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

learn more… | top users | synonyms

2
votes
1answer
455 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
0answers
49 views

Bounding the off-diagonal entries of a hermitian matrix [on hold]

I asked this question on stackexchange and didnt receive any answers, am therefore posting it here. The Pauli matrices are $I = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$, $X = ...
0
votes
1answer
266 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
0answers
36 views

How to solve this nonlinear optimization problem?

I have a nonlinear optimization problem with linear constraints. How to solve this? $\sigma_i$ and $\rho_i$ are the optimization variables. ...
1
vote
1answer
444 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 ...
1
vote
1answer
90 views

A bound on the number of bilinear functions needed in order to obtain the minmax

For $n\in\mathbb N$, let $\Delta(n)=\{x\in\mathbb R^n:x_i\geq 0, \sum_ix_i=1\}$ be the set of probability vectors in $\mathbb R^n$. Is there a function $m:\mathbb N\to\mathbb N$ such that for any ...
-1
votes
0answers
16 views

stochastic dual coordinate ascent VS [closed]

I am using vl_feat library to train an SVM. The function I am using (vl_svmtrain http://www.vlfeat.org/matlab/vl_svmtrain.html) uses stochastic dual coordinate ascent to minimize the HINGE LOSS as I ...
0
votes
0answers
35 views

Limit of argmin of sum

Suppose that I know $f_n\rightarrow f$ and $g_n\rightarrow g$ are both continuous maps from a Complete Riemmanian Manifold $X$ to $\mathbb{R}$ which converge pointwise almost everywhere. Then is it ...
4
votes
2answers
70 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 ...
2
votes
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 ...
0
votes
0answers
35 views

Separation on discrete set

Consider the set $L = \prod_{i=1}^n\{1,0\}$, i.e. L consists of the element of n-tuples whose entries are 0 or 1. Also we can regard $L$ as a subset of $R^n$. Define linear functions $f(x)= a_1x_1+ ...
1
vote
0answers
27 views

Optimizing sum of approximate and exact functions

This is a research question that I had asked in Math.SE about a month ago, but even after putting a bounty on it, I did not get any answers. I have two real values functions, where one ...
0
votes
0answers
18 views

A possible condition on a set of vectors to be uniquely determined by their gramian?

Suppose you are given a gramian matrix $G=W^TW\in R^{k\times k}$, $W\in R^{d\times k},\;d>k$. The given gramian $G$ determines, up to a unitary transformation, the columns of $W$, i.e. an ...
4
votes
0answers
77 views

Designing Character Other Than 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 ...
0
votes
0answers
14 views

Finding orthogonal basis with constraint

Is there any fast algorithm that output an orthogonal basis $e_i,i\leq n$ of $R^n$ with $e_i\in V_i$? Where $V_i,i\leq n$ are given linear subspaces of $R^n$. And is there any condition on $V_i,i\leq ...
0
votes
0answers
52 views

Unsupervised classification [migrated]

In a typical supervised learning problem, one observe $(X,Y)$ where $Y$ is a categorical variable. We confront the such a problem that $Y$ is hidden and instead $(X,Z)$ is observed where both ...
2
votes
1answer
271 views

Is there any way to compare between diagonals of a resolvent and a Cauchy transform?

Say $A$ is a symmetric matrix of $n$ dimensions. Then let the ``resolvent" of $A$ be the matrix valued function $R_A(z) = \frac{1}{z-A}$ and its Cauchy transform be the real valued function $C_A(z) = ...
0
votes
1answer
120 views

sum of fractional functions optimization problem

Consider the following sum of fractional functions optimization problem $$ \begin{array}{l} \mathop {\min }\limits_{\bf{x}} \,\,\,\sum\limits_{i = 1}^p {\frac{1}{{{\bf{a}}_i^T{\bf{x}} + {b_i}}}} \\ ...
0
votes
0answers
22 views

