**0**

votes

**0**answers

19 views

### Stochastic gradient descent interleaved with deterministic optimization

I wish to solve
$\min_{x, y_k} \frac{1}{n} \sum_{k=1}^n f_k(x, y_k)$.
where $f_k$ are all smooth and convex.
Using standard stochastic gradient descent (SGD), each iteration I sample a k from $\{1, ...

**2**

votes

**0**answers

54 views

### Condition number after preconditioning

Suppose $A$ and $P$ are symmetric, positive definite matrices and that we factor $P^{-1}=EE^\top.$ Is it true that the condition number of $PA$ is upper-bounded by the condition number of ...

**0**

votes

**0**answers

22 views

### is the minimum envelope of two inrersecting convex functions convex? [on hold]

let C1 and C2 two convex curves representing a cost function and intersecting each other once (say). how to prove that there exists at least one point for which C2 > C1?

**6**

votes

**2**answers

190 views

### What methods do we have to understand the spectrum of matrices with restricted entries?

Consider questions of the form (or the "most probable value of" version of these questions rather than the "largest possible"),
What is the largest possible spectral radius of a $n \times n$ matrix ...

**0**

votes

**0**answers

60 views

### Multiple convex sets hyperplane separations

Hyperplane separation theorem states that if there is a bounded convex set $X$, a convex set $Y$, then there is a hyperplane that separates $X$ from $Y$.
That is, there is a statement that gives ...

**5**

votes

**2**answers

224 views

### Minimum of squared sum minus sum of squares

I know that
$$
\min_{\|x\|_2=1=\|y\|_2} \left(\sum_{k=1}^nx_ky_k\right)^2-\sum_{k=1}^nx_k^2y_k^2 \geq -1/2
$$
with equality whenever $|x_k|=\frac{1}{\sqrt{2}}=|y_k|$ for two coordinates.
I'm ...

**2**

votes

**1**answer

62 views

### Rate of convergence for cyclic gradient descent

I'm trying to solve the optimization problem
$\min_x \frac{1}{n} \sum_{i=1}^n f_i(x)$
where $f_i$ are (strongly) convex, smooth, lower semi-continuous, etc.
However, I am not able to do conventional ...

**0**

votes

**0**answers

23 views

### Practical bundle methods

What is the most widely applied bundle method currently available? I am trying to solve a large convex optimization as accurate as possible with dimension being roughly 5000+.... Thank you.

**2**

votes

**1**answer

42 views

### smooth minimization of piecewise linear convex function

Is it possible to apply Nesterov's smooth minimization of non smooth function on a problem of the form
$$\mathop {\min }\limits_{\lambda \in {R^m}} \mathop {\max }\limits_{\sigma \in {{\left\{ ...

**1**

vote

**0**answers

27 views

### For a non-convex function f, how to find a function g such that $g\circ f$ is strictly convex? [migrated]

The following function $f(x)={1\over (1+e^{-x})}$ is non-convex but $\ln(f(x))$ is convex. Given a non-convex function $f$, can we find a function $g$ such that $g\circ f$ is strictly convex? If yes, ...

**2**

votes

**1**answer

46 views

### When is a convex program continuous in its constraint vectors?

Consider $$F(z)=\min ae^{-x}+b e^{-y} s.t. x\ge 0, y\ge 0\text{ and } x+y=z$$
I checked if this function is continuous, but it is not at $z=0$. $F(z)=2\sqrt{ab}e^{-z/2}$ when $z\ne 0$, and ...

**2**

votes

**0**answers

50 views

### l1 Quadratic Programming [closed]

Within a SQP- algorithm it can happen that the constraints of the quadratic sub- problems are infeasible. In order to overcome this infeasibilities, a l1 penalty method can be used according to ...

