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3 votes
1 answer
132 views

How to maximize the variance of a subset of integers?

$\DeclareMathOperator{\Var}{Var}$Given the set of numbers $\Omega := \{1, \ldots, n\}, n \in \mathbb{Z}^+$, how can I choose a subset, $A$ of $\Omega$ , such that $\min(\Var(A), \Var(\Omega \setminus ...
Hasan Zaeem's user avatar
1 vote
0 answers
29 views

Integral hull of a polyhedron Q is polyhedron

Let $Q \subseteq R^n$ be a rational polyhedron and let $Q_I=Convexhull(Q \cap Z^n)$. By finite basis theorem, we have $Q=P+C$ for some rational polytope $P$ and finitely generated cone $C$ where $C=R_+...
Sowbarnika R's user avatar
-1 votes
0 answers
41 views

Is it possible to backtrack an optimization solver? [closed]

I have an optimization problem and was using a linear programming optimizer to find solutions. However, I find that past a certain size, the problem becomes "infeasible" and has no solutions....
Bamboozle's user avatar
-2 votes
0 answers
24 views

Conditions for a cubic function to be quasiconcave or quasiconvex [closed]

I would like to understand under what conditions a cubic function $f(x)=ax^3+bx^2+cx+d$ can be considered quasiconcave or quasiconvex. Specifically, I am interested in finding conditions on the ...
nuobei tang's user avatar
1 vote
1 answer
82 views

Solution to a quadratically constrained quadratic program with unit ball constraint

I am working on a quadratically constrained quadratic program (QCQP) of the form:$$ \min_{x} \quad \frac{1}{2} x^T P x + q^T x + r$$ $$ \text{subject to} \qquad x^{T}x \leq 1 $$ where $P \in S^{++}_{...
nuobei tang's user avatar
0 votes
0 answers
37 views

Bounding the error of a truncated moment problem

Let $\{x_{i}\}_{i=1}^{\infty}$ be a non-increasing sequence of non-negative real numbers, and let $\{y_{j}\}_{j=1}^{B}$ be a non-increasing sequence of non-negative real numbers, where $B$ is a finite ...
CWC's user avatar
  • 433
0 votes
0 answers
39 views

Optimizing over convex polynomials

I have a minimization problem which reads $\min\limits_P J(P)$, where the minimum is over convex polynomials in $n$ variables, with degree at most $d$, and $J$ is a function taking polynomials as ...
JackEight's user avatar
  • 101
2 votes
1 answer
91 views

Does approximately null gradient imply approximately global minimum for convex functions?

Let $f: \mathbb{R}^{n} \rightarrow \mathbb{R}_{+}$ be a non-negative and differentiable convex function which vanishes in a non-empty convex set $\Omega$ - possibly unbounded. Usually, when one ...
R. W. Prado's user avatar
0 votes
0 answers
30 views

Question on the closed proper convex functions

I'm confused with the definition of the closed proper convex functions when reading the paper https://people.orie.cornell.edu/aslewis/publications/00-dykstras.pdf It appears that, when a function $f$ ...
Fawen90's user avatar
  • 1,389
1 vote
0 answers
48 views

What makes the generalized projection different than metric on a Banach space?

I have came across the notion of generalized projection in Banach spaces, introduced by Ya. Alber and has seen many iterative algorithms being solved by using this projection. It helps in finding the ...
PPB's user avatar
  • 85
0 votes
0 answers
35 views

Describing the boundary of the feasible direction cone to a convex open subset of $\mathbb{R}^n$ at a boundary point: connection via subdifferential?

Let $U\subset \mathbb{R}^n$ be a convex, open set with nonempty boundary. Let $x_0\in \partial{U}.$ We can describe $U$ locally near $x_0$ as a super level set of a suitable continuous concave ...
Learning math's user avatar
0 votes
0 answers
21 views

Easy instance of set cover

I am trying to prove that a natural greedy algorithm solves the following instance of the set cover problem: for a set of elements $e\in U$ with a set of weights $w_e$, we define the cost of a subset ...
Tom Solberg's user avatar
  • 4,049
1 vote
0 answers
29 views

Change in active constraints when perturbing the objective of a QP

Suppose I have a quadratic program (with positive semidefinite cost matrix) with affine (polytopic) constraints. It is known that the solution to this is piecewise affine, with the ``pieces'' defined ...
xJ8v4KtZr2's user avatar
4 votes
1 answer
171 views