Dennis More' Superlinear Convergence_refrences request

Why in the proof of superlinear convergence of restricted broyden class (for the unconstrained optimization) we need the bounded deterioration condition for the approximation of all the true hessian ...
1
vote
0answers
27 views

Which algorithm is most efficient for a specific QP problem

I have a QP problem of the following kind: $\min_{\alpha\in\mathbb{R}^n}\frac{1}{2}\alpha^T M \alpha - p^T\alpha$ s.t. $l\leq \alpha \leq u$ The matrix $M$ is symmetric and positive definite and of ...
5
votes
2answers
395 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 ...
1
vote
0answers
18 views

Concavity of maxima [closed]

Suppose we have the following optimization problem : $\min\limits_x kf(x) + g(x)$ where $f$ is a decreasing convex function in $x$ and $g$ is an increasing convex function. Can we say that $x^*$ is ...
4
votes
0answers
132 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 & ...
1
vote
0answers
32 views

Minimum of sum of modulus [closed]

How can I evaluate the minimum of $$ \left|7x-1\right|+\left|7y-5\right|+\left|7z-1\right| $$ if $x,y,z$ are non negative reals such that $ x+y+z=1$ and $y^2 \le xz$? Is there a standard way to ...
3
votes
1answer
112 views

Global minimum of nonlinear least square

We have a continuous and differentiable function $f(\cdot)$ that maps from $R^n$ to $R^n$. We are trying to solve a nonlinear least square problem: Minimize $J(x)=\Vert f(x)-z\Vert^2$ subject to box ...
9
votes
4answers
538 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 ...
0
votes
1answer
100 views

Is the linear production game a convex game?

In cooperative game theory, the linear production game (LPG) is defined by letting the characteristic function have the form of a linear programming problem. Does anyone know if the LPG is a convex ...
1
vote
1answer
43 views

On optimizing a function whose projection and projected vector go through a linear transformation

Assume the two sets of vectors $\{\mathbf{a}_1,\ldots,\mathbf{a}_N\}$ and $\{\mathbf{b}_1,\ldots,\mathbf{b}_N\}$ of equal length. My goal is to find the optimum matrix $\mathbf{C}$ to the following ...
0
votes
0answers
36 views

Bounds on the curvature of a sequence of convex functions

Let $\{f_n\}$ be a sequence of (real-valued) smooth convex functions on $[0,1]$, with $f_n(0) = f_n(1) = 0$ for all $n$. Let $t_n \in [0,1]$ be the minimizer of $f_n$ and assume that $M_n:= f_n(t_n) ...
-1
votes
1answer
51 views

Does element-wise concavity guarantee joint concavity?

I have a function of two variables, and I have checked that along one direction (fixing another variable), it is a monotonically increasing and concave function. Whereas in another direction (fixing ...
2
votes
2answers
185 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 V\ ...
10
votes
2answers
284 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 ...
1
vote
0answers
55 views

Log convexity for the norm of a vector-valued function

Log convexity of various functions defined on the space of Hermitian matrices plays an important role in matrix analysis and probability theory. Given $v \in \mathbb{C}^n$, $D$ a diagonal matrix with ...
0
votes
0answers
22 views

Maximizing modular function subject to supermodular constaint

I'm trying to solve a constrained optimization problem with submodular functions and get some nice properties of the solution. Unfortunately, I think I am in a setting where Topkis' theorem does not ...
0
votes
0answers
20 views

Bounding the error of the optimal solution from an approximated objective

Let $f$ be a smooth convex function defined in a bounded region $X$ and its Hessian is bounded $m I \le |\frac{\partial^2 f(x)}{\partial x^2}| \le M I$ for some $M>m>0$. Let $x^*=argmin_{x\in X} ...
0
votes
0answers
45 views

Find optimal value for a regularization parameter in generalized eigenvalue problem

Consider the generalized eigenvalue problem : $ \Sigma_{XY} \Sigma_{YX} {W} = \lambda \Sigma_{XX} {W} $ where $\Sigma_{XX} $ and $\Sigma_{XY}$ are sample covariance matrices are of the matrices ...
0
votes
0answers
42 views

Nuclear norm maximization

I am trying to solve a nuclear norm maximization problem: $$\arg \max_{Q \in O(n)} \|WQV^T\|_*$$ where $Q$ is an $n \times n$ orthogonal matrix and $W$ and $V$ are real $d \times n$ matrices. I've ...
2
votes
1answer
249 views

Is this a log-concave function?

