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1answer
77 views

Constrained minimum maximal distance.

Let $C$ and $D$ be two convex sets. And suppose $C\cap D\neq \emptyset$. Let $x^*$ is the solution to the optimization problem: $$\min_{x\in C} \max_{y \in D} |x-y|^2$$ Is it true that $x^* \in D$. ...
1
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1answer
148 views

Regularizing a Convex function with itself

Hi, This is a problem that has being bothering me the last few days. Assume a convex function $f(x): {\mathbb R}^n \rightarrow {\mathbb R}$ with a unique minimizer $x^{\star}$. Now consider the ...
2
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0answers
140 views

Quadratic optimization with parameter in constraint

Disclaimer: I posted the same question on math.stackexchange. However, the FAQ suggests to post research-level questions in this forum. Question: Given a function $q: \mathbb R^{N\times N}\mapsto ...
0
votes
1answer
246 views

Minimum conditional expectation of complement of event given conditional expectation of event?

Suppose $X$ is a pdf over $[0,m]$ and $Y$ is a binary experiment on $X$ such that $P(Y=1|X)$ is continuous, and we have that $\mathbb{E}[X|Y=1] = \mu_y$ and $\mathbb{E}[X] < \mu_y$. Is it always ...
1
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0answers
92 views

minimizing the sum of euclidean norms

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 ...
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2answers
101 views

Feasibility of a given set of homogenuous nonconvex quadratic inequality constraints

Let $C_1$,$C_2$,...$C_N$ be $M \times M$ indefinite hermitian matrices. What can we say about the following quadratic constriants \begin{align} w^{H}C_1w>0 \\\ w^{H}C_2w>0 \\\ ...~~~~~~~~~~ \\\ ...
2
votes
1answer
357 views

Maximizing supermodular functions

I have a real supermodular objective function which I want to maximize with constraint. The constraint is on the size, like |A|=k . I am wondering if anyone can give me more information about a ...
0
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0answers
81 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
1answer
194 views

minimization of a function when the feasible set is an unbounded cone

I have the following semi-infinite programming problem: I need to minimize a strictly convex real-valued function $f:\mathbb R^n\to\mathbb R$ subject to infinite linear constraints. I know in advance ...
1
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1answer
82 views

Expected rank - computable approximations

I'm interested in finding the expected rank of some random matrix $A$ (I don't want to specify its distribution right now, since my question makes sense in general). Computing $\mathbb{E} \ ...
2
votes
1answer
275 views

lipschitz constant of a multivariate function

I have a function $f:\mathbb{R}^{50} \rightarrow \mathbb{R}$ and I need to compute the Lipschitz constant of $f$ to solve an optimization problem using a specific algorithm. Does any one have ...
1
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2answers
152 views

What is the dual of an semidefinitely representable (SDR) cone?

The Question Let $V\simeq \mathbb{R}^r$ be an $r$-dimensional vector space with the usual Euclidean inner product. Let $\mathcal K\subset V$ be a cone defined as $$ \mathcal K=\Big\{x\in ...
6
votes
2answers
150 views

On the convexity of element-wise norm 1 of the inverse

Question first asked on math.stackexchange here: http://math.stackexchange.com/questions/317209/on-the-convexity-of-element-wise-norm-1-of-the-inverse On the convexity of element-wise norm 1 of the ...
2
votes
0answers
70 views

Fastest 'Oracle' Algorithm for satisfying a single linear constraint on a convex set?

In this paper by Arora, Hazan, and Kale (http://www.cs.princeton.edu/~arora/pubs/MWsurvey.pdf) a method is given for using the Multiplicative Weights Update algorithm to quickly solve Linear Programs ...
3
votes
1answer
178 views

Is the Binomial Expectation of a Multivariate Convex Function Convex in the Vector p?

