Questions tagged [convexity]

For questions related to convex functions and convex sets, especially as they relate to optimization problems.

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Why is there a separate area for PSD constraints and PSD variables in the Conic Benchmark Format?

This question pertains to the Conic Benchmark Format (CBF) for specifying a convex optimization problem. Here's a link to the specification. In the CBF specification, there are separate areas for ...
Robert Bassett's user avatar
2 votes
1 answer
133 views

Convexity of p power of the q norm (0<p<1, q>1)

I encountered a minimization problem involving the following function: $f(\mathbf{x})=\|\mathbf{x}\|_q^p$ Here, $q>1$ and $0<p<1$. Naturally, each entry of $\mathbf{x}$ is greater than $0$. I ...
Piko Mone's user avatar
4 votes
1 answer
148 views

Does minimizing the upper bound due to Jensen's inequality yield an equivalent solution?

$\DeclareMathOperator*{\argmin}{\arg\!\min}$Consider the convex function $f : X \to \mathbb R$, where $X \subseteq \mathbb R^n$ is a convex set. Define the functions $g_\ell : X^m \times \Delta \to \...
mhdadk's user avatar
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2 answers
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Relaxing non-affine equality constraints in convex optimization

Consider the convex function $f$. In section 4.2.1 in these lecture notes, the author writes: 4.2.1 Relaxing non-affine equality constraints For functions $g_i(x)$, $i \in \{1,\dots,d\}$ that are ...
mhdadk's user avatar
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Is it possible to make a posynomial concave using a change of variables?

Note: this question was already posted on Math.SE but received no answers, so I'm re-posting it here for better reach. Consider the following posynomial with respect to the variables $x_1,\dots,x_n$: ...
mhdadk's user avatar
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Is a posynomial concave under the following conditions?

Update In case anyone is interested, this question has been answered here. Consider the following posynomial with respect to the variables $x_1,\dots,x_n$: \begin{align} f(x_1,\dots,x_n) &= \sum_{...
mhdadk's user avatar
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2 votes
1 answer
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Question About Fritz John Theorem and Slater Constraint Qualification

Background Information I am studying constraint qualifications. Here are two theorems leading to my question: Theorem 1$\space\space\space\space$ [Fritz John Theorem] Suppose that $f, g_1, \dots, g_k$...
Beerus's user avatar
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Changing the order of $\sup$ and $\inf$

I have a problem in the following form \begin{align} \begin{array}{cll} \sup_{\theta \in \mathrm{dom}(f)} & \inf_{z \in \mathbb{R}^n} & \underbrace{f(\theta)}_{\text{concave in $\theta$}} + \...
independentvariable's user avatar
3 votes
2 answers
181 views

When Biconvex function is Pseudoconvex function?

Is a Biconvex function f(x,y):=yg(x), (where g is a convex function, y>=0), Pseudoconvex function?
Abbas Khademi's user avatar
2 votes
1 answer
64 views

Does this kind of "partition" have a name?

Consider a convex polyhedron $A$. Assume we have subsets $A_1,\ldots,A_n$ of $A$ that are themselves covex polyhedra and are mutually disjoint except maybe sharing an edge, and that their union gives $...
pele's user avatar
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2 answers
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Is upper incomplete gamma function convex?

Considering the definition of upper incomplete gamma function: $\Gamma(a, x) =$ $\int_{x}^{\infty}t^{a-1}e^{-t} dt$ Given that $a$ is fixed and $0 < x < a$, can we prove the function is convex ...
Javidit's user avatar
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3 votes
1 answer
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On a clarification on usage of inequalities in convex programming

The inequality $x^3\leq y$ is not convex. But $0<x$ added to the above provides a convex region. My question is whether in convex programming it is allowed to use both inequalities together and use ...
Turbo's user avatar
  • 131
2 votes
1 answer
113 views

Convex not strictly convex!

