Questions tagged [convex-optimization]
Optimization with convex constraints and convex objectives; notions related to convex optimization such as sub-gradients, normal cones, separating hyperplanes
0
votes
0answers
36 views
Concavity of a function after binomial transform and maximization
Suppose $f(x)$ is a one-dimensional discrete concave function, and $X(n,p)$ is Binomial random variable. Is the function $g(x)= \max_n \mathbb{E} f(x+X(n,p))$ also discrete concave?
3
votes
0answers
54 views
Optimizing a multivariate symmetric (permutation-invariant) function
Let $\ell$ and $d$ be two integers such that $\ell \le d$.
I would like to find the global maxima of the following symmetric function $f\colon (0,1]^n \to \mathbb{R}$,
$$f(x_1, \ldots, x_n) := \sum_{\...
1
vote
0answers
20 views
Second order necessary and sufficient conditions for convex nonsmooth optimization
For convex smooth optimization, first and second order necessary and sufficient conditions are well known. Does such standard second order necessary and sufficient conditions exist for convex ...
0
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0answers
9 views
How to Create Point-Optimal Objective Functions
Here is a problem that has originated from some IP research i'm working on.
You are given a polyhedron $P$ in standard matrix inequality form of $Ax \le b$, $x \in \mathbb{R}^n$ as well as a point $...
0
votes
0answers
59 views
Is dynamic programming suitable for a specific optimization problem?
Let $c,\,\mathcal{P}_0,\,\mathcal{P}_1,\,\mathcal{P}_2,\ldots$ be a sequence of positive real numbers.
Let $N\in\{1,\,2,\,3,\ldots\}$ and let $t\in\{0,\,1,\,2,\ldots\}$, with $N$ and $t$ fixed.
...
0
votes
0answers
33 views
optimisation problem for unknown function
I have two variables function $z=f_{a,b}(x,y)$ ($a,b$ - some parameters), however I don't know its formula (I can compute its value for given $x,y$ and $a,b$). I would like to find minimum of this ...
-1
votes
0answers
43 views
Maximum or minimum value
I appologize if the question is too elementary to MathOverFlow, but there was no answer or comment when I posted it to MathStackExchange. I just deleted the question on MathStackExchange and turned to ...
0
votes
0answers
26 views
Convergence of a fixed-point algorithm for a concave objective function
Let's suppose we have an objective function $\max_\limits{x} \sum_\limits{i} f_i(x_i)$ with the constraint that $\ x_i \geq 0, \sum_\limits{i} x_i = 1$.
Each function $f_i$ is continuous and ...
0
votes
1answer
69 views
Closed form solutions for maximal subsets of convex polytopes
I'm looking for any known exact results about inscribing simple convex bodies inside a convex polytope. The most famous is the Löwner-John ellipsoid, but as far as I understood in general there is no ...
1
vote
0answers
71 views
Matrix completion in $2\times2$ case by nuclear norm minimization to guarantee rank $1$?
Does fixing diagonal entries and minimizing nuclear norm under weighted sum of entries conditions produce a rank $1$ matrix? I think the answer for this is no.
At least could it be true in $2\times2$ ...
2
votes
1answer
101 views
Polygon of convex arcs
Convex polygons in the plane $R^2$ arise in linear programming where the constraints are linear. The objective linear function attains its maximum at a vertex of the feasible region(if exists).
Assume ...
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0answers
21 views
A question related to parametric linear programming
Consider the following parametric linear problem:
\begin{align}
\min z(t)=c^T x\\
Ax=b(t)\\
0\leq x\leq u.
\end{align}
We know $z(t)$ is a piecewise linear function. Let $x(t_1)$ and $x(t_2)$ be the ...
0
votes
2answers
136 views
Is it possible to “solve” iterative (convex/non-convex) optimization problems via learning (one-shot)?
I posted a following question in MSE, but I think it should be posted here in MO. Since I don't know how to transfer the post from MSE to MO, I have pasted the question below. Thank you in advance and ...
1
vote
2answers
55 views
Alternative characterization of epi-convergence
I am struggling with the proof of a property of epi-convergence.
