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1
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0answers
21 views

Convex interaction energy

Does anybody know examples of absolutely continuous probability measures $\mu_0,\mu_1$ on $\mathbb{R}^n$ with finite 2nd moments such that $$ \frac{d^2}{dt^2}\left(\int_{\mathbb{R}^n\times ...
1
vote
1answer
53 views

Minimize a strictly convex quadratic function subject to linearly equality and nonnegativity constraints in finite time?

I am wondering if we can minimize a strictly convex quadratic function in finite time, subject to linearly equality and nonnegativity constraints. Thanks!
0
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0answers
40 views

minimum distance between sets and relation with functions [on hold]

Let $A, B$ be two convex and closed subsets of $\mathbb{R}^n$. We would like to the minimum distance between these two sets. i.e., we want to find a solution for the following problem. $$ \min ...
0
votes
0answers
25 views

Maximizing the product of projections of a vector on another vectors

I want to get the $N\times1$ complex vector $\mathbf{x}$ which maximizes this real valued function $f=\mathbf{x}^{H}\left (\mathbf{a}_{1} \mathbf{a}_{1}^{H}\mathbf{x}\mathbf{x}^{H}\mathbf{a}_{2} ...
2
votes
5answers
188 views

Distance between two sets

Let $A, B$ be two convex and closed subsets of $\mathbb{R}^n$. We would like to the minimum distance between these two sets. i.e., we want to find a solution for the following problem. $$ \min ...
0
votes
1answer
23 views

optimization of inverse matrix with constraint on matrix elements

everyone! I have this optimization problem with constraint. D and T are symmetric matrice, where T is known and D is the unknown parameter. x and v are two known p-dimensional vectors. The objective ...
2
votes
2answers
73 views

How to minimize the Bregman divergence on a convex hull spanned from a set of vectors?

everyone. It has been well known that the following minimization problem of a Bregman divergence with linear inequality can be solved by successively projecting the current point to each constraint ...
22
votes
2answers
747 views

An Interesting Optimization Problem

You are given n non-negative integers $a_1, a_2 ,, a_n$. In a single operation, you take any two integers out of these integers and replace them with a new integer having value equal to difference ...
1
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0answers
25 views

Searching for the maximum of a (strictly convex) two-dimensional distribution via maximization over a series of arbitrarily specified 1D intervals

Let $f$ be a strictly convex two-dimensional distribution with a maximum $M$ at some unknown position $(x_m,y_m)$. Starting from the origin, $(x_0,y_0) = (0,0)$, we need to find $(x_m,y_m)$, however ...
0
votes
0answers
32 views

Homotopy with non piece-wise linear boundary

in the middle of a long proof I encounter the following problem. Let $E$ be a closed and convex set in $\mathbb R^n$ such that for all $\vec x\in E$ it holds that $\sum_ix_i=1$. (We can understand ...
3
votes
2answers
171 views

Gradient descent-like optimization on a convex landscape with noisy sampling

This is a rewrite of the original positing (below), and is crossposted to ...
2
votes
1answer
83 views

Block Covariance Matrix - Positive Definite? (Quadratic Optimization) [closed]

I have a covariance matrix C. I have then formulated an quadratic optimization problem that involves the following matrix in the quadratic form: [ C C ] [ C C ] However, the quadratic solver ...
1
vote
0answers
72 views

Extreme points of a set related to semidefinite cone

Let $X \in \mathbb{R}^{n \times n}$ be symmetric matrix. Consider the following set $$ \mathcal{C} = \{ X: X \succeq 0, \quad 0 \le X_{ij} \le 1, \forall i,j\} $$ What are the extreme points of this ...
3
votes
1answer
105 views

SDP formulation of noisy low rank matrix completion

Exact low rank matrix completion using nuclear norm minimization can be formulated as a semidefinite program (SDP). Following the notation in the paper, a convex problem for noisy matrix completion ...
0
votes
0answers
63 views

An attempt to solve “Maximization of a total variation distance subject to another total variation distance in Markov chain”

I have been trying to solve Maximization of a total variation distance subject to another total variation distance in Markov chain. As a recall, suppose we have a pair of correlated random variables ...
0
votes
0answers
46 views

No Strong Duality In Spite of Slater's Condition

I was reading some course notes here. On Page 8, it says: Note that strong duality holds here (Slater's condition), but the optimal value of the last problem is not necessarily the optimal ...
0
votes
1answer
118 views

Find the following transformation $G$

I asked this question 6 days ago on math.stackexchange.com (http://math.stackexchange.com/questions/656585/find-the-following-transformation-g). I didn't get any answers yet, so I'm posting here. I'm ...
0
votes
1answer
81 views