Let $(a_k)$ be a log-concave positive decreasing sequence. Is $\sum\limits_{k=1}^n a_k(1-e^x)^{k-1}$ log-concave in $x<0$, for each natural $n$?
0
votes
0answers
46 views

Question on solving an optimization problem using Variable splitting and ADMM

Tell me if I have found the right approach to the following optimization problem: $$ min_{x} \frac{1}{2}\left \| Ax-b \right \|_2^2 \\ s.t. \ \ \Phi v=x \ , \ {x^T(1-x)}=0 $$ $A$ and $\Phi$ ...
1
vote
1answer
73 views

Envelope theorem for second derivative

I am maximizing a function $f(x,z)$ on $x$ ($z$ is treated a parameter in the maximization). The function $f$ is strictly concave on both variables. I know how to use the envelope theorem for the ...
1
vote
0answers
35 views

Is this QCQP convex or nonconvex

\begin{equation} \begin{split} \min_{x\in \mathbb{R}^n}\:f(x)=(1/2)x^{T}Q_0x+c_0^T x \end{split} \end{equation} s.t. $$ g_i(x)=\frac{1}{2}x^T Q_ix-lmax_i\leq0,i\in\{1,...,m/2\} $$ $$ ...
4
votes
1answer
148 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 + ...
1
vote
0answers
41 views

Projecting on a convex compact polytope with special form

Let $E$ be a large sparse $l$-by-$n$ matrix ($l$ and $n$ can be in the billions...) with coefficients in $\{-1, 0, 1\}$: the first row of $E$ is the vector $(1,0,0,\ldots,0) \in \mathbb R^n$, and ...
1
vote
0answers
72 views

Projecting on a a special polyhedron

Let $X$ be an $n$-by-$p$ matrix and consider the closed convex polyhedron $$\mathcal P_X := \{y \in \mathbb R^n | \|X^Ty\|_\infty \le 1\}.$$ Notice that $\mathcal P_X$ is symmetric about the origin. ...
5
votes
1answer
267 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 ...
11
votes
1answer
311 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 ...
1
vote
0answers
36 views

convergence of unconstrained convex optimization

I encounter an optimization problem. The simplified version is like following: Denote function $F(x):\mathbf{R}^n\rightarrow\mathbf{R}$, where $F(x)$ is a smooth lower bounded convex function (i.e. ...
1
vote
2answers
63 views

Sensitivity analysis in minimum norm problems under a linear constraint

Suppose $\Delta$ is some nice topological space, say compact, and Hausdorff. Let $A:\Delta \rightarrow \mathbb{R}^{m\times n}$ be a continuous $m\times n$ matrix valued map. Let $b\in \mathbb{R}^{m}$ ...
1
vote
0answers
46 views

Derivatives of Minkowski function?

Let $A\subset \mathbb R^n$ and $M$ be the convex hull of the set $A$, e.g., $M:=Conv(A)$. The Minkowski function on $M$ is defined as follows \begin{align*} &f: \mathbb R^n \to \mathbb R\\ ...
0
votes
0answers
60 views

Well-posedness of gradient flows

For a convex lower-semicontinuous functional on a Hilbert space $I\colon H\rightarrow\mathbb{R}$, it is shown in Evans' PDE that the Hilbert-space-valued ODE $$\begin{cases}\mathbf{u}'(t)\in-\partial ...