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

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24
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2answers
979 views

An Interesting Optimization Problem

You are given n non-negative integers $a_1, a_2 ,, a_n$. In a single operation, you take any two integers out of these integers and replace them with a new integer having value equal to difference ...
17
votes
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 ...
11
votes
1answer
304 views

Minimize sum of $\ell_2$ norm and linear combination, on simplex

Let $\Delta_n := \{x \in \mathbb{R}^n | x \ge 0, \sum_{1 \le i \le n}x_i = 1\}$ be the $n$-simplex. For $a, b \in \mathbb R^n$, with $\Delta_n \not \ni a$, consider the problem of computing the ...
10
votes
2answers
280 views

More general than semidefinite program?

I was TAing my convex optimization class and explaining that Linear Programs are a special case of Second Order Cone Programs, which are themselves special cases of Semidefinite Programs. My question ...
9
votes
3answers
831 views

Is group theory useful in any way to optimization?

For what I have seen, optimization uses a lot of linear algebra and convex analysis, but I have not seen any group theory being used, so I was curious about it. Is group theory useful in any way to ...
9
votes
4answers
530 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 ...
8
votes
3answers
316 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 ...
8
votes
2answers
269 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 ...
7
votes
2answers
242 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 ...
6
votes
2answers
215 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 ...
6
votes
0answers
117 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
159 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 ...
5
votes
1answer
297 views

What it is the volume of the unit ball section of the cone of positive definite matrices?

Let $PD_{n}$ be the cone of positive definite $n \times n$ real matrices and let $B$ be the unit sphere in $n \times n$ dimensions. What is the volume of $PD_{n} \cap B$? EDIT: Let's assume that $B$ ...
5
votes
2answers
326 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 ...
5
votes
2answers
389 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 ...
5
votes
2answers
110 views

convert a special case of nonlinear fractional programming into a convex problem

Is it possible to convert a fractional problem (maximization) with objective function equal to the ratio of a concave function and convex function ? This question sound impossible but I have read this ...
5
votes
2answers
726 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\\ && ...
5
votes
1answer
155 views

Reference request: Continuity of unique maximizer of linear functional on convex set

Does anyone know reference for a theorem of the following sort: Proposition: Let $K \subset\mathbb {R}^n$ be a compact convex set, and assume that $$f(w):=\operatorname{argmax}_{x\in K}w(x) $$ is ...
5
votes
1answer
99 views

How to prove the existence of the polytope in $\mathbb{R}^d$ with a given number of faces, minimizing the isoperimetric ratio?

This is the isoperimetric type question. We know that in $\mathbb{R}^d$, balls are the sets that minimize the isoperimetric ratio $\frac{S^{d}}{V^{d-1}}$, where $S$ is the surface area and $V$ is the ...
5
votes
1answer
404 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 ...
5
votes
1answer
199 views

optimization problem, any solution?

The objective is as follows: $\min_{\mathbf{F}} a Tr(\mathbf{F} \mathbf{F}^H) - Re\{\mathbf{b}\mathbf{F}^H \mathbf{C} \mathbf{F} \mathbf{d}\}$ $s.t.\ \ \ Tr(\Sigma \mathbf{F} \mathbf{F}^H)<p$ ...
5
votes
1answer
211 views

Minimize Frobenius norm

My question is the following: Suppose $M$ is an $n \times n$ symmetric real matrix. I want to find an $n \times n$ symmetric real matrix X such that $|| X -M||_F$ is minimized with the constraint ...
5
votes
0answers
71 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) = ...
5
votes
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 ...
5
votes
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 ...
4
votes
2answers
340 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 ...
4
votes
2answers
104 views

Convex optimization with full subdifferential information

Can anyone direct me to any algorithms or theorems that describe the difficulty of solving a non-smooth convex optimization problem for the special case where the full subdifferential is available? ...
4
votes
2answers
300 views

Minimax theorem on a non convex domain

A minimax theorem is a theorem which states that under certain conditions on $\mathcal{X}$, $\mathcal{Y}$ and $f$: $$ \inf_{x \in \mathcal{X}}{\sup_{y \in \mathcal{Y}}{f(x,y)}} = \sup_{y \in ...
4
votes
2answers
91 views

Questions concerning convergence rate of Iterated Projections

Assume for simplicity $C_1,C_2,...,C_n\subset \mathbb{R}^m$ to be closed and convex subsets with $\underset{i=1}{\overset{n}{\bigcap{}}}C_i\neq\emptyset$. Let $x^0\in\mathbb{R}^n$ and define the ...
4
votes
1answer
175 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 ...
4
votes
1answer
63 views

About optimization with Renyi divergence

Can someone link me to some pedagogic example of computing the Renyi divergence between two discrete/continuous distributions? Like examples where someone has been able to obtain a neat closed form ...
4
votes
1answer
71 views