Update: Linear programming problems (LP) have a convex space, precisely vector space, such as a convex feasible region as pointed out by @prubin. Also, they may have either unique or multiple ...
A.Omidi's user avatar
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FOC point vs Stationary point in local optimization

In this SIAM Review paper the authors are giving the following necessary condition for a point being a local maximum of a convex function: Let $F: \mathbb{R}^n \mapsto \mathbb{R}$ be convex. If $x$ ...
independentvariable's user avatar
2 votes
0 answers
89 views

How to figure out integer variables in the relaxation set?

Suppose, there is mixed-integer programming as follows: $(1)$ $$\begin{aligned} \min&\quad c^{\top} x\\ \text{s.t.}& \quad A x \geq b \\ &\quad B x \geq d \\ &\quad x \geq 0 \\ &...
A.Omidi's user avatar
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4 votes
1 answer
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How to know if a combinatorial optimization problem is linear or not?

I want to know if a combinatorial problem like the knapsack problem is linear or not. And how do we know if a given problem is convex or not?
MAJID majid's user avatar
3 votes
2 answers
366 views

How to make following constraint a convex one?

I would like to write a constraint as follows, where $x,y>0$ are optimization variables, and $a,b,c,d,A$ are positive constants. How to make it a convex constraint? \begin{equation} \frac{{ax}}{{\...
qinqinxiaoguai's user avatar
2 votes
3 answers
249 views

Convex optimization on the unit hypercube with gradients and a bounded minimum

I'd like to find the minimum of a smooth, continuous function inside the unit hypercube (the dimensionality of which could go into the hundreds or even thousands). The function is convex (Hessian $\...
sk29910's user avatar
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How to convexify log(convex) function?

I have the following optimization problem: \begin{align}\max_x&\quad\log_2(1+|a+bx|^2+cx^2)\\\text{s.t.}&\quad0\le x\le1\\&\quad(1-x^2)\ge\text{constant}\end{align} where $a$ and $b$ are ...
Shayan zargari's user avatar
8 votes
2 answers
348 views

Is $\min \ x^3 \ \mathrm{s.t.}\ x \geq 0$ a convex problem?

The problem $$\min \ x^3 \ \mathrm{s.t.} \ x \geq 0$$ is sometimes said to be a convex optimization problem. $f(x) = x^3$ is not a convex function. However, in the domain of $x\geq 0$ it is convex. So ...
independentvariable's user avatar
3 votes
0 answers
142 views

Proving convexity for a function with summation and integer variable

I would like to show that the function $f$ is convex in $\rho\in [0,1)$ under $s\in \mathbb{Z}^+$. When I use Sympy packages of Python to find $\displaystyle\frac{\partial^2 f(\rho)}{d\rho^2}$. I get ...
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Decomposition of Polyhedra

There is no doubt that clear examples consolidate the understanding of concepts being learnt. I am new to finding the structure and decomposition of a polyhedra. Suppose that we have the system $$ \...
holala's user avatar
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1 answer
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Find a dual problem with one dual decision variable to the problem of finding the orthogonal projection of a given vector

Given the set $T_{\alpha}=\{x\in\mathbb{R}^n:\sum x_i=1,0\leq x_i\leq \alpha\}$ For which $\alpha$ the set is non-empty? Find a dual problem with one dual decision variable to the problem of finding ...
convxy's user avatar
  • 405
4 votes
1 answer
136 views

Optimization of strongly convex functions with approximate evaluations of gradient and Hessian

Suppose I want to find the minimum of a differentiable, strongly convex function $f:\mathbb{R}^n\to\mathbb{R}$ with constant $\mu>0$. That is, for all $x,y\in\mathbb{R}^n$, I have that: $$f(y) \geq ...
user134977's user avatar
3 votes
1 answer
114 views

How to evaluate the convexity of an optimal control problem?

Can we consider an optimal control problem, a convex optimization problem like static optimization problems? If it is possible, under what conditions, will this problem be a convex problem? For ...
Amin's user avatar
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1 vote
0 answers
54 views

$\nabla_y\nabla_vf(x^*)\geq0$ for any concave $f$ if and only if $y=-v$

$f:\mathbb R^3\to\mathbb R$ is an arbitrary concave function. $H$ is a plane. $v$ is a given vector on $H$. $x^*=\max_{x\in H} f(x)$ Prove that $\nabla_y\nabla_vf(x^*)\geq 0$ if and only if $y=-v$. I ...
High GPA's user avatar
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1 answer
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Quasi-convex function must be "partially monotonic"?

