**1**

vote

**1**answer

149 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

**0**answers

39 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. ...

**4**

votes

**2**answers

374 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

**2**answers

104 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, ...

**3**

votes

**1**answer

62 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 $p,q$ ...

**1**

vote

**1**answer

38 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 ...

**1**

vote

**1**answer

179 views

### optimization of inverse matrix with constraint on matrix elements

everyone! I have this optimization problem with constraint.
$D$ and $T$ are symmetric matrices, where T is known and D is the unknown parameter.
$x$ and $v$ are two known p-dimensional vectors.
The ...

**0**

votes

**1**answer

173 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 ...

**0**

votes

**0**answers

21 views

### Is there a unique saddle value for a convex/concave optimization? [migrated]

I asked this question previously in math.stackexchange and couldnt receive any answer. The question is fundamental, though I couldn't find a reference or a proof somewhere.
Here is the question:
...

**0**

votes

**1**answer

279 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

**0**answers

72 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

**1**answer

165 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 ...

**1**

vote

**0**answers

38 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

53 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

62 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

51 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

61 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 ...

**5**

votes

**0**answers

144 views

### Characterizing matrices with rank constraint

Given matrix $M\in\Bbb Z_{\geq0,\leq b}^{n\times n}$, is there a nice method to characterize $$\mathscr{D}[M,b]=\{Q\in\Bbb R_{\geq0,\leq b}^{n\times n}:\mathsf{rank}(M-Q)= \mathsf{rank}(Q),\quad ...

**3**

votes

**1**answer

82 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 ...

**1**

vote

**1**answer

71 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 ...

**15**

votes

**8**answers

910 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 ...

**1**

vote

**0**answers

34 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

61 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 ...

**0**

votes

**0**answers

34 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 ...

**8**

votes

**3**answers

283 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

90 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) = ...

**2**

votes

**1**answer

93 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 ...

**0**

votes

**1**answer

50 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

52 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 ...

**2**

votes

**1**answer

163 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 $f$ ...

**0**

votes

**2**answers

99 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 ...

**5**

votes

**1**answer

168 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 ...

**1**

vote

**1**answer

67 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 ...

**0**

votes

**0**answers

41 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 ...

**3**

votes

**1**answer

108 views

### A difficult combinatorial optimization problem

Let $\mathcal{J}$ be a closed, bounded, compact, convex set in $\mathbb{R}^L$.
(Notations: vector $\mathbf{x}$ is denoted in bold letters and its $i^{th}$ co-ordinate is denoted as $x_i$. ...

**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
...

**0**

votes

**0**answers

75 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 ...

**7**

votes

**2**answers

200 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

52 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

97 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

45 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

55 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

156 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

270 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

82 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 ...

**1**

vote

**2**answers

124 views

### investigating positivity/negativity of a function [closed]

I'm investigating if and how the positivity or negativity of a multivariable function can be proved. Consider $y_{1},y_{2},y_{3}\in\mathbb{R}$ and the following function
...

**7**

votes

**3**answers

340 views

### The distribution of the shortest path through $n$ points

In the big picture, I'd like to know: if I sample $n$ points uniformly at random in the unit square, what is the probability that the shortest path that visits each one of them is very small?
More ...

**1**

vote

**2**answers

125 views

### The set of matrices with same spectral radius

I am working on an optimization problem over the set of positive matrices (that is, matrices where all entries are positive numbers) that have the same spectral radius. My main problem is how to ...

**2**

votes

**1**answer

174 views

### Multiplicative gradient descent?

The normal gradient descent is additive: $w_{t+1}=w_t-\lambda_t\nabla f(w_t)$, but is there a multiplicative gradient descent that looks something like $w_{t+1}=w_t[-\lambda_t\nabla f(w_t)]$?
I know ...