Obstacles!

Consider a closed Riemannian manifold $(M,g)$ and a positive function $\psi: M \to R$. Fix a point $p \in M$, I have been struggling to construct a solution to the heat equation, $\partial_t u = ...

**4**

votes

**1**answer

220 views

### Minimizing action squared versus action

I have a very basic question in the calculus of variations:
Suppose I want to minimize the functional
$$A[r, r'] = \int_\Omega L(r, r') dx $$
When is it possible to say that extremals of $A$ agree ...

**9**

votes

**1**answer

3k views

### “You can't push a rope”

"You can't push a rope" is a wisdom saying that some engineering teachers pass along to their students. Since I'm not an engineer, I can only guess at what they mean, but it sounds to me like code ...

**4**

votes

**1**answer

277 views

### Going in the direction of the gradient

First, a motivating example. Suppose $f(x)$ is convex, differentiable, with a single minimum $x^*$.
Then the differential equation $$\dot{x}(t) = -\nabla f(x(t))$$ drives $x(t)$ to $x^*$.
Now my ...

**4**

votes

**0**answers

126 views

### Convexified threshold of a function

Upd. The question in a nutshell: find convex set on plane which is «closest» to a given non-convex set.
It is given integrable function $0\leq f(x,y)\leq 1$ with bounded support: $f(x,y)=0$ when ...

**5**

votes

**1**answer

221 views

### A Lagrangian problem with a countable family of local extrema ?

Dear MO contributors,
let $r > 0, L > 0$. I am interested in maximizing the integral:
$$
\int_0^{2\pi} \frac{f(\alpha)^2 f'(\alpha)^2}{\sqrt{f(\alpha)^2 + f'(\alpha)^2}} \ \mathrm{d} \alpha
$$
...

**0**

votes

**0**answers

140 views

### Differentiability of minimax objective function with respect to a decision variable

I have the following optimization problem:
$$\text{find } x= \min_{a} \max_{\lambda\in\Lambda} |R(h\lambda)|$$
where $\Lambda$ is some finite, fixed set of complex numbers and $R(z)$ is a ...

**2**

votes

**2**answers

207 views

### Equitable division of a contiguous resource

I have come across the following result regarding equitable division of a resource, which is a simple and immediate consequence of linear programming complementarity (in the infinite-dimensional ...

**5**

votes

**1**answer

265 views

### Geometric applications of Ekeland's variational principle

I'm looking for geometric applications of Ekeland's variational principle in order to see it at work in a context I'm familiar with. Let me recall the principle itself:
Definition. Let $(X,d)$ be a ...

**11**

votes

**3**answers

480 views

### finding the most-isolated point in a high-dimensional cube

I have a set of points {$x_1,\ldots,x_n$} located in the d-dimensional unit cube $[0,1]^d$. $n$ is about 1000 and $d$ is about 25. I'd like to find
$\max_{\omega\in [0,1]^d}\min_{i=1,\ldots,n} ...

**1**

vote

**3**answers

2k views

### How to get the largest subset of a set of sets of intervals with no overlapping intervals

Given: Set of Set of Intervals. Example {{(1,2), (3,4)}, {(1, 3)}, {(13,14)}}
Wanted Result: Largest Subset, in which no Interval overlaps with another from every Set pairwise.
Example:
Input ...

**2**

votes

**2**answers

322 views

### My overdetermined linear system gives both bad and good estimates. Why ?

Hello to everyone.
What the question means is that different ways of
expressing the same relation between the data and unknown variables produce
really weird fit results:
The problem:
I have the ...

**0**

votes

**1**answer

243 views

### Is it known that every PDF continuous in all $R^n$ has a maximum? [closed]

I'm working with maximum a posteriori estimation and managed to show that every probability density function that is continuous in all $R^n$ always has at least one global maximum. I've search around ...

**10**

votes

**1**answer

247 views

### Network flows with capacities on pairs of edges

Take a standard network flow problem: a directed graph with nonnegative capacities on each edge, a source $s$, a sink $t$. We all know how to find the maximum flow from $s$ to $t$.
Now add edge-pair ...

**1**

vote

**1**answer

136 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

133 views

### Proving that a constructed curve solves an optimization problem

Caveat up-front: I'm not a mathematician, so please excuse any stupidity/ignorance that follows. First, let me explain what I'm trying to do: I want to choose a function that maximizes the following ...