$f(x)$ is quasi-convex, $$x^*\in\arg\min_{x\in C}f(x).$$ How to prove that, for any $a\in C$, $f(x) $ is weakly monotonic in the direction of $(x^*-a)$? Is this simple result a part of an ancient ...
High GPA's user avatar
  • 179
6 votes
1 answer
211 views

Convexity of the variance of a mixture distribution

$X$ is a random variable that is sampled from the mixture of uniform distributions. In other words: $$X \sim \sum_{i=1}^N w_i \cdot \mathbb{U}(x_i, x_{i+1}),$$ where $\mathbb{U}(x_i, x_{i+1})$ denotes ...
independentvariable's user avatar
4 votes
1 answer
183 views

Can we get closed form solution for such a problem?

\begin{align}\min&\quad\sum_{i=1}^N\frac{A_i}{x_i}\\\text{s.t.}&\quad\sum x_i \le X\\&\quad x_i \ge 0\end{align} wherein $A_i>0, (i\in\{1,\dots,N\})$ is constant, $x_i, (i\in\{1,\dots,N\...
qinqinxiaoguai's user avatar
1 vote
0 answers
93 views

Question on quadratically constrained quadratic program

If the constrained optimization problem is a quadratically constrained quadratic program of the form \begin{align}\min&\quad x^HQx-a(x+x^H)+b|z^Hx|^2\\\text{s.t.}&\quad\|x\|^2\le1\end{align} ...
Kali's user avatar
  • 19
5 votes
3 answers
2k views

Linear programming convexity

Is it possible for a linear programming model to be non-convex ? If it is, please, provide a simple 2 variables example and explain why it is non-convex. EDIT 1: I have been wondering, maybe the ...
G Oliveira's user avatar
2 votes
1 answer
161 views

Simple nonlinear programming using convexity analysis and KKT

I want to solve the following two-variate nonlinear programming using KKT conditions: $$ \begin{align} \begin{split} \max \quad & 15 \sqrt{x_{1}} + 16 \sqrt{x_{2}} \\ \text{s.t.} \quad &...
Edward's user avatar
  • 391
3 votes
1 answer
240 views

Convexity Analysis

For the following function, I am testing the convexity in $\lambda$. All parameters are in $\mathbb{R}^+$. $$\frac{\left(- \lambda \left(b + \frac{p}{\beta}\right) + 1\right) \left(\left(1 + \frac{p}{...
tcokyasar's user avatar
  • 1,249
8 votes
2 answers
211 views

how to penalize a shortfall of a sum of absolute values

I have a model where there is a constraint on the sum of absolute values, and I would like to add a penalty on the shortfall from the constraint. More specifically: \begin{align*} \text{maximize}\ &...
Henry's user avatar
  • 542
5 votes
1 answer
113 views

Concavity: Definition similar to sub-modularity

We know that for $b \geq a$, and some $s \geq 0$, a concave function $f$ satisfies: $f(a+s) - f(a) \geq f(b+s) - f(b)$. This is not a frequent definition of concavity, but can be found, e.g., here....
independentvariable's user avatar
9 votes
1 answer
560 views

Compute the distance from a point inside a convex set to the boundary of the set

Problem Let $\mathcal C = \{ X \in \mathbb{R}^n \mid g(X) \leq 0\}$ where $g$ is convex, and let $X_c \in \mathcal{C}$. Is there any algorithm to compute the distance from $X_c$ to the boundary of $\...
C Marius's user avatar
  • 507
6 votes
2 answers
847 views

Existence of Optimal Solution

Assume we are solving $\min\{f(x) \ | \ x \in S \}$. If $f: \mathbb{R}^n \mapsto \mathbb{R}$ is a proper closed convex function, and $S$ is a non-empty closed convex set, does this imply that the ...
independentvariable's user avatar
3 votes
2 answers
277 views