Let $\mathbf{p}=(p_1,\dots,p_m)$ be a vector in $[0,1]^m$ and let $\mathbf{X}=(X_1,\dots,X_m)$ be a vector of independently-distributed binomial random variables such that $X_i\sim ...
1
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0answers
597 views

Robust optimization in matlab using fmincon [closed]

I am trying to implement the following optimization (from this paper) in Matlab using fmincon: $\min_\omega\omega'\Sigma\omega$ subject to $\min_Ur_p \geq r_0$ where $\Sigma$ is a positive definite ...
1
vote
1answer
143 views

Subgradient of Minimum Eigenvalue

Consider three $N \times N$ Hermitian matrices $A_0$, $A_1$, $A_2$. Consider the function \begin{align} f(t_1,t_2)=\lambda_{\text{min}}(A_0+t_1A_1+t_2A_2) \end{align} where $\lambda_{\text{min}}$ ...
1
vote
1answer
753 views

Schur complement and negative definite matrices

Hello, My question regards to the Schur complement lemma. Consider the matrix $M=\left( \begin{array}{cc} A & B\\\ B^T & C \end{array}\right) $. According to the lemma $M\geq0$ iff $C>0$ ...
1
vote
1answer
74 views

Numerical optimisation for multivariate Gaussians

Hi, I want to calculate $ f_{\mathbf x}(x_1,\ldots,x_k)\, = \frac{1}{(2\pi)^{k/2}|\boldsymbol\Sigma|^{1/2}} \exp\left(-\frac{1}{2}({\mathbf x}-{\boldsymbol\mu})^T{\boldsymbol\Sigma}^{-1}({\mathbf ...
5
votes
0answers
141 views

Numerical linear algebra: how to compute $B^TC^{−1}B$ efficiently

Hi, my question is similar to this one. I have to compute $B^TC^{−1}B$, where $C$ is a strictly positive definite $n\times n$ matrix and $B$ is $n\times m$. The matrix $C$ is huge ($n$ up to a ...
0
votes
1answer
132 views

Minimal point of the intersection of convex sets.

I am trying to find out if there is any known result in convex optimization that implies the following statement: "A minimal point of the intersection of $N > 2$ convex sets in $\mathbb{R}^2$ is ...
1
vote
2answers
172 views

Convex optimization problem to QPP

Briefly, have the following problem: \begin{equation} \sum_{i = 0}^n a_i \ (max [ F_i( \bar x ), 0 ] )^2 \rightarrow min, \\\\ s.t.\\\\ A \bar x \leq b \end{equation} where $ F( \bar x ) $ is a ...
2
votes
2answers
256 views

A certain type of Quadratic Constrained Quadratic Programming Problem (QCQP)

Let $P_1$, $P_2$ be two hermitian matrices. Can anyone comment about the following (QCQP) \begin{equation} \min_{z}~z^{H}z \\\ ~~subject~to ~z^{H}P_1z+1\leq 0,~z^{H}P_2z+1\leq 0 \end{equation} I am ...
0
votes
1answer
227 views

Is unconstrained integer convex optimization problem NP-hard?

Does anyone know a reference to the answer if unconstrained integer convex optimization problem (i.e. $\min_{x\in \mathbb{Z}^N} F(x)$, $F$ is convex and $N$ is NOT fixed) is NP-hard? Thank you in ...
0
votes
2answers
319 views

Is minimum of convex envelope the same as minimum of the original function?

Hello everyone my question is: $Question:$ Consider a function $f:X \rightarrow \mathbf R$ where $X$ is a convex subset of $\mathbf{R}^n$. The convex envelope of $f$ over $X$ is defined as the ...
0
votes
1answer
545 views

Finding linearly independent columns of a large sparse rectangular matrix

I have a problem that necessitates solving a large non-negative least-squares problem. My matrix A is large, sparse, highly rectangular (num rows >> num cols) and nearly binary. However, A is not ...
2
votes
1answer
247 views

How to Find a Matrix Closest to a Given One Under Certain Constraints

I was reading a paper about BFGS and met the following problem: $\min_B \|B-B_k\|$, s.t. $B=B^{\top}, Bs_k=y_k, s_k^{\top}y_k>0$ and $B$ is positive definite. Here $B_k$ is a symmetric positive ...