**28**

votes

**1**answer

1k views

### Optimization problem arising from the study of zeta zeros

Motivation: The following problem arose in [1] while studying the vertical distribution of the zeros of the Riemann zeta-function. At the time, my collaborators and I were unable to solve it and I ...

**4**

votes

**1**answer

233 views

### An optimization involving (random) graphs

Suppose we have a graph on $n$ nodes. We would like to assign to each node either a $+1$ or a $-1$. Call this a configuration $\sigma \in \{+1,-1\}^n$. The number of $+1$s that we have to assign is ...

**1**

vote

**1**answer

170 views

### Covering max flow arcs by arc disjoint paths

Let $(N,A,s,t,u)$ be a network with node set $N$, arc set $A$, source $s\in N$, sink $t\in N$ and capacity vector $u\in\{1,2,\ldots,T\}^A$, and let $x=(x_a)_{a\in A}$ be a maximum $(s,t)$-flow. Is it ...

**2**

votes

**0**answers

197 views

### Copositive matrix?

I want to check under what conditions a matrix of the form $\alpha J -Q$ is copositive, where $J$ is the all-ones matrix and $Q$ is doubly nonegative (i.e. entrywise nonnegative and positive ...

**2**

votes

**1**answer

159 views

### Lipshitz Constant of the convex extension of a submodular function

The title says it all :)
Given a submodular function (take the rank function of a matroid, for a concrete example) $f:\{0,1\}^n\rightarrow \mathbb{R}$, we can extend it to a convex function ...

**0**

votes

**0**answers

240 views

### A question regarding Danskin's theorem

Suppose $\phi({\bf x},{\bf z})$ is a continuous function of two arguments,
$\phi: {\mathbb R}^N \times Z \rightarrow {\mathbb R}$,
where $Z \subset {\mathbb R}^m$ is a (non-empty) compact convex ...

**0**

votes

**1**answer

228 views

### Constrained Optimization: Matrix Inverse in Objective/Constraints?

Made a crucial mistake in the problem formulation; please delete.

**7**

votes

**2**answers

401 views

### Eigencircles of n x n matrices?

An eigenvalue of a 2 x 2 matrix satisfies the equation
$$ \left(\begin{array}{cc} a & b \\\\ c & d \end{array} \right)\left( \begin{array}{c} x \\\\ y \end{array}\right) = \lambda \left( ...

**1**

vote

**1**answer

202 views

### f_n(x,p) converge uniformly to nice f(x,p); do zeros of f_n(.,p) converge uniformly to zeros of f(.,p)?

Fix compact intervals $X, P \subseteq \mathbb{R}$.
Let $f_n : X \times P \to \mathbb{R}$ be a sequence of $C^2$ functions converging uniformly to a $C^2$ function $f$. The first and second ...

**1**

vote

**1**answer

229 views

### Solving a QCQP problem with sparse regularization

I want to solve the follong QCQP problem:
$$
\mbox{Minimize}\quad\beta^TA\beta+\mu\Omega(\beta)$$
$$
\mbox{s.t.}\quad\beta^TB\beta=1 \quad\mbox{and}\quad\beta\ge0
$$
where $A$ and $B$ are both ...

**1**

vote

**2**answers

375 views

### Sherali-Adams relaxation

I am trying to find a book or a paper, which explains, how and why the Sherali-Adams relaxation method works. The original paper (1990) is difficult for me to understand. I need a more basic ...

**0**

votes

**3**answers

890 views

### Solving a non-convex quadratically constrained quadratic program

I have the following quadratic optimization problem: $\min_{\vec{x}} |\vec{x}|^2$ subject to $\vec{x}^T G_j \vec{x} \geq 1$, $j = 1 \ldots m$, where the $G_j$ are positive semidefinite. $|\vec{x}|$ is ...

**7**

votes

**2**answers

350 views

### More general form of inequality?

I have proved a simple Lemma that I need for a larger result, and I was wondering whether it is actually another more famous result in disguise.
The lemma says that for any set of vectors in ...

**3**

votes

**0**answers

148 views

### stochastic control / geometric mean

Consider the following problem:
Given $\Omega$ and $U$ two symmetric definite positive matrices, choose a matrix $K$ to minimize the expectation $x' \Omega x + x'K'UKx$ when $x$ follows the invariant ...

**2**

votes

**1**answer

134 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 ...