Maximizing trace subject to two equality constraints

I am looking at the following optimization problem $$\begin{align} \underset{{\bf X}}{\text{maximize}} \qquad&\mathrm{tr}({\bf AX})\\ \text{subject to} \qquad& \mathrm{tr}({\bf X}) = 1,\\ &...
usergh's user avatar
  • 43
7 votes
2 answers
242 views

Prove that $ n \leq d+1 $ under ordering constraints in $\mathbb{R}^d$

Let $x_1, \dotsc, x_n \in \mathbb{R}^d$ and $\theta_1, \dotsc, \theta_n \in \mathbb{R}^d$ be vectors such that for every $k \in [n]$, the following inequality holds: $$ \langle x_k, \theta_k \rangle &...
Alireza Bakhtiari's user avatar
0 votes
1 answer
57 views

Self-concordant barrier for the epigraph of $f(x,y) = x^p y^{1-p}$?

The problem Assume $p > 1$. Consider the function $$f(x,y) = x^p y^{1-p}, \qquad x,y > 0.$$ Note that $$ f'' = p(p-1)x^{p-2}y^{-1-p} \begin{bmatrix} y \\ & x \end{bmatrix} \begin{bmatrix} 1 &...
Sébastien Loisel's user avatar
0 votes
2 answers
97 views

Optimization algorithms for Kronecker approximation of high-dimensional covariance matrices

I'm working with a high-dimensional covariance matrix and exploring Kronecker product approximations to make it computationally manageable. Here's the setup: I have a graph $G$ represented by a $D\...
JJbox's user avatar
  • 1
0 votes
1 answer
236 views

Solving a 0-1 quadratic matrix inequality

I am working on a binary optimization problem. So far I have derived the following constraint functions. \begin{align} \begin{bmatrix} \left( P + \sum_{i=1}^n (\sum_{j=1}^n x_{i, j} \alpha_j) e_i e_i^...
zycai's user avatar
  • 11
1 vote
0 answers
46 views

Intuition for proximal point method using L2 regularization

To minimize a function $f$, the proximal point method is defined as $$x_{k+1} := \operatorname*{argmin}_x f(x) + \frac{1}{2\eta}\|x - x_k\|^2.$$ What's the intuition for why we want to use L2 ...
optimal_transport_fan's user avatar
2 votes
4 answers
212 views

Efficient algorithm for graph problem

Let $D=(V,E)$ be a directed graph, $S,T\subset V$ and $f:V\rightarrow \{1,\ldots, k\}$ a positive, bounded weight-function and $l\in \mathbb{N}$, find a path $v_1,\ldots, v_l\in V$ with $v_1\in S$ and ...
Martin Clever's user avatar
1 vote
1 answer
60 views

Optimal transport for sum of two costs

Let $X$ be a finite set and $\sigma_0$, $\sigma_1$ two fixed measures on $X$ with $\sigma_0(X)=\sigma_1(X)$. A transportation plan is a measure $\mu$ on $X\times X$ whose projections on the first and ...
user95282's user avatar
  • 1,074
1 vote
1 answer
106 views

Iterated optimal transport

Suppose we are interested in two consecutive transport plans (in the Kantorovich formulation). That is, we are given finite sets $X$, $Y$ and $Z$, endowed with probability measures $\mu_X$, $\mu_Y$ ...
tex.support's user avatar
0 votes
0 answers
69 views

Degree of reflectional symmetry of (unbounded) convex polyhedra in Euclidean spaces

Let $U \subset \mathbb{R}^m$ be an open domain. I'm trying to come up with a measure of its degree of reflectional symmetry and I have a question. The post in two-part, where in PART I I introduce the ...
Learning math's user avatar
7 votes
0 answers
249 views

Proving this function is convex

Let $C$ be a symmetric positive definite matrix such that $0\leq c_{ij} \leq 1$, $c_{ii}=1$, and define $f$ as $$f(x)=\sum_{i}x_{i}\log(\sum_{j}c_{ij}x_{j})$$ for positive vectors $x$ (in fact let's ...
Tom Solberg's user avatar
  • 4,049
1 vote
0 answers
39 views

In search for optimal solution for a minimization problem

I am trying to solve the following optimization problem: $\begin{align}\min_{\mathbf{c}_{k}} \sum_{k=1}^{K}\frac{q_k}{c_k} \\ \text{s.t. } c_k &\geq t_k \quad \forall k \tag1\label1 \\ \sum_{k=1}^{...
Wireless Engineer's user avatar
1 vote
0 answers
143 views

Integer points inside the high-dimensional ball (asymptotics)

Let $N(\alpha, n)$ denote the number of integer points inside the origin-centered ball of radius $\alpha \sqrt n$ in $n$ dimensions, where $\alpha \in (0,\infty)$ is some fixed constant. Precisely: $$...
DJA's user avatar
  • 435
0 votes
0 answers
52 views

What are the injective embeddings of R^d into the cone of (semi-) positive definite matrices of dimension d?

