Convex Optimization is a special case of mathematical optimization. It includes Linear Programming and least-squares.

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What's the difference between this proximal method and subgradient projection?

http://stanford.edu/~boyd/papers/pdf/prox_algs.pdf In the link above it is proposed that the nonsmooth separable resource allocation problem $$\min \sum f_i(x_i) \ \ \text{s.t.} \ \ \textbf{1}^Tx = ...
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20 views

Subgradient of the optimal value of a linear program with respect to its parameters

Consider the linear program $f(c)=\min\{c'x\mid x\in\mathbb{R}^n,Ax=b,x\geq0\}$. Are there any results on what is $\partial f/\partial b$?
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21 views

Is $K =\{ S: \exists \text{ positive diagonal} D, D^TSD \;\text{diagonally dominant}\}$ convex?

I am doing some convex cone optimization and wonder whether the following set $K_1$ is convex or not. Assume the following matrices are all in $\mathbb{R}^{n\times n}$ and symmetric. Let the set of ...
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31 views

Convex envelopes of bivariate functions

In order to convexify my nonlinear non-convex program I need convex envelopes for the function $(x/y)^2$, both x,y are positive. I am only aware of the convex envelopes of the type $xy$ from here ...
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35 views

how to find the edges emanating from a given vertex in a polyhedron

Suppose my polyhedron $P$ is defined as $P={ x\in \mathbb{R}^n \mid Ax=b, x\geq0 }$ I have $x_0$, which is a vertex of $P$. How to find the edges emanating from $x_0$? In other words, I want to find ...
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1answer
57 views

Is the expectation of log-concave function still log-concave?

I know the expectation preserves the concavity (or convexity), but I was wondering is it still true that the expectation of log-concave function still log-concave; to be more precise, Let ...
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1answer
39 views

The Legendre-Fenchel transform of $BV$ semi-norm

I am reading a numerical paper in which it calls some "easy" facts from convex analysis but I can't justify it... Let $\Omega\subset \mathbb R^2$ be open. Define for a function $u\in L^1(\Omega)$ and ...
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9 views

Gomory's cut typical running time until the constraint is fractional

I was considering the following problem. Say we are given an linear programming problem $$ \max c^Tx $$ $$ Ax \le b $$ $$ x \ge 0$$ Where instead I consider $i^{th}$ the optimal solution $X_i$ of ...
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1answer
31 views

How do I prove that objective function is not convex

Here is my objective function. \begin{equation} \begin{array}{c} \underset{\mathbf{x},\mathbf{y}}{\text{minimize}}\hspace{4mm} \big(\lambda_1\left( \mathbf{y}^T V^{(1)}\mathbf{x} \right)^2 + ...
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17 views

Convex signal reconstruction for convex generator function?

Let $f : \mathbb{R} \mapsto \mathbb{R}$ be a convex (not affine) function and suppose that $y = f(x)$. We want to reconstruct the input $x$ for a given $y$, and the standard approach would be to ...
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73 views

proving that $\text{aff}C-\text{aff}C\subset\text{aff}\,(C-C)$

In proof of Theorem 6.4.1 of Auslender's book about asymptotic cones, the author assumes that $\text{rge}\,A\subset\text{aff}\,C$ and for $\epsilon>0$ claims that ...
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2answers
147 views

Rank reduction to satisfy Barvinok's upper bound & Rank of a set notation

After reading n times the four first sections of the 4th Chapter of J.Dattorro's book (Convex Optimization & Euclidean Distance Geometry). I am confused between yes or no, every extreme point of ...
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0answers
17 views

Rewriting a strictly convex quadratic equality constraint

I have the following constraint $$ x^TAx+b^Tx=c $$ where $A$ is a positive definite matrix. Is there any way to take advantage of the strict convexity of this expression to reformulate the constraint ...
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0answers
13 views

Find the nearest point of $y$ from a probability simplex. [duplicate]

I need to compute the nearest point of $x$ from a probability simplex. Formally, I want to ask if there is a close form for the solution to the following optimization problem: for $y\in \mathbb{R}^k$, ...
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1answer
26 views

How do I show that the set $S$ in $\mathbb R^3$ defined by linear inequalities is a $3$-simplex?