Reverse optimization of a minimum cost flow network

Given an undirected graph $(V,E)$, with $W$ as the weight of each edge, and a convex cost function $F(X)$, such as $|X|^k$ ($k>1$). The cost to send $x$ unit of flow through edge $e_i$ is defined ...
4
votes
2answers
108 views

Minimax theorem on a non convex domain

A minimax theorem is a theorem which states that under certain conditions on $\mathcal{X}$, $\mathcal{Y}$ and $f$: $$ \inf_{x \in \mathcal{X}}{\sup_{y \in \mathcal{Y}}{f(x,y)}} = \sup_{y \in ...
-1
votes
1answer
71 views

Uniqueness of the solution to a quadratic problem [closed]

Consider a positive definite matrix $\boldsymbol H$, the known vectors ${\boldsymbol b}$ and ${\boldsymbol a}_i$. Now the minimization problem is casted with respect to the vector ${\boldsymbol x} $ ...
0
votes
1answer
96 views

Eigenvalues of a given parametrized matrix.

Let $\mathbf{A}$ and $\mathbf{B}$ be two complex rank-one $N\times N$ positive semi-definite matrices. Let the matrix $\mathbf{C}$ be defined as \begin{align} ...
1
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1answer
164 views

Interior of a dual cone

Let $K$ be a closed convex cone in $\mathbb{R}^n$. Its dual cone (which is also closed and convex) is defined by $K' = \{ \phi\ |\ \phi(x) \geq 0,\ \ \forall x \in K\}$. I know that the interior of ...
1
vote
1answer
97 views

Lagrange multiplier and semidefinite programming

suppose we have a primal semidefinite programming. for finding its dual we use Lagrange multiplier $w_i$ for each semidefinite constraint. If the Lagrange multiplier be zero for one constraint what we ...
1
vote
0answers
121 views

How to solve such an optimization problem efficiently?

Given a symmetric positive semi-definite matrix $\mathbf Y$ and a convex set $\mathcal M$ which is a subset of all symmetric positive semi-definite matrices (consider a simple case of $\mathcal M$: a ...
2
votes
2answers
84 views

Projection onto rotated box

Does anyone know if there is an efficient way to find the projection of an arbitrary point $z$ onto a rotated box, i.e. onto the set $\Omega=\{x \mid a \leq Ux \leq b\}$ where $U$ is a unitary matrix? ...
3
votes
1answer
128 views

Equivalent method for maximum likelihood estimation of covariance parameters

My goal is to estimate the parameters of a covariance matrix $\Omega$, by maximizing the following log-likelihood function: $$\log L(\vec\tau, \rho, \sigma \mid W, X) = -m\ln(\left | \Omega \right |) ...
0
votes
0answers
67 views

minimizing the sum of euclidean norms with box constraints

minimizing the sum of euclidean norms with box constraints I am a graduate student in computer science, making a thesis on uncertainty geometry. During my thesis I came across the following ...
1
vote
0answers
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 ...
1
vote
0answers
91 views

Lipschitz continuity of solution set mapping of a parametric convex optimization problem

I have a parametric convex optimization problem: \begin{array}{cl} \underset{x}{\text{minimize}} & f\left(x,z\right)\\ \text{subject to} & g\left(x\right)\leq0 \end{array} where $x$ is the ...
2
votes
1answer
102 views

Circumscribed ellipsoid of minimum Hilbert-Schmidt norm

Let $K\subseteq \mathbb{R}^n$ be a full-dumensional convex body. The Löwner ellipsoid of $K$ is the unique ellipsoid of smallest volume containing $K$. My question is about a related object: the ...
-1
votes
1answer
43 views

Determining the sign of each element of the optimal of a strict convex function

The problem is: Let $\vec{x}\in\mathbb{R}^d$ be the variable and $f(\vec{x})$ be a scalar function that is globally strictly convex in $\mathbb{R}^d$. We assume the unique optimum of $f$ to be ...
2
votes
1answer
136 views

quadratic programming on hypercube

I want to maximize a quadratic form $\mathbf x^T\mathbf Q\mathbf x$ and also want to find out which vector $\mathbf x$ maximizes the quadratic form when $\mathbf Q$ is an $n\times n$ positive ...
1
vote
1answer
162 views

Question on convex optimization and dual norms [closed]

I have the following questions about dual norms : How do you prove that the dual of the dual norm is in fact the original norm? This is what I have so far: If I have $||y||_* $ as the norm dual of ...
2
votes
2answers
106 views

Survey on Compared Running Time: Ellipsoid Method vs. Simplex Method

If you look through papers on the Ellipsoid Method, there is a large agreement, that the Ellipsoid Method, although theoretically polynomial, is in practice way slower than the Simplex Method. ...
2
votes
1answer
113 views

maximizing convex quadratic form over the intersection of unit sphere and positive orthant

