Optimization with convex constraints and convex objectives; notions related to convex optimization such as sub-gradients, normal cones, separating hyperplanes

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59 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
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1answer
135 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
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2answers
164 views

Constructing a quasiconvex function [closed]

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, \...
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1answer
61 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 {f_2}\...
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0answers
148 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 ...
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0answers
96 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
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1answer
100 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 ...
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74 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
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0answers
68 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 $x_1,x_2\...
4
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1answer
103 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 ...
4
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1answer
195 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
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0answers
178 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
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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 \...
2
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0answers
95 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
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3answers
323 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
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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) = ...
1
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1answer
104 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 \mathbb{R}^{...
0
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1answer
93 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 ...
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0answers
129 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
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0answers
139 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
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2answers
161 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
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1answer
148 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
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1answer
462 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 ...
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0answers
180 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
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1answer
72 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 $$\sum_{i=1}^{k}\left[-4a_{i}^{3}+\left(3n-...
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8answers
2k 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
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1answer
185 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 $0<q_{j}...
2
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1answer
180 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 $f$ ...
8
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2answers
280 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 ...
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0answers
142 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
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0answers
118 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 $$g(\mathbf{y}):=\max_{\mathbf{x}\in\...
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0answers
50 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 \mathcal{...
-3
votes
1answer
91 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
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0answers
212 views

Reference request: functional analysis results used in Taubes paper (1980)

I am studying Taubes paper 'Arbitrary N-vortex solutions to the first order Ginzburg-Landau equations'. I am looking for a reference of three following theorems: Let $f(x)$ be a convex funtional ...
5
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2answers
846 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
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2answers
351 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
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1answer
119 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 ...
9
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4answers
541 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 ...
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2answers
129 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 $$f\left(y_{1},y_{2},y_{3}\...
3
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1answer
136 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 $x_i$. $\mid\...
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2answers
150 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 ...
5
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2answers
426 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 ...
3
votes
1answer
194 views

Linear dependency of real numbers with integer coefficients adding up to zero [closed]

Let $x = (x_1, \dots, x_n)$ be a vector of real number. I was asking myself if there was an efficient way of telling whether there exists a non-zero vector of integers $z \in \mathbb Z$ such that both ...
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0answers
96 views

Convex Optimization related problem

Suppose two non-negative convex functions $f$ and $g$ be given. We want to solve the following optimization $$\max_{g\leq\epsilon}f.$$ Now suppose that both $f$ and $g$ can be upper-bounded by a ...
2
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0answers
339 views

On increasing the penalty term in convex optimization with regularization

Given the two strictly convex (unique solution) optimization problems as: $$Problem\:1:\min_{X} f(X)+\|X\|_{F}^2 \hspace{2cm}Problem \:2:\min_{X}f(X)+n\|X\|_2^2$$ where $X\in\mathbf{S}_{++}^{n}$ (...
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0answers
58 views

Reference request: Edmond's Algorithm for integer hull

I'm looking for a good reference for the algorithm (supposedly by Edmonds) to compute the integer hull of a polytope, not by cutting plane methods but by starting with a set of integer points and then ...
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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 ...
2
votes
1answer
101 views

Is first term of my cost function convex?

I have an optimization problem in the form of [\begin{array}{l} \mathop {{\rm{Minimize}}}\limits_{\bf{X}} \,\,\,2\left| \delta \right|\sqrt {{\rm{Tr}}\left( {{\bf{A}}{{\bf{X}}^2}} \right)} {\rm{ - ...
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0answers
66 views

Finding gradient of an optimization

I am trying to find the gradient of the following optimization problem and then add to objective, but I got some trouble in computing. Could you please help me? Assume that we have an optimization ...
1
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0answers
57 views

Characterization of the optimal solution in relative entropy minimization

The following optimization problem is related to relative entropy and to the limit of the iterative proportional fitting procedure. For $1 \leq i,j \leq n$ and fixed $w_{ij} \geq 0$, and fixed $a_i, ...