Consider the set $S$ in $\mathbb R^3$ defined by the inequalities: $x+y+z \ge 1$ $-x+y+z \le 1$ $x-y+z \le 1$ $x+y-z \le 1$ How can I show that $S$ is a $3$-simplex ? (Convex hull of $3 + 1$ ...
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1answer
28 views

Functional Lifting in Optimisation - Reference Request

I'm looking to learn about the use of (functional) lifting applied to a non-convex optimisation problem to give a (larger) convex problem. Unfortunately, I'm having a great deal of trouble finding ...
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49 views

Minimizing the max function

Suppose we have the single-variable function $$f(x) = \max_k \{f_k(x)\}$$ where each $f_k$ is convex and smooth (and known beforehand). We want to minimize it over some bounded interval. We can, in ...
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17 views

Find $U \subset D $ with $|U|$ minimal s.t $v = (1/2,1/2,1/2)$ belongs to the convex hull of $U$.

Consider the convex hull $\text {conv} \{e_1,e_2,e_3,(1,1/2,1/2),(1/2,1,1/2), (1/2,1/2,1)\}$ in $\mathbb R^3$ and the vector $v = (1/2,1/2,1/2)$. I want to compute $U \subset ...
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34 views

strengths and weaknesses of analytical method

I was wondering if anyone could suggest any books or paper that explain/discuss the advantages and drawbacks of analytical methods for optimization. Also, if we have a convex objective function ...
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2answers
122 views

Prove a closed ball in $\mathbb R^3$ has an infinite number of extreme points.

How do I show that a closed ball in $\mathbb R^3$ has an infinite number of extreme points ? (Closed ball is written as $S = \{(x,y,z) \in \mathbb R^3 | \sqrt {x^2 + y^2 + z^2} \le R \}$) I know ...
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24 views

How to define the nuclear norm of a tensor

As we know,the nuclear norm of a matrix $X$ is defined as this: $$||X||_{*}=\sum{\sigma_{i}}$$ where $\sigma_{i}$ is the singular value of $X$. But how to define the nuclear norm of a tensor ...
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1answer
48 views

Let $S = \{ (x,y,z) \in \mathbb R^3 : x^2 + y^2 + z^2 \le 1\}$. Prove $(1-\lambda)x + \lambda y \in S$ for $x=\lambda'y$, $\lambda' < 0$.

Let $S = \{ (x,y,z) \in \mathbb R^3 : x^2 + y^2 + z^2 \le 1\}$. I've verified that $x,y \in S$ implies $(1-\lambda)x + \lambda y \in S$ when $x,y$ are linearly independent using Pythagoras and when ...
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1answer
18 views

Compute the extreme points of a polyhedron $P$ and write $(1/2,1/2,1/2)$ as a linear combination of these.

Compute the extreme points of a polyhedron $P$ and write $(1/2,1/2,1/2)$ as a linear combination of these. I want to compute all the extreme points of the set $P$ (polyhedron) in $\mathbb R^3$ ...
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2answers
46 views

Can $\text{conv} \{ e_1,e_2,e_3, (1/2, 1/2, 1) , (1/2, 1, 1/2) , (1, 1/2, 1/2) \}$ be reduced to a convex hull of a subset of these?

How do I see whether $$\text{conv} \{ e_1,e_2,e_3, (1/2, 1/2, 1) , (1/2, 1, 1/2) , (1, 1/2, 1/2) \}$$ can be reduced to a convex hull of a subset of these vectors? That is, if $D = \{ e_1,e_2,e_3, ...
1
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1answer
38 views

Simple question on maximizing a family of concave functions

Given $x \in [0,1]$, let $f_k(x)$ be concave function in $x$ for all $k= 0,1,...,N$. Is the following two maximization formulations equivalent? $$ \max_{0 \le k \le N} f_k(x) =?= \max \{f_0(x), ...
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5answers
87 views

Let $B=\{ (x,y,z) \in \mathbb R^3 : x^2 + y^2 + z^2 \le 1\}$. Prove existence of a unique point $v_0 \in B$ that is closest to $v \notin B$?