We need the following definitions:
For a sequence of sets $(C^\nu)_\nu$ in $\mathbb R^n$, the outer limit is the set $\limsup_\nu C^\...
1
vote
0answers
117 views
The perturbation of a convex function can also be convex?
$ W^{1,\infty}(D)\ni f:D\to\mathbb R, (x,y)\mapsto f(x,y)$, is a strictly increasing on both dimensions (i.e. if $x_1>x_2$ then $f(x_1,y)>f(x_2,y)$), lipschitz continuous function defined on a ...
3
votes
0answers
56 views
Legendre transform on signed measure space
Let $X$ be an open set in $\mathbb{R}^n$ and $M(X)$ be the space of finite signed measures defined on $X$. $L(p)$ is a lower-semicontinuous convex functional defined on $M(X)$. My question is: (1) ...
1
vote
1answer
52 views
Efficient way to compute eigenvalue decomposition for following problem
I have an optimization problem
$$\begin{array}{ll} \text{minimize} & Tr(X^TAX) \\ \text{subject to} & X^TX=I
\end{array}$$
where $A\in R^{n \times n}$ and it is symmetric positive definite, ...
2
votes
1answer
51 views
Given the following two assumptions, How to conclude $\lim_{k \rightarrow \infty}|\nabla g_r(y^{k})|=0 \ $?
Please refer attached 6-page short paper for details.
Let $M(y)=y+2r\nabla g_r(y)$.
Given $\ g_r(M(y^k))>g_r(y^k)+r|\nabla g_r(y^k)|^2, \forall k. \ $ (equation 36 in the paper)
and $ \ \lim_{k ...
0
votes
0answers
23 views
Projecting a polyhedral cone onto its intersection with the infinity-norm ball
For a point in a convex polyhedral cone, $x\in \mathcal{C} = \{\sum_{i=1}^m \alpha_i r_i \vert \alpha_i \geq 0, r_i \in \mathbb{R}^n \}$, is there an efficient algorithm to project $x$ onto the ...
1
vote
0answers
58 views
Solve simple stochastic variational inequality
Let $Z$ be a random variable with finite mean $\mathbb E[Z]$ and let $\phi:(-\infty,\infty) \rightarrow (-\infty,\infty]$ be a convex l.s.c function which is differentiable at $z=1$ with gradient $\...
1
vote
0answers
26 views
Dual representation of problems involving $f$-divergences
Studying some problems arising in decision-making under model uncertainty, I'm led to consider the following problems.
Let $\mathbb E_P$ and $\mathbb V_P$ denote the expectation and variance ...
0
votes
1answer
112 views
How to solve this optimization problem efficiently? [closed]
Let, $D\in\mathbb{C}^{1\times M}$ is a row vector with $M$ elements
$V\in\mathbb{C}^{3^M\times M}$ is a given matrix
$T$ is a scalar (real and $>1$)
$\textbf{The problem at hand is as follows:}$
...
3
votes
1answer
61 views
Echange of Infimum Integral with Pointwise Infimum
Setup
Suppose that $U$ is a subset of $L^{\infty}_{\mu}(\mathscr{F})\cap L^1_{\mu}(\mathscr{F})$ defined by
$$
f\in U \Leftrightarrow g(f(x))\leq M \mbox{ and } f \in L^{\infty}_{\mu}(\mathscr{F})\...
3
votes
1answer
151 views
it's convex sequence inequality
A sequence $a_0,a_1,\dots,a_n$ of real numbers is called concave if $a_{0}=0$, and for each $0<i<n$, we have $a_i\geq\dfrac{a_{i-1}+a_{i+1}}{2}$.
Find the largest $c(n)$ such that for every ...
2
votes
1answer
116 views
Advantages of hyperbolic programming over semidefinite programming?
What are the advantages of a hyperbolic program over a semi definite program? SDPs can be used to represent a wide variety of algebraic constraints. Are there constraints that can be represented in a ...
2
votes
0answers
63 views
convex approximation for a non convex function
Consider the function
$f\left( {{x_1},...,{x_M},{y_1},...,{y_N}} \right) = \left( {\sum\limits_{j = 1}^M {{\alpha _j}{x_j}} } \right)\left( {{e^{ - \sum\limits_{i = 1}^N {{\beta _i}{y_i}} }}} \right)$...