Self-concordant function for dual cone

I wonder if there is any existing result for self-concordant function in the literature about the following question. Suppose $f$ is a self-concordant barrier function of a proper cone $K$ (pointed, ...
4
votes
1answer
144 views

Nonlinear least square with quadratic equality constraint

I am looking for an appropriate method or hint to solve the following constrained nonlinear least square problem: $\operatorname{argmin}_X \sum_{i\in I} \|\mathbf{X}_i - \mathbf{X}_{i+1}\|_2^2 + ...
4
votes
1answer
99 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
votes
2answers
211 views

A certain type of constrained Rayleigh-Ritz ratio

Let $\mathbf{A_1}$ and $\mathbf{A_2}$ be two hermitian matrices. Consider the problem \begin{align} \max_{\mathbf{u}^H\mathbf{u}=1}~\mathbf{u}^H\mathbf{A}_1\mathbf{u} \\\ ...
4
votes
1answer
142 views

Constrained optimization (QCLP) over $x$ with the constraint $x = Az$

I have a problem that looks very much like a (norm-constrained) linear program, but with an extra constraint that is unusual for me. The problem is, given a matrix $A$ and a vector $w$, $$ \min_{x ...
4
votes
2answers
524 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 ...
4
votes
0answers
62 views

Designing Better “Temperature” for Simulated Annealing on Combinatorial Optimization

Many research on designing temperature for simulated annealing is carried out. We wonder if there is any research on designing general feature of the Hamiltonian used in Simulated Annealing. For ...
4
votes
0answers
129 views

Closed-form solution of a linear programming question

Among all the probability matrices \begin{equation*} P = \left(\begin{array}{cccc} p_{00} & p_{01} & \ldots & p_{0,J-1} \\ p_{10} & p_{11} & \ldots & p_{1,J-1} \\ \vdots & ...
4
votes
0answers
45 views

How does one go from convexity to submodularity?

If I have a function which is convex in the hypercube, $[-1,1]^n$ then when would it imply that its restriction to $\{-1,1\}^n$ is submodular? It would be helpful is someone can share some specific ...
4
votes
0answers
104 views

An inequality from the “Interlacing-1” paper

This question is in reference to this paper, http://annals.math.princeton.edu/wp-content/uploads/annals-v182-n1-p07-p.pdf (or its arxiv version, http://arxiv.org/abs/1304.4132) For the argument to ...
4
votes
0answers
136 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 ...
4
votes
0answers
191 views

When is the sum of a weak-$*$ closed convex cone and a subspace also weak-$*$ closed?

Let $X$ be a Banach space. Suppose $C \subset X^*$ is a convex cone and $V \subset X^*$ is a subspace, and suppose both $C$ and $V$ are closed in the weak-$*$ topology. Are there any general ...
4
votes
0answers
413 views

An intuition for three different types of subgradients (proximal, regular, limiting)

I'm having a bit of difficulty getting my head around the different types of subgradients we're currently covering in a nonsmooth optimisation class I'm taking. These subgradients are (assume $x \in$ ...
3
votes
1answer
197 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 ...
3
votes
1answer
127 views

Computational complexity of low rank SDP

Suppose we are given a general SDP of the form with an additinal rank requirement \begin{array}{rl} {\displaystyle\min_{X \in \mathbb{S}^n}} & \langle C, X \rangle_{\mathbb{S}^n} \\ \text{subject ...
3
votes
2answers
86 views

Is the prox-residual monotone?

$\newcommand{\scp}[2]{\langle #1,#2\rangle}\newcommand{\id}{\mathrm{Id}}$ Let $f$ and $g$ be two proper, convex and lower semi-continuous functions (on a Hilbert space $X$ or $X=\mathbb{R}^n$) and let ...
3
votes
1answer
548 views

SDP formulation of noisy low rank matrix completion

Exact low rank matrix completion using nuclear norm minimization can be formulated as a semidefinite program (SDP). Following the notation in the paper, a convex problem for noisy matrix completion ...
3
votes
1answer
302 views

quadratic programming on hypercube

I want to maximize a quadratic form $\mathbf x^T\mathbf Q\mathbf x$ and also want to find out which vector $\mathbf x$ maximizes the quadratic form when $\mathbf Q$ is an $n\times n$ positive ...
3
votes
1answer
104 views

Finding a semigroup that maximizes the trace of a sum of matrices

Let $H$ be a finite semigroup containing $n$ elements from a compact group $G$. I am trying to solve $$\max_{h_i,\ h_j\ \in\ H} \operatorname{tr} \sum_{i,\ j\ \leq\ n} \rho(h_i^{-1} h_j)(A_j ...