How can we characterize the set of all injective functions from $\mathbb{R}^d$ to the set of all symmetric positive definite matrices of dimension d?
Drmanifold's user avatar
1 vote
0 answers
73 views

Convexity and subdifferential monotonicity

Do you know any reference where I can find some results in this sense: Consider $W:K\to [0,\infty)$ is a functional defined on a convex cone $K\subset X$, where $X$ is a Banach space. Then the ...
Bogdan's user avatar
  • 1,759
1 vote
0 answers
41 views

Is there any software package to find the vertices of convex polytope where the inequality constraints are bounded by variable?

I know of this package lcon2vert that computes vertices from given inequality and equality constraints describing a bounded polyhedron. Here the bounds of constraints only accept numerical values, i.e....
Soumyabrata hazra's user avatar
1 vote
1 answer
63 views

Question on the relation between the Lagrangian Multiplier $\mathcal{L}=r+\lambda g(r,\theta_1,\dotsc,\theta_{N-1})$ and the Hessian of $r$

I want to minimize the radius $r=\sqrt{x^2_1+x^2_2+\dotsb+x^2_N}$, with the constraint $g(r,\theta_1,\dotsc,\theta_{N-1})=0$. Here $g(r,\theta_1,\dotsc,\theta_{N-1})$ is a function defined in the $N$-...
Guoqing's user avatar
  • 375
13 votes
0 answers
710 views

Minimizing total variation under constraint

For $p\in[0,1]$, we write $\mathrm{Ber}(p)$ to denote the Bernoulli measure on $\{0,1\}$; that is, $\mathrm{Ber}(p)(0)=1-p$, $\mathrm{Ber}(p)(1)=p$. For $n\in\mathbb{N}$ and $p=(p_1,\ldots,p_n)\in[0,1]...
Aryeh Kontorovich's user avatar
1 vote
0 answers
45 views

Inequality Involving Concave Monotonic Function

Assume that $ f: \mathbb{R} \to \mathbb{R}_+ $ is a concave, non-decreasing and positive function. Let $\mathbb{X}$ be a finite set consisting of $ 0\leq x_1 \leq x_2 \leq x_3 \leq \ldots \leq x_n$. ...
Alireza Bakhtiari's user avatar
0 votes
0 answers
51 views

Minimizer of forward and reverse Kullback-Leibler divergence with sum constraints on marginals

Consider minimization of the Kullback Leibler divergence between two discrete distributions $p$ and $q$: \begin{align*} D_{KL} \left( p \parallel q \right) = \sum_i p_i \log \left( \frac{p_i}{q_i} \...
TalTal The Eighth's user avatar
2 votes
0 answers
58 views

An s-convex function lying between two convex functions

Let $f: \mathbb R_{+} \to \mathbb R_{+}$ be an $s$-function in the second sense, i.e., $$ f(\lambda x +(1-\lambda)y) \leq \lambda^s f(x) +(1-\lambda)^s f(y)$$ for every $\lambda \in (0,1)$. Assume ...
MAY's user avatar
  • 55
0 votes
0 answers
48 views

A question on a quantitative form of Farkas' lemma

Suppose A is an $m \times n$ matrix whose entries are non-negative integers and $\mathbf{b}$ is a vector with rational entries. A version of Farkas lemma implies that if the equation $$A\mathbf{x}=\...
Keivan Karai's user avatar
  • 6,214
0 votes
0 answers
54 views

How to deal with minimizing a flat objective function

Problematic (Debiased Sinkhorn barycenter, proposed by H.Janti et al.): Let $\alpha_1, \ldots, \alpha_K \in \Delta_n$ and $\mathbf{K}=e^{-\frac{\mathrm{C}}{\varepsilon}}$. Let $\pi$ denote a sequence ...
Tung Nguyen's user avatar
1 vote
0 answers
99 views

Minimum of the maximum element frequency given the family size and the universe size

[Crossposted at math.stackexchange]. Consider families of sets $\mathcal{F}$ with size $n = |\mathcal{F}|$ and universe $U(\mathcal{F})$ with size $q = |U(\mathcal{F})|$. I have written and solved ...
Fabius Wiesner's user avatar
0 votes
1 answer
150 views

When are infimal convolutions contractions?