Convexity of a function

I would like to show that this function $$2x^2 + 8y^2$$ is convex or concave by using the definition $$f(θx+(1−θ)y) \le θf(x)+(1−θ)f(y)$$ From my understanding, using the Hessian matrix, I believe ...
george's user avatar
  • 135
5 votes
0 answers
2k views

Convexity of the projection of a convex set

Question: A set $S \subset \mathbb{R}^m \times \mathbb{R}^n$ is convex. Using the fact that affine maps preserves convexity prove that $S(y) = \{x \in \mathbb{R}^m\mid (x,y)\in S \}$ and $\hat{S} = ...
independentvariable's user avatar
8 votes
2 answers
431 views

Convex Optimization: Separation of Cones

I am trying to solve Exercise 2.39 at Boyd and Vandenberghe's Convex Optimization book. In one source, the answer is given as: 2.39 Separation of cones. Let $K$ and $\tilde K$ be two convex cones ...
independentvariable's user avatar
4 votes
1 answer
310 views

How to express this constraint?

I have the constraint \begin{align}\max&\quad\gamma\\\text{s.t.}&\quad a\ge\gamma b\\&\quad\gamma\le 1\end{align} where $\gamma$ is an optimization variable and $a$ is a function of some ...
KGM's user avatar
  • 2,265
9 votes
1 answer
99 views

Problem solvable $\Rightarrow$ subproblems solvable if feasible region closed, convex?

Let $c \in \mathbb{R}^n$, $M \subseteq \mathbb{R}^n$ such that the problem \begin{align}P:\quad\min_{x \in \mathbb{R}^n}&\quad c^\intercal x\\\textrm{s.t.}&\quad x \in M\end{align} is solvable....
zxmkn's user avatar
  • 213
8 votes
1 answer
269 views

Convexity/Concavity of Average Number of Jobs in M/M/1 Queue?

I am working on a problem involving the average number of jobs $L$ in an M/M/1 queue with arrival rate $\lambda$, service rate $\mu$. For traffic intensity $\rho = \frac{\lambda}{\mu}$, $$ L = \frac{\...
brentertainer's user avatar
10 votes
1 answer
341 views

Solving convex programs defined by separation oracles?

General question: What software can solve convex programs defined by a separation oracle? The objective function is concave, and the feasible set is a polytope. By a separation oracle I mean that I ...
Elle Najt's user avatar
  • 203
13 votes
2 answers
1k views

Is This Constraint Convex?

I have a constraint that I believe to be convex and not affine which I think means that I can implement a relaxation. I will first define the full constraint, and then build up my (informal) reasoning ...
GrayLiterature's user avatar
9 votes
2 answers
540 views

Convexity of a QP

In quadratic programming (QP), you encounter an objective of the following form: $$x^TQx + c^Tx$$ and often it's desirable to know if the QP is convex. One method to check for convexity is by ...
Josh Allen's user avatar
9 votes
3 answers
743 views

Examples of problems with non-convex constraint functions but convex feasible region

I'm looking for examples of (classes of) problems with a non-convex, non-linear formulation, but convex feasible region. That is, a problem of the sort: $$ \begin{array}{lll} \text{minimize} & ...
Robert Schwarz's user avatar
12 votes
2 answers
605 views

Dedicated solver for convex problems

Are you aware of a fast solver (open source or commercial) for convex NLPs that is faster than IPOPT? I'm interested in problems in the 50K+ variable range, both dense and sparse. Ideally, it would be ...
Nikos Kazazakis's user avatar
18 votes
3 answers
6k views

Can an integer optimization problem be convex?

I'm trying to wrap my head around an apparent paradox that I've come across while trying to learn more about optimization algorithms: On one hand several sources state that convex optimization is ...
Skander H.'s user avatar
  • 2,139
16 votes
1 answer
486 views

Convexity of Variance Minimization

$X$ is a discrete random variable taking value $x_n$ with probability $1/N$ for $n=1, \ldots,N$. I would like to set the $x_n$ values in an optimization problem. My objective is to minimize the ...
independentvariable's user avatar