**1**

vote

**1**answer

334 views

### conjugate function for matrix mixed norm

I am familiar with the conjugate function of the vector norm, which uses the concept of dual norm and is defined as follows:
...

**1**

vote

**0**answers

221 views

### Matrix conditions under which spectral radius is smaller than 1?

Hello everyone,
I would like to find out which conditions are necessary so that the spectral radius $\rho(M)<1$ where $M$ represents the following matrix:
$M = \left( \begin{array}{ccc}
W & 0 ...

**6**

votes

**5**answers

3k views

### Is that true all the convex optimization problems can be solved in polynomial time using interior-point algorithms

Just a new guy in optimization. Is that true all the convex optimization problems can be solved in polynomial time using interior-point algorithms?

**2**

votes

**1**answer

230 views

### Generalized unique nearest point problem for a compact, convex set in a strictly convex Banach space.

If $X$ is a Real Banach space with strictly convex norm, it is known that for any non-empty compact, convex set $K$ and point $x_0\notin K$, there exists a unique point $z_0\in K$ minimizing the ...

**1**

vote

**1**answer

276 views

### Projection onto a quadratic cone?

Consider a constraint of the form
$$ f(x) := x^T A x = 0 $$
where $A \in \mathbb{R}^n$ is symmetric but may be singular and indefinite. The constraint set $C$ is a (nonconvex) cone, since for any ...

**2**

votes

**1**answer

376 views

### faces of a polytope

Let $a_1,a_2,..,a_n\in\mathbb{R}^m$, $n>m$ and the points are in general position meaning, no hyperplane contains more than $m$ points. Define a polyhedron $$P=\{x\in\mathbb{R}^m:a_1'x\leq0, ...

**1**

vote

**1**answer

143 views

### optimization related to sdp

I have the dual solution of an sdp problem and strong duality hold in this case, I have the dual feasible solutions . From the Dual feasible solutions can i get the primal feasible solution?

**6**

votes

**5**answers

1k views

### Application of polynomials with non-negative coefficients

Question 1: Are there any deeper applications (in any field of mathematics) of polynomials (with possibly more than one variable) over the real numbers whose coefficients are non-negative? So far I ...

**8**

votes

**1**answer

200 views

### Algorithm for matching in the power set lattice

Suppose that we have two probability distributions, $f$ and $g$ on the subsets of a finite set $X$, i.e. $f, g: P(X) \to [0,1]$, with
$$
\sum_{A \subseteq X} f(A) = \sum_{A \subseteq X} g(A) = 1.
$$
...

**2**

votes

**1**answer

1k views

### Sufficient conditions for gradient descent convergence

I have an unconstrained optimisation problem with convex objective function $f(x)$. Suppose I have access only to some function of the gradient $\hat{\nabla}= g(\nabla f)$, and I take gradient steps ...

**3**

votes

**2**answers

654 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 ...

**5**

votes

**0**answers

125 views

### Minmax problem for polygons

Let $\text{Pol}_n$ be the set of all convex polygons on a plane with $n$ vertices. For $P\in \text{Pol}_n$ denote by $\text{Tr}(P)$ the set of all triangles which vertices are some vertices of $P$. I ...

**2**

votes

**1**answer

131 views

### Relations between a set of inner products of vectors

Suppose we have n normalized vectors on an arbitrarily large Hilbert space $|A_1\rangle,\dots,|A_n\rangle$, $\langle A_i|A_i\rangle=1$ for every i. And there're $\frac{n(n-1)}{2}$ inner products ...

**7**

votes

**2**answers

322 views

### Entropy conjecture for distributions over $\mathbb{Z}_n$

Suppose we have two independent random variables $X$ (with distribution $p_X$) and $Y$ (with distribution $p_Y$) which take values in the cyclic group $\mathbb{Z}_n$. Let $Z = X +Y$, where the ...

**3**

votes

**1**answer

276 views

### Stronger bound for a modified Lyapunov Equation

In regard to the stability analysis and control properties of the linear system $\dot{x}=Ax$.
Consider the solution $P$ of the continuous Lyapunov equation $AP+PA^T+Q=0$, where $A,Q,P
\in
...

**2**

votes

**1**answer

202 views

### Nearest trio of neighbours for non-intersecting ellipses

Hi,
I'm working on a problem which is to find the closest trio of neighbours for a set of arbitrarily placed non-intersecting ellipses. As a new user I'm not allowed to include image tags but I've ...