Suppose $B=\{ (x,y,z) \in \mathbb R^3 : x^2 + y^2 + z^2 \le 1\}$. Consider a point $v \notin B$. How do I prove that there exist a unique point $v_0 \in B$ that is closest to $v$. Also, how do I ...
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1answer
30 views

How to introduce auxiliary variables to make the objective function separable?

$$\min_{X}\|X_{(1)}\|_{*}+\|X_{(2)}\|_{*}+\|X_{(3)}\|_{*}+\lambda\|Ax-b\|_2^2$$ where $X$ is a three order tensor, $X_{(i)}$ is a matrix whose column are the mode-$i$ fibers of $X$(i=1,2,3),$x$ is ...
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1answer
33 views

MLE of an increasing nonegative signal from convex optimization book

I need help solving this problem from Stephen Boyd's Convex Optimization additional exercise. Its question 6.6 from additional exercise. Maximum likelihood estimation of an increasing nonnegative ...
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2answers
352 views

What numerical methods are known to solve $L_1$ regularized quadratic programming problems?

What numerical methods are suitable to solve the following problem $$\min_x \tfrac{1}{2}x^T A x + b^Tx + \lambda ||x||_1$$ where $x,b\in\mathbf{R}^n$, and $A\in \mathbf{R}^{n\times n}$ is positive ...
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1answer
28 views

what does separable convex program mean?

In literature,a separable program is formulated like this: $$\min_{x_{1},...x_{n}}\sum_{i=1}^{n}f_{i}(x_{i})$$ where $f_{i}$ is a closed proper convex function. My question is what does 'closed' ...
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0answers
41 views

Can I perform Maximum likelihood via optimization?

I have two $3 \times 3$ matrices $\mathbf{a}$ and $\mathbf{f}$. $\mathbf {f}$ is completely known to me. Also $a_{ij} \in [+1,-1]$ \begin{equation} \mathbf{f} = \left( \begin{array}{ccc} f_{11} ...
0
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1answer
42 views

Iterative method to compute only the positive eigenvalue's and corresponding eignevectors of a very large matrix?

I have a very large dense matrix (~10000 X ~10000) which is not full rank . I want to compute only the positive eigenvalues and corresponding eigenvectors instead of computing all of them. I have ...
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1answer
19 views

Can a Convex QCQP Problem with an additional linear constraint be converted to a SOCP?

I have a quadratically constrained quadratic programming problem that I massaged into the form $$ \begin{aligned} & \underset{x}{\text{minimize}} & & x^T Q x \\ & \text{subject to} ...
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1answer
23 views

Optimization: maximizing nonconvex sum of product of constraints

I'm wondering if there is any way to convexify, approximate, and/or simplify the following problem. $\max. \sum_{k \in K} \prod_{i \in I} (a_{ik} x_{ik} + b_{ik})$ s.t. $x_{ik} \in [0,1]$ where ...
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2answers
37 views

Derivative of a function of trace

Suppose $X$ is a diagonal matrix, $X \in \mathbb{R}^{m \times m}$. Let $f\colon\mathbb{R} \to \mathbb{R}$ be a twice differentiable function. Find the following $$\nabla^2_X f(tr(X))$$ where ...
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0answers
11 views

How to generate feasible $H$-conjugate descent search directions in convex subset

If we want to minimize a quadratic function $f(x)=c^Tx+\frac12x^THx$ (where $H$ is a symmetric positive-semidefinite matrix) in a convex subset $C\subset\mathbb{R}^n$, then is it possible to generate ...
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1answer
29 views

chain rule with scalars, vectors, and matrices

Consider two differentiable functions, $f : \mathbb{R}^{n \times n} \to \mathbb{R}$ and $g : \mathbb{R}^2 \to \mathbb{R}^{n \times n}.$ In general, for some $x \in \mathbb{R}^2$, what is the gradient ...
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Is there any software to solve this large scale convex optimization problem?