Let $X$ be a separable Fréchet space and $\varphi,\psi:X\to \mathbb{R}$ be a lower semi-continuous and convex function with $\psi$ bounded below and coercive. Consider the infimal convolution $$ \...
Math_Newbie's user avatar
0 votes
0 answers
56 views

How explicit the optimiser of this optimisation problem can be?

Provided the given parameters as follows : $\mu\in\mathbb R, \sigma\in\mathbb R_+$ are constant, $\kappa, r, \alpha, \beta: \mathbb R_+\to\mathbb R_+ $ are measurable functions such that $\kappa(y)\...
GJC20's user avatar
  • 1,334
7 votes
2 answers
178 views

Separating domains in $\mathbb{R}^{2n}$ by a real algebraic variety

Suppose $\Omega_1$ and $\Omega_2$ are two disjoint unbounded domains in $\mathbb{R}^{2n}$, $n \in \mathbb{N}$. Can there be conditions on $\Omega_1$ and $\Omega_2$ so that these two domains can be ...
Soumya Ganguly's user avatar
0 votes
0 answers
115 views

Software for computing polytopes

As can be inferred from the title, I want to do some computation on the facets representation of the polytopes given the vertices. My advisor recommended me Polymake, which is indeed useful even with ...
AlexiosF's user avatar
6 votes
1 answer
234 views

Stopping criteria for damped Newton iterations with backtracking line search

Are there better criteria than the Armijo criterion for damped Newton iteration with backtracking line search, when the objective is standard self-concordant? (See Boyd and Vandenberghe.) Let $F(x)$ ...
Sébastien Loisel's user avatar
1 vote
0 answers
79 views

Touring a sequence of convex polygons with minimal energy

There is a known problem of touring a sequence of $n$ polygons: given a starting point $s$, an ending point $t$ and a sequence of polygons $P_1,\dots,P_k$ with a total of $n$ vertices, find points $...
ssss nnnn's user avatar
  • 177
0 votes
0 answers
56 views

Convex optimization of the Lovász extension of a submodular function

I have a finite set of $n$ elements $A$, and a submodular function $f:2^A\rightarrow R$. Let $g:[0,1]^n\rightarrow {R} $ be the Lovász extension of $f$. I want to solve the following optimization ...
Tomer Ezra's user avatar
5 votes
1 answer
176 views

Efficient counting of integer solutions to linear system

In my research, I have a particular 18x18 matrix $\mathbf{A}$ which defines the linear system $\mathbf{A}\cdot \mathbf{x} \leq \mathbf{-1}$ over the nonnegative integers. And I'm interested in ...
user326210's user avatar
0 votes
0 answers
72 views

Minimizing the Spectral Norm of the Hadamard Product of a Quadratic Form Using CVX

I am trying to use CVX to minimize the spectral norm of the Hadamard product of two matrices, one of which is in quadratic form. Specifically, I am trying to minimize $\|{\bf A} \odot {\bf XX}^H\|_2$, ...
usergh's user avatar
  • 43
6 votes
0 answers
48 views

Strengthening the Kovner-Besicovich theorem: Does every unit-area convex set in the plane contain a centrally symmetric hexagon of area $2/3$?

The Kovner-Besicovich theorem states that every convex set $S$ in the plane contains a centrally symmetric subset $C$ of at least $2/3$ the area of $S$, and that this bound is sharp for triangular $S$....
RavenclawPrefect's user avatar
0 votes
0 answers
38 views

Approximate local minima for sum of inverse trigonometric functions

Let $\{a_1, a_2, ..., a_N\} \in [0, 1[^N$, I would like to approximate the minimum of the function $$f(x) = x \sum_{i=1}^N \left(\sin(x)^2 - \sin(a_i x)^2 \right)^{-2} $$ in the domain $x \in {]0, \...
BCasale's user avatar
2 votes
1 answer
170 views

Equivalence of minimizing trace and determinant over matrix quadratic form in multivariate regression

Consider the multivariate regression model $$Y = XB + E$$ where $Y$ is $n \times p$ and corresponds to the dependent variables, $X$ is $n \times k$ and corresponds to the independent variables, $B$ is ...
respectableuser1's user avatar
0 votes
0 answers
40 views

Iterating partially-unconstrained optimization with projection

Let $f:H\to \mathbb{R}$ be a strictly convex Fréchet differentiable, coercive function on a separable Hilbert space $H$ and let $C_1,C_2\subseteq H$ be closed and convex. I want to optimize $$ \tag{(A)...
ABIM's user avatar
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