2
votes
1answer
92 views
Convergence of a stochastic sequence?
I am reading this paper related to an algorithm for nonsmooth optimization problems. After many simplifications, I was able to formalize the method as follows: let $\Bbb B $ denote the unit ball in $\...
0
votes
1answer
68 views
maximization of a log norm function
Considering the following optimization program:
$$
maximize \ \ \ \log \left( \|x\|_\infty \right)
$$
$$
subject \ to \ \ Ax\leq b, \ x \geq 0
$$
can we rewrite this program as a convex ...
1
vote
1answer
69 views
LASSO problem but with a maximization instead of minimization
I have the following optimization problem (like the LASSO problem but with maximization instead of minimization):
$\mathbf{maximize}_{\boldsymbol{\alpha}} \|\mathbf{x} -\mathbf{A}\boldsymbol{\alpha}\|...
0
votes
1answer
49 views
Quasiconvexity property of quasinorms
Schatten $p$ norm is convex when $p\geq1$ holds and if $p\in(0,1)$ it is quasinorm.
If $p\in(0,1)$ then is Schatten $p$ norm quasi convex? I am interested in definition of quasi convexity here https:...
3
votes
0answers
35 views
Level sets of strongly convex and smooth functions
Let $f: \mathbb{R}^N \to \mathbb{R}$ be a $\alpha$-strongly convex and $\beta$-strongly smooth function, i.e.,
$$ f(x) + \langle\nabla f(x), y- x\rangle + \frac{\alpha}{2}\|y-x\|^2
\leq f(y) \leq f(x) ...
4
votes
2answers
235 views
Fast projection onto a subspace
Given an $n$-dimensional vector $\mathbf{c}\in [0,1]^n$, let $\Delta_{\mathbf{c}}$ be the set of points $\{\mathbf{x}\in [0,1]^n: \langle \mathbf{c},\mathbf{x} \rangle \le 1\}$, where $\langle \mathbf{...
1
vote
0answers
36 views
Minimum Preserving Transformations [closed]
If $f:X\rightarrow Y$, $g:Y\rightarrow Y$ are functions and $g$ is monotone increasing function then
$$
\operatorname{argmin}_{x \in X} f(x)
=
\operatorname{argmin}_{x \in X} g\circ f(x) .
$$
X and Y ...
2
votes
1answer
77 views
Why the convexity condition on the definition of a face of a convex set?
A face of a closed convex set $X\subseteq\mathbb{R}^n$ is defined to be a set $F\subseteq X$ such that:
$F$ is convex.
Every line segment from $X$ whose interior meets $F$ is contained in $F$.
Is ...
1
vote
2answers
172 views
Maximizing a function that is sum of gaussians
Let $\mathbf{x}_1,\dots,\mathbf{x}_n$ be given $n$ vectors in $\mathbb{R}^d$. Define the function
\begin{align}
\mathcal{K}(\mathbf{x},\mathbf{y})=
\alpha\exp(-\frac{||\mathbf{x}-\mathbf{y}||^2}{2\...
5
votes
2answers
157 views
Is there any efficient solution of the matrix equation AXB + (AXB)' + cX = D
I am trying to find the symmetric solution $X\in \mathbb{R}^{p\times p}$ of following matrix equation:
$AXB + (AXB)^T + cX = D$
where $A,B\in \mathbb{R}^{p\times p}$ are symmetric positive ...
9
votes
1answer
427 views
Is KL divergence $D(P||Q)$ strongly convex over $P$ in infinite dimension
By KL divergence I mean $D(P||Q) = \int dP \log(\frac{dP}{dQ})$. I am looking for the conditions under which this strong convexity is true and possible references. I could not find an answer for ...
0
votes
0answers
65 views
Iterative methods for minimizing sequences
Let $\mathbb{X}$ be a Banach space equipped with some norm $||\cdot||_\mathbb{X}$ and $F:\mathbb{X}\to\mathbb{R}$ be some linear functional. Suppose we are given a set $A\subseteq\mathbb{X}$ which is ...