I want to solve the following large scale convex problem: $min\ \ ||A$u-b$||_2^{2}+ ||$U$_{(1)}||_*+||$U$_{(2)}||_*+||$U$_{(3)}||_*$ where U is a three order tensor, U$_{(i)}$ is a matrix whose ...
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0answers
29 views

Optimising a piecewise convex function

I have a function $f(x)$ defined for $x\in \mathbb{R}^+$ that is decreasing, piecewise convex and continuous. The 'pieces' of $f$ are exponential and the rate of decrease for each piece reduces as we ...
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1answer
37 views

Homework on matrix and convex set [closed]

Suppose that $A,B\in\mathbb{R}^{n\times n}$ and both symmetric. Define $$ H=\{\sigma\in\mathbb{R} \mid A+\sigma B \text{ is semi-positive definite}\} $$ Assume that there exist ...
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2answers
30 views

Optimization problem: smallest euclidean distance with positive entries constraints

Suppose there is the simple function: \begin{align} f(x,y,z) &= (x-a)^2 + (y-b)^2 + (z-c)^2 + (x+y-S-z - d)^2 \end{align} where $a,b,c,d$ are nonnegative constants, and $S$ is an integer. I ...
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1answer
28 views

a quadratic function with a solution

I am studying Convex Optimization and my book says that if I have the function $y=x^TAx+2b^Tx$ and the solution $x^*=-A^\dagger b$, then $y$ can be reduced to $y^*=-b^TA^\dagger b$ $\quad$ ($\dagger$ ...
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18 views

Properties of a proper cone

Let $K$ be a proper cone. I need to prove following properties: if $x \preceq_K y$ and $u \preceq_K v$, then $x+u \preceq_K y+v$ if $x \preceq_K y$ and $y \preceq_K z$, then $x \preceq_K z$ if $x ...
0
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1answer
91 views

Convex optimization approximation

Consider the optimization problem $\mathcal{P}_0$ \begin{equation*} \begin{aligned} & \underset{x\in \mathbb{R}^2}{\text{minimize}} & & \left\| x-p \right\|^2 \\ & \text{subject ...
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1answer
33 views

Prove a convex function

I have to prove that if $f:A \to \mathbb{R}$ is convex and $c \ge 0$ then $c \cdot f:A \to \mathbb{R}$ is convex. I know that function $f:A \to \mathbb{R}$ is convex if for $\forall x,y \in A$ and ...
0
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1answer
21 views

Is the following function concave? (or log of it)

I have a function $f(x_1, x_2, ..., x_M) = \displaystyle \prod_{i = 1}^N \frac{(\sum_{j = 1}^{M} a_jx_jI_{ij})^2}{\sum_{j = 1}^{M} a_jx_j}$ in domain $\{{\bf x} \in {\bf R}^m \setminus {\bf 0} \ ...
2
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1answer
40 views

How to solve the convex optimization problem [closed]

$$\min (\|X_{(1)}\|_{*}+\|X_{(2)}\|_{*}+\|X_{(3)}\|_{*})+u\|Ax-b\|_2^2+v\|Cx\|_2^2$$ where $X$ is a three order tensor, $X_{(i)}$ is a matrix whose column are the mode-$i$ fibers of $X$(i=1,2,3),$x$ ...
0
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0answers
43 views

how to prove convexity of the function below?

For a graph $G$ consider the following function, $$f=\sum_{(i,j) \in G ,(i,k) \notin G } \max(0,c+ \left\|e_i-e_j\right\|_2^2-\left\|e_i-e_k\right\|_2^2)$$ where $e_i \in\mathbb R^n$ ($n$ dimension ...
0
votes
1answer
22 views

Minimizing nonsmooth single variable functions?

What options is available if one wants to minimize a nonsmooth convex function of one variable? Subgradients would work, but there has to be some nice way of utilizing that we're only searching in 1d. ...
0
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2answers
37 views

Convergence of a sequence of projections

Let $C \subset \mathbb{R}^n$ be a compact, convex set, and $P \in \mathbb{R}^{n \times n}$ be a positive definite matrix ($P \succ 0$). Consider the projection $\Pi_P: \mathbb{R}^n \rightarrow C$ ...