For a positive semi-definite matrix $C$, I want to find the solution to the following problem: $\arg\max_{h\geq 0} h^T C h\quad$ s.t. $\quad h^T h\leq 1$ Any pointers are welcome.
3
votes
1answer
153 views

Fast algorithm for maximizing smallest eigenvalue of linear combination of hermitian matrices

I have an engineering back ground. Due to work, I came across this problem \begin{align} &\max_{\lambda,y_i\in \mathbb{R}}~\lambda \\\ ...
1
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0answers
45 views

Checking feasibility with eigenvalue and linear constraints

Is there an efficient algorithm to check the simultaneous feasibility of linear and eigenvalue constraints? For example, given $ \lambda_1 \ge \lambda_2 \ge 0$, is the following problem efficiently ...
1
vote
0answers
99 views

Modifying a QP to incorporate more constraints

Consider the following problem: $$\min \sum_{i=1}^n (Y_i - Z^{(i)})^2 \\ \text{subjected to}~ \epsilon_k^{\top}(X_j-X_k) \leq Z^{(j)}-Z^{(k)} ~ \forall k,j = 1 \ldots n. $$ where $\epsilon_1, ...
3
votes
1answer
111 views

optimize spectral radius

Hi I would like to solve the following optimization problem. Let $A$ be an $n \times n$ nonnegative real matrix where $A^{-1}$is an M-matrix. Let $D=\text{diag}\{d_{1}, \dots , d_{n}\}$ be a ...
1
vote
0answers
69 views

Interior point optimisation using big M for L1 norm on linear system using Dikin's Affine method

I am a 4th year undergrad surveying student studying computations, specifically $L_{1}$ norm minimisation of residuals in large data sets. To start with (and probably to finish with) I'm using a set ...
2
votes
1answer
123 views

optimization over positive semidefinite matrices

I wonder what is the most explicit characterization that can be given for the solution to the ($N$-dimensional) problem of maximizing the criterion $$ -\textrm{trace}[AS^{-1}] - b^\top Sb $$ over ...
1
vote
1answer
120 views

Maximum Dispersion of a Connected Geometric Graph

Let $\left\{\mathbf{p}_1,\dots, \mathbf{p}_k\right\}$ be a set of points in $n$-dimensional Euclidean space, and let the second moment of these points be defined as: $ U=\sum \limits_{i=1}^{k} ...
2
votes
0answers
113 views

An intuition for three different types of subgradients (proximal, regular, limiting)

I'm having a bit of difficulty getting my head around the different types of subgradients we're currently covering in a nonsmooth optimisation class I'm taking. These subgradients are (assume $x \in$ ...
1
vote
1answer
132 views

Semi Definite Relaxation for a Quadratic Feasibility Problem using CVX

Consider the following Semi-Definite Feasibility problem \begin{align} \max_{\mathbf{Z}}~0 \\\ \mathrm{trace}(\mathbf{Z})\leq \rho \\\ \mathrm{trace}(\mathbf{S}_1\mathbf{Z}) \geq \alpha \\\ ...
4
votes
2answers
95 views

A certain type of constrained Rayleigh-Ritz ratio

Let $\mathbf{A_1}$ and $\mathbf{A_2}$ be two hermitian matrices. Consider the problem \begin{align} \max_{\mathbf{u}^H\mathbf{u}=1}~\mathbf{u}^H\mathbf{A}_1\mathbf{u} \\\ ...
3
votes
0answers
195 views

Coordinate mirror descent

Let $f$ be a jointly convex function of 2 variables say $x,y$. I am interested in solving the optimization problem $$\min_{x,y\in\Delta} f(x,y)$$ where $\Delta$ is a $d$ dimensional simplex. An ...
2
votes
2answers
265 views

convergence of the infima of convex functions

Can one give a reference to a result like this: If a sequence of convex functions $f_{n}$ on $\mathbb{R}$ converges pointwise to a non-monotonic function $f$, then ...
0
votes
0answers
97 views

Min of a real-valued Fourier transform

Let $P$ be a compact, convex, symmetric, $d$-dimensional body in $\mathbb R^d$, and let $\mu$ be a (necessarily) symmetric probability measure on $P$, so that $\mu_P(x) = \mu_P(-x)$, for all $x \in ...
0
votes
1answer
206 views

Nonconvex optimization problem

I have a nonconvex optimization problem. It is actually optimizing a linear objective function over a set of linear constraints and a set of nonlinear, non convex constraints. Is this problem ...
0
votes
2answers
109 views

Rewrite optimization objective

Hi, I wanted to ask, under which conditions can one rewrite the optimization objective $\min_x f(x)\;\;\;s.t.\;\;\;g(x) \leq s$ as $\min_x g(x)\;\;\;s.t.\;\;\;f(x) \leq t$ I have particular ...