3
votes
0answers
78 views
Fenchel conjugate on a Hadamard manifold
Let $M$ be a Hadamard manifold and let $F:M\to\mathbb{R}$ be a real-valued convex function on $M$. What would be the Fenchel-Young conjugate of $F$?
In general for a real locally convex vector space $...
3
votes
1answer
189 views
concavity of $\log [ (1+\frac{x_0}{1+x_0+x_1/2+x_2/4}) (1+\frac{x_1}{1+x_0/4+x_1+x_2/2}) (1+\frac{x_2}{1+x_0/2+x_1/4+x_2}) ]$
Let $f:~ [0,1]^3 \rightarrow \mathbb{R}$ be
$$
f(x_0,x_1,x_2)= \log \left[ \left(1+\frac{x_0}{1+x_0+\frac{x_1}{2}+\frac{x_2}{4}}\right) \left(1+\frac{x_1}{1+x_1+\frac{x_2}{2}+\frac{x_0}{4}}\right) \...
4
votes
3answers
142 views
Quasi-concavity of $f(x)=\frac{1}{x+1} \int_0^x \log \left(1+\frac{1}{x+1+t} \right)~dt$
I want to prove that function
\begin{equation}
f(x)=\frac{1}{x+1} \int\limits_0^x \log \left(1+\frac{1}{x+1+t} \right)~dt
\end{equation}
is quasi-concave. One approach is to obtain the closed form ...
1
vote
0answers
36 views
Lower semicontinuity of proximal gradient descent sequence
I'm using an alternate optimization scheme to minimize a function $F(a,b,c) = G(a,b,c) + H(c)$ where $G$ is continuous and differentiable, $H$ is not differentiable but lower semi-continuous and ...
0
votes
1answer
139 views
Strict complementary slackness for semidefinite programs with strong duality
By a theorem of Goldman and Tucker it is known that if a linear program (LP) has a finite valued optimal solution, then there is an optimal primal/dual pair $(x,z)$ satisfying not only complementary ...
3
votes
1answer
107 views
Strong polynomial algorithm for linear programming
What is the current state of finding a strong polynomial algorithm for linear programming? Is there any reference?
2
votes
1answer
110 views
Quasi-concavity of $f(x)=(1-\frac{x}{1000})\log_2(1+2^x)$ on $[0~1000]$
I want to prove that function $f:[0~1000]\rightarrow R$, $$f(x)=(1-\frac{x}{1000})\log_2(1+2^x)$$ is quasi-concave. Any idea how to do the proof? I already tried to prove that any super-level set is ...
4
votes
1answer
316 views
Minimize the variance of a Boltzmann distribution
N.B.: Sorry for cross-posting from https://stats.stackexchange.com/posts/347804/edit (I realized it was the wrong venue for the question, but couldn't find an easy way to transfer the question here).
...
1
vote
0answers
48 views
Relative interior of a normal cone at a face of a convex polytope?
Suppose $A$ is a nonempty convex polytope in $\mathbb{R}^n$. Suppose $F$ is a face of $A$.
Consider the normal cone of $A$ at $F$:
$C_A(F)=\{v\in\mathbb{R}^n:v\cdot x\geq v\cdot y\ \forall\ x\in ...
1
vote
1answer
166 views
Is this parametrized semidefinite program convex?
I am considering an optimization problem of the form:
\begin{equation}
\begin{split}
f(s) &= \min_{X} \mathrm{tr}(C(s)X) \\
&\;\;\;\;\;\;\;\;\;\;\; X \ge 0, \\
&\;\;\;\;\;\;\;\;\;\;\; \...
0
votes
0answers
44 views
Is this set of polynomial constraints convex? Can I optimise it?
Basically I want to optimise a problem with a constraint on the (variable dependant) roots of a polynomial, which I would like to be assigned depending on other constraints.
I reduced the problem to ...
0
votes
1answer
47 views
Linearly constrained saddle-point optimization
Let $f(x,y)$ be a smooth (twice differentiable) saddle function (convex in $x$ and concave in $y$), where $f \colon X \times Y \rightarrow \mathbb{R}$, and $X \subset \mathbb R^n$, $Y \subset \mathbb ...