**2**

votes

**1**answer

77 views

### Functions that are easy to compare to a norm

Let $X$ be a subset of $\mathbb{R}^d$, let $\|\cdot \|_p$ be a norm with $1\leq p\leq\infty$, and let $f:\mathbb{R}^d\to\mathbb{R}$ be a function. I'm trying to find examples of $X$, $p$, and $f$ for ...

**0**

votes

**0**answers

45 views

### Characterization of the maximizer of a function based on a parameter's value

Consider a smooth, continuously differentiable, and jointly concave function $f(x,y,z;a)$, where $x,y$ and $z$ are decision variables and $a$ is a problem parameter.
I have two optimization problems. ...

**3**

votes

**1**answer

73 views

### generalized mean inequality extension

from generalized inequality, we now that for $p>q$, we have $M_p(\mathbf{x})\ge M_q(\mathbf{x})$. now I am curious to know if we can find a constant $\alpha(p,q)$ which is only function of $p,q$ ...

**3**

votes

**2**answers

119 views

### Constructing a quasiconvex function

Let $C\subset\mathbb{R}^2$ be a nonempty convex set. A function $f:C\rightarrow\mathbb{R}$ is called
convex if
$$
f(\lambda u+(1-\lambda)v)\leq\lambda f(u)+(1-\lambda)f(v), \quad\forall u,v\in C, ...

**1**

vote

**1**answer

42 views

### monotonicity alike functions

assuming we have two smooth function ${f_1},{f_2}:{R^N} \to R$,
under what condition, we have
${f_1}\left( {{{\bf{x}}_1}} \right) \ge {f_1}\left( {{{\bf{x}}_2}} \right) \leftrightarrow ...

**0**

votes

**0**answers

87 views

### Hadamard / matrix product adjoint

First of all I would like to thank everyone over here at mathoverflow for their amazing generosity and help (for both pros and newbies like myself).
I apologize if this question seems dumb; I'm a new ...

**1**

vote

**0**answers

41 views

### On compactness in Sion's minimax theorem

Sions minimax theorem (wiki, paper) can be stated as follows:
Let $X$ be a compact convex subset of a linear topological space and $Y$ a convex
subset of a linear topological space. Let $f$ be a ...

**1**

vote

**1**answer

64 views

### Nested convex optimization

Suppose I have a convex optimization problem of the form $$\min_x f(x) ~~s.t.\\x\in X$$. Say that $f(x)$ and its (sub)gradient are not given in a closed form, but are determined by solving a convex ...

**1**

vote

**0**answers

67 views

### A version of isotone projection cones

We write $a \succeq b$, where both $a, b \in \mathbb{R}^n$, as a shorthand for $a_i \ge b_i$ for all $1 \le i \le n$. Let $C$ be a closed convex cone in the first orthant of $\mathbb{R}^n$ and denote ...

**5**

votes

**0**answers

56 views

### Finding the optimal mixture of two convex functions

I am trying to find an efficient way to solve the problem $$\min_{p,x_1,x_2} p\cdot f(x_1)+ (1-p) \cdot f(x_2)~~~~~ s.t.\\p\cdot g_1(x_1) + (1-p)\cdot g_2(x_2)\leq 1 \\ 0\leq p \leq 1$$ where ...

**3**

votes

**1**answer

65 views

### Analysis of first-order methods for constrained convex optimization with approximate oracles

In many first-order optimization methods an oracle is needed whose action enforces the constraint/regularizations. For example, in projected gradient descent, conditional gradient method, and proximal ...

**3**

votes

**1**answer

88 views

### Maximizing Frobenius Norm of Commutator (an opposite Procrustes problem)

I was wondering if anybody has any suggestions on the following problem:
Let $S$ be an $n\times n$ positive definite symmetric matrix. I wish to find an $n\times n$ orthogonal matrix $R$ which ...

**5**

votes

**0**answers

163 views

### Characterizing matrices with rank constraint

Given matrix $M\in\Bbb\{0,1\}^{n\times n}$, is there a nice method to characterize $$\mathscr{D}[M]=\{Q\in\Bbb\{0,1\}^{n\times n}:\mathsf{rank}(M-Q)= \mathsf{rank}(Q),\quad M-Q\in\Bbb\{0,1\}^{n\times ...

