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0
votes
1answer
30 views

Limits of argmin ratios and sums

In my research on convergence properties of certain Bayesian methods I have encountered $\mathop{\mathrm{arg\,min}}$ limits of the forms \begin{equation} \lim_{n\to\infty} \mathop{\mathrm{arg\,min}}_{\...
0
votes
1answer
301 views

Maximize a sum of log of sum

For a matrix $c (m\times n)$ of non-negative constants, find values of $\lambda_1, \lambda_2, \ldots, \lambda_n$ that satisfy $\sum_{k=1}^n \lambda_k = 1$, $\lambda_k \ge 0 \, \forall k$ and maximize ...
10
votes
0answers
214 views

“Small” maps from sphere to sphere

Start with a continuous map $f:S^{n+k} \rightarrow S^n$ (round unit spheres). The graph of $f$ lives in $S^{n+k}\times S^n$ and suppose it has a surface area (as a subspace of co-dimension $n$). Now ...
6
votes
0answers
118 views

A specific case of the $p$-center problem

Given a fixed positive integer $m$, let $\cal{S}$ be the subset from $\mathbb{R}^m$ defined as $\cal{S} = \{u \in \mathbb{R}^m \mid \forall i \in \{1, \dots, m\}, u(i) > 0$ and $\sum_{i=1}^m{u(i) = ...
6
votes
0answers
556 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 $...
5
votes
0answers
76 views

Basin of Attraction

I have a function $F$ which is defined as follows: $$ F(x) = \sum_{i=1}^N f(z_i^T x) $$ where ${z_i}$ are known $m \times 1$ vectors, $x$ is an $m \times 1$ vector, and for $t\in \mathbb{R}$, $f(t) = \...
4
votes
0answers
211 views

maximize non-convex composite function

I want to maximize a composite function over a convex set \begin{equation} \begin{aligned} & \underset{\mathbf{p}}{\text{maximize}} & & f(\mathbf{p})-g(\mathbf{p})\\ & \text{subject to}...
3
votes
0answers
449 views

Minimum weight bipartite graph clique covering

I was wondering if anyone here could give me any pointers as to how to solve the following problem. Let $B=(L,R,E)$ be a bipartite graph, and $\forall u\in L\cup R$, let $c_u$ be a cost associated to ...
3
votes
0answers
193 views

Could SVD be used to optimize the partial inner-products?

Suppose a set $N$ of $n$ distinct points in $m-$dimensional space is given in $X\in\mathbb{R}^{n\times m}$. Also, suppose a subset $L\subset N$, $|L|=l<m<n$, with $m-$dimensional coordinates in ...
2
votes
0answers
67 views

Characterizing the optimimum over the space of probability measures

Consider the following optimization problem: \begin{equation} \max_{\mu \in \mathcal{M}} \int \log\left( \int e^{\alpha U(x,y)} d\mu(y) \right) d\nu(x) \end{equation} where $\mathcal{M}$ is the space ...
2
votes
0answers
161 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. ...
1
vote
0answers
27 views

Difference between Chebyshev first and second degree iterative methods

Consider linear equation $Au = f$. We want to solve it with iterative method (assuming $A$ is good). First order iterative method is: $$ u^{k+1} = u^k - \alpha_{k+1}(Au^k - f), $$ The second degree ...
1
vote
0answers
28 views

About a particular definition of a Hessian of a function of tuples of matrices

Say I have a function $L : (W_1,..,W_{H+1}) \rightarrow \mathbb{R}$ i.e it takes a tuple of $n$ matrices of different dimensions and computes a number from them. Then I see being defined a ...
1
vote
0answers
21 views

Deterministic global solution to find the Optimal-knot placements for continuous piecewise linear functions to fit nonlinear data

I have been searching lately for a deterministic global technique to linearize a nonlinear function with continuous piecewise linear regions. I've a univariate nonlinear function y=f(x). where f(x) ...
1
vote
0answers
33 views

solution of an infinite horizon optimization problem

Give the following formulation: $\min_{\{x_s(t):\forall s,t\}} \sum_{s \in \mathcal{S}} \mathbf{1}\left(\lim_{T\rightarrow \infty} \frac{1}{T} \sum_{t=1}^T \frac{y_s(t)}{x_s(t)}\leq 1\right)$ $s.t. ...
1
vote
0answers
121 views

Likelihood convexification

I am doing constrained vector optimization using a non-convex non-linear likelihood function. My problem is of the following form: $$\begin{align*}\hat Q &= \underset{\vec Q}{\arg\min} -\log \...
1
vote
0answers
51 views

Modified Orthonormal Procrustes Problem

In the general orthonormal Procrustes problem, we want to find an orthonormal matrix $C$ to minimize $\|Y-XC\|_F^2$, where $Y$ is a known $n\times q$ matrix, $X$ is a known $n \times m$ matrix, and $C$...
1
vote
0answers
48 views

Changing a nonlinear equality constraint into some conic inequality plus rank constraint

If we have a constraint optimization problem in which one of our constraint is $\prod\limits_{k = 1}^N {\left( {x - {a_k}} \right) = 0} $ . How could this nonlinear equality condition be changed into ...
1
vote
0answers
114 views

range of the difference-of-two-qubit-$4 \times 4$-density-matrix-determinants

The determinant of a two-qubit $4 \times 4$ density matrix--that is, a Hermitian, nonnegative definite matrix with unit trace--lies between $0$ and $(\frac{1}{2})^8$. (A "pure state" has determinant ...
1
vote
0answers
123 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
178 views

Constructing an $\epsilon$-net for a Lipschitz subspace of $L^2$

Let $X$ be a subset of $L^2([0,1])$ which contains only Lipschitz function. Also, the member of $X$ are uniformly bounded $$ |x(t)| < M, \text{ for all $x \in X$ and $t \in [0,1]$}. $$ Let $F: X \...
1
vote
0answers
101 views

Trying to get an idea of the maths I could use for this optimization problem

Firstly, apologies if some of the notation or terminology is odd, or if I am defining functions that have standard notation associated with them already - I am not familiar with the concepts in this ...
1
vote
0answers
138 views

An $L^{\infty} Version of Principal Component Analysis?

