Stack Exchange Network

Stack Exchange network consists of 174 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

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
votes
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 ...
0
votes
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 ...