**1**

vote

**0**answers

37 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 ...

**2**

votes

**0**answers

64 views

### How to solve the following generalized quadratic programming problem [closed]

I want to solve a generalized form of a quadratic programming problem
$$\min_x \left(\sqrt{x^TPx}+\sqrt{x^TQx}\right)^2+c^Tx$$,
$$\textrm{ s.t. } Ax\le b.$$ Here, $P$ and $Q$ are both positive ...

**8**

votes

**3**answers

287 views

### Lower bound for spectral radius on $\operatorname{GL}(n,\mathbb{Z})$

Consider the group of matrices $G =\operatorname{GL}(n,\mathbb{Z})$ with integer entries and determinant $\pm 1$. For each matrix $D \in G$, the product of the eigenvalues of $D$ is equal to $\det D ...

**6**

votes

**0**answers

94 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) = ...

**0**

votes

**0**answers

36 views

### Least square problems with binary variables

I want to solve the heat equation $T_t(x,t) = - L_x . T(x,t) + F(x,t)$ in an edge-weighted graph where $L_x = \sum_i x_i e_{ij}$ is weighted Laplacian matrix of the graph. Then I conclude to the ...

**1**

vote

**1**answer

76 views

### What's the most efficient way to solve this euclidean projection on non-negative affine space constraint?

I've come across this convex optimization problem in my research where I need to project a matrix $X_0$ onto a non-negative affine space constraint and box constraints. Concretely,
$X \in ...

**0**

votes

**1**answer

51 views

### accelerate convex optimization by proximal projection

I am using level method to solve non-smooth convex programming problem (where the objective function is given by an oracle from another program ):
http://www2.isye.gatech.edu/~nemirovs/Lect_EMCO.pdf
...

**1**

vote

**0**answers

64 views

### Proximal Mapping of composition with linear operator

Let $A$ be an orthogonal matrix. Then the proximal mapping $prox_{f \circ A}(x)$ can be evaluated efficiently by
$$
(I + \partial (f \circ A))^{-1}(x) = prox_{f \circ A}(x) = A^T prox_{f}(A x),
$$
as ...

**4**

votes

**0**answers

110 views

### Two quadratic programming problems always same answer? [closed]

Was exploring quadratic programming optimization and for two types of problems the answers seemed to always equal.
Problem 1:
Minimize $\tfrac{1}{2} \mathbf{x}^T Q\mathbf{x}$
Subject to $ A ...

**0**

votes

**2**answers

104 views

### Does removing some constraints in convex program change the optimal solution? [closed]

Suppose I have a convex program which has only two variables, the objective function is strictly convex, and the constraints are linear functions.
I think removing all non-tight constraints doesn't ...

**1**

vote

**1**answer

72 views

### An optimization problem in complex space

Consider the following optimization problem
$$
\min \| \textbf{Ax-B}\|
$$
$$
s.t.|x_i|=1,i=1,...,n
$$
where $\textbf{x}\in \mathbb{C}^{n}$ is the optimization varaible, $x_i$ is the $i$-th ...

**5**

votes

**1**answer

183 views

### Algorithm to minimally connect line segments in Euclidean plane

Suppose you have finitely many line segments in the Euclidean plane. How do you "connect them to form one chain of line segments of minimal length?"
More formally and generally, what I'm looking for ...

**0**

votes

**0**answers

53 views

### Restricted strong convexity for biconvex functions

Recently, it has been shown (arxiv paper) that non-convex functions with the restricted strong convexity (RSC) property has the interesting property that their local minima lie within a small ball of ...

**0**

votes

**1**answer

59 views

### Finding maximum of a function with unfixed number of variables

Can anybody solve this:
For a constant positive integer $n\geq6$
find $k$ and positive integers $a_{1},a_{2},...,a_{k}$
that maximize the expression
...