I have a $k$ by $n$ matrix $A$, with $k \ll n$. In case it helps, the $k$ rows are orthonormal. I'm interested in finding a $k$ by $k$ orthogonal matrix $M$ so as to maximize the $L^{\infty}$ norms ...
1
vote
0answers
217 views

On the convergence of a special fixed point iteration

The problem is actually a quadratically constrained quadratic program. And the formulation is: $max: \frac{1}{2}x^TQx + d^Tx$ $s.t. x\in R^{n,+} ,\sum_{i\in I_p}x_i^2=1, p=1..k$ where $d\in R^{n,+}$,...
0
votes
0answers
52 views

deriving concave upper bounds of a domain constrained nonconvex function over a simplex

Consider a nonconvex function $h(X)=f(X^\dagger AX)$, where $X\in C^{r \times n}$, $A$ is a positive semidefinite matrix, and $f$ satisfies the following two properties: \begin{align} &f(W): H_{+}^...
0
votes
0answers
41 views

Limit of argmin of sum

Suppose that I know $f_n\rightarrow f$ and $g_n\rightarrow g$ are both continuous maps from a Complete Riemmanian Manifold $X$ to $\mathbb{R}$ which converge pointwise almost everywhere. Then is it ...
0
votes
0answers
43 views

Largest instance of highly nonlinear benchmark functions (e.g. Rastrigin function)

What is the largest instance size (number of variables) ever numerically solved for highly nonlinear (continuous, not combinatorial) optimization benchmarks functions, such as Rastrigin, Schwefel or ...
0
votes
0answers
31 views

Dennis More' Superlinear Convergence_refrences request

Why in the proof of superlinear convergence of restricted broyden class (for the unconstrained optimization) we need the bounded deterioration condition for the approximation of all the true hessian ...
0
votes
0answers
68 views

Optimization with vectors

I am trying to solve the following optimization problem as a small part of a research project, and I do not know if there exists closed form solutions. My linear algebra is very rusty and I am looking ...
0
votes
0answers
175 views

A contractive mapping which I don't understand

Given a matrix $Y$ and a vector $c$ define the following iteration $\hat{c} = f(c)$, where each element of $\hat{c}$ is given by $$\hat{c}_{\ell} = \frac{\sum_k Y_{k,\ell}\frac{1}{|c_{\ell}|^2+|c_{k}|...
0
votes
0answers
52 views

Separable Least squares - is there a notion of conjugate directions?

I have a general question. Suppose I have the following to optimize $$\|Y-A(\mathbf{x})B(\mathbf{y})\|^2$$ where $Y$ is a vector, $A(\mathbf{x})$ is a matrix that depends on a vector $\mathbf{x}$ in a ...
0
votes
0answers
129 views

Optimization over space of probability measures

Consider an optimization problem as follows: $$ \min\mathbb E_w[f_0(w)] \mathrm{\,\,\,\,\,\ s.t.\,\,\,\,} E_w[f_i(w)]\leq 0 ,\,\,\, i=1,\dots, k $$ where the maximum is taken over $\mathscr M$, ...
0
votes
0answers
85 views

numerical solver for stochastic optimal control problems

can any one recommend numerical solver (c/c++ library preferred) for stochastic optimal control problems? For deterministic optimal control I found something like that: http://abs-5.me.washington.edu/...
0
votes
0answers
75 views

Sufficient optimality condition for a non-smooth quasiconvex problem

The result of relaxing to an integer program is the following optimization problem: $$\min_{\textbf{x}} \sum_{i=1}^n \alpha_i h(x_i)\quad subject \; to \quad A\textbf{x} = \textbf{0}$$ where $\textbf{...
0
votes
0answers
115 views

The role of subgradient in programming with nonsmooth functions

It is obvious that there is similarity between subgradient and gradient. The subgradient of smooth functions is reduced to gradient. I have two questions. The first is does there exist subgradient ...
0
votes
0answers
62 views

equivalence between primitive and dual

Hi everyone! I have a problem about the duality gap of the primitive problem and the dual problem. This problem comes from a probabilistic model named Lagrangian UVM. http://www.freeimagehosting.net/...
0
votes
0answers
836 views

optimization of a separable function

Hello everyone, this is a optimization problem whose objective function is separable: $$F(x)=\sum_{i=1}^n\frac{\theta_i^2}{4}\sum_{j=1}^m\left(1+\rho x_ik_j+\sqrt{(x_ik_j+\rho)^2+1-\rho^2}-\frac{2\...
0
votes
0answers
361 views

Optimization of a matrix with an objective function (for ML)

Hi. I need to do max. likelihood for an objective likelihood function L (minimize it), and the target is a matrix. ie: $$min_KL(K)$$ For example: K is, let's say, of size 3x3 and with initial ...
0
votes
0answers
94 views

minimizing the integral of a function over square sets.

Hi! I'm interested in some problems, but to be honest i'm not sure of the field they belong to. Let $h(x,y)$ be a bivariate function on $X^2$, where $X$ is some nice topological space (for instance $...
-1
votes
0answers
24 views

Multi-objective optimization for large matrices

I have a large matrix 102400 x 600 to optimize for two different criteria (maximum likelihood over a large dataset and another more complicated one). In practice, it represents a factors loading ...