**16**

votes

**8**answers

964 views

### When do people actually use the maximum entropy distribution?

One of the standard problems in convex optimization is the calculation of the maximum entropy distribution that satisfies some set of criteria. For example, if $\mathbf{x} \in \mathbb R^n$ is an ...

**0**

votes

**0**answers

83 views

### Matrix equation

Let $A$ be $k\times n$ matrix i.e., $A=(a_{1},\ldots, a_{n})$ where $a_{j} \in \mathbb{R}^{k}$, $rank(A)=k$ and $1\leq k \leq n$. Let $q=(q_{1},\ldots, q_{n})\in\mathbb{R}^{n}$ be such that ...

**2**

votes

**1**answer

166 views

### Necessary conditions for optimality in Banach spaces

Let $X$ denote the non-negative "orthant" of the Banach space $L^2$ (or whatever you call the set of functions in $L^2$ that are non-negative), and let $C$ be a closed, convex subset of $X$. Let $f$ ...

**7**

votes

**2**answers

207 views

### Removing constraints in convex optimization

Say I have a convex optimization problem of the form $$\min_x f(x) ~~ s.t.\\ g_1(x)\leq0,\\\vdots \\g_n(x)\leq 0$$ with all functions convex. Suppose that $x^*$ is a unique optimizer to my problem and ...

**1**

vote

**0**answers

64 views

### What is the purpose of the definition of “metric regularity”/“regularity modulus”?

A set mapping $F:X \rightrightarrows Y$ is said to be metrically regular for $\overline{x}\in X$ and $\overline{y} \in Y$ if there exists a $\kappa\in(0,\infty)$ for which
$$
d(x,F^{-1}(y))\leq ...

**3**

votes

**0**answers

101 views

### This function looks quasiconvex, can't understand why

Suppose that $\mathbf{C}$ is a given matrix with non-negative entries in $\mathbb{R}^{m\times n}$ and $d$ is a given scalar, and let $g(\mathbf{y})$ be defined by ...

**1**

vote

**0**answers

46 views

### Is there a unique tilted measure with specified marginals?

Suppose $\mathcal{A},\mathcal{B}$ are finite sets and $\mu_{A,B}(a,b)$ is a probability measure on the product set $\mathcal{A}\times \mathcal{B}$ so that $\mu_{A,B}(a,b)>0$ for each $a\in ...

**-3**

votes

**1**answer

59 views

### Convert constraint to do convex optimization or use Lagrange multiplier method [closed]

$w_1, w_2, w_3 ... w_n$ are the weights I need to find
I have the following constraint:
$|w_1| + |w_2| + .. |w_n| <= 5$
That is the sum of the absolute values of the weights has to be less than ...

**2**

votes

**1**answer

210 views

### Linearly constrained eigenvalue problem

Suppose I'd like to:
\begin{align}
\mathop{\text{min}}_\mathbf{x} && \mathbf{x}^T\mathbf{A}\mathbf{x} \\
\text{subject to:} && \mathbf{x}^T \mathbf{M} \mathbf{x} = 1\\
&& ...

**4**

votes

**2**answers

276 views

### Convex Sets and Nearest Neighbors

For a set $S \subseteq \mathbb{R}^n$ and a point $x \in \mathbb{R}^n$, let $c_S(x)$ be the point $s \in S$ that minimizes $\|x-s\|$ if such a point exists and is uniquely determined. It is known that ...

**3**

votes

**1**answer

88 views

### Can one always find sparse solutions to an $\ell^1$-minimization problem?

Consider $A\in\mathbb{R}^{m \times N}$ and $b \in \mathbb{R}^m$, with $m<N$. Is it true that the optimization problem
$$\min \|x\|_1 \quad s.t. \;\; A x = b,$$
admits an $m$-sparse solution in ...