Questions tagged [convexity]
For questions related to convex functions and convex sets, especially as they relate to optimization problems.
46
questions
1
vote
0answers
74 views
How to figure out integer variables in the relaxation set?
Suppose, there is mixed-integer programming as follows:
$(1)$
$$\begin{aligned}
\min&\quad c^{\top} x\\
\text{s.t.}& \quad A x \geq b \\
&\quad B x \geq d \\
&\quad x \geq 0 \\
&...
3
votes
1answer
60 views
How to know if a combinatorial optimization problem is linear or not?
I want to know if a combinatorial problem like the knapsack problem is linear or not. And how do we know if a given problem is convex or not?
3
votes
2answers
218 views
How to make following constraint a convex one?
I would like to write a constraint as follows, where $x,y>0$ are optimization variables, and $a,b,c,d,A$ are positive constants. How to make it a convex constraint?
\begin{equation}
\frac{{ax}}{{\...
2
votes
3answers
85 views
Convex optimization on the unit hypercube with gradients and a bounded minimum
I'd like to find the minimum of a smooth, continuous function inside the unit hypercube (the dimensionality of which could go into the hundreds or even thousands). The function is convex (Hessian $\...
1
vote
1answer
176 views
How to convexify log(convex) function?
I have the following optimization problem: \begin{align}\max_x&\quad\log_2(1+|a+bx|^2+cx^2)\\\text{s.t.}&\quad0\le x\le1\\&\quad(1-x^2)\ge\text{constant}\end{align} where $a$ and $b$ are ...
6
votes
2answers
95 views
Is $\min \ x^3 \ \mathrm{s.t.}\ x \geq 0$ a convex problem?
The problem $$\min \ x^3 \ \mathrm{s.t.} \ x \geq 0$$
is sometimes said to be a convex optimization problem. $f(x) = x^3$ is not a convex function. However, in the domain of $x\geq 0$ it is convex. So ...
3
votes
0answers
88 views
Proving convexity for a function with summation and integer variable
I would like to show that the function $f$ is convex in $\rho\in [0,1)$ under $s\in \mathbb{Z}^+$. When I use Sympy packages of Python to find $\displaystyle\frac{\partial^2 f(\rho)}{d\rho^2}$. I get ...
1
vote
0answers
70 views
Decomposition of Polyhedra
There is no doubt that clear examples consolidate the understanding of concepts being learnt. I am new to finding the structure and decomposition of a polyhedra. Suppose that we have the system
$$ \...
1
vote
1answer
51 views
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
...
4
votes
1answer
90 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:
$$f(y) \geq ...
3
votes
1answer
70 views
How to evaluate the convexity of an optimal control problem?
Can we consider an optimal control problem, a convex optimization problem like static optimization problems? If it is possible, under what conditions, will this problem be a convex problem? For ...
1
vote
0answers
47 views
$\nabla_y\nabla_vf(x^*)\geq0$ for any concave $f$ if and only if $y=-v$
$f:\mathbb R^3\to\mathbb R$ is an arbitrary concave function.
$H$ is a plane. $v$ is a given vector on $H$.
$x^*=\max_{x\in H} f(x)$
Prove that $\nabla_y\nabla_vf(x^*)\geq 0$ if and only if $y=-v$.
I ...
2
votes
1answer
67 views
Quasi-convex function must be "partially monotonic"?
$f(x)$ is quasi-convex,
$$x^*\in\arg\min_{x\in C}f(x).$$
How to prove that, for any $a\in C$, $f(x) $ is weakly monotonic in the direction of $(x^*-a)$?
Is this simple result a part of an ancient ...
6
votes
1answer
142 views
Convexity of the variance of a mixture distribution
$X$ is a random variable that is sampled from the mixture of uniform distributions. In other words:
$$X \sim \sum_{i=1}^N w_i \cdot \mathbb{U}(x_i, x_{i+1}),$$
where $\mathbb{U}(x_i, x_{i+1})$ denotes ...
4
votes
1answer
176 views
Can we get closed form solution for such a problem?
\begin{align}\min&\quad\sum_{i=1}^N\frac{A_i}{x_i}\\\text{s.t.}&\quad\sum x_i \le X\\&\quad x_i \ge 0\end{align}
wherein $A_i>0, (i\in\{1,\dots,N\})$ is constant, $x_i, (i\in\{1,\dots,N\...
1
vote
0answers
84 views
Question on quadratically constrained quadratic program
If the constrained optimization problem is a quadratically constrained quadratic program of the form \begin{align}\min&\quad x^HQx-a(x+x^H)+b|z^Hx|^2\\\text{s.t.}&\quad\|x\|^2\le1\end{align} ...
4
votes
3answers
736 views
Linear programming convexity
Is it possible for a linear programming model to be non-convex ? If it is, please, provide a simple 2 variables example and explain why it is non-convex.
EDIT 1:
I have been wondering, maybe the ...
1
vote
1answer
98 views
Simple nonlinear programming using convexity analysis and KKT
I want to solve the following two-variate nonlinear programming using KKT conditions:
$$ \begin{align} \begin{split}
\max \quad & 15 \sqrt{x_{1}} + 16 \sqrt{x_{2}} \\
\text{s.t.} \quad &...
3
votes
1answer
182 views
Convexity Analysis
For the following function, I am testing the convexity in $\lambda$. All parameters are in $\mathbb{R}^+$.
$$\frac{\left(- \lambda \left(b + \frac{p}{\beta}\right) + 1\right) \left(\left(1 + \frac{p}{...
8
votes
2answers
107 views
how to penalize a shortfall of a sum of absolute values
I have a model where there is a constraint on the sum of absolute values, and I would like to add a penalty on the shortfall from the constraint. More specifically:
\begin{align*}
\text{maximize}\ &...
5
votes
1answer
106 views
Concavity: Definition similar to sub-modularity
We know that for $b \geq a$, and some $s \geq 0$, a concave function $f$ satisfies:
$f(a+s) - f(a) \geq f(b+s) - f(b)$.
This is not a frequent definition of concavity, but can be found, e.g., here....
9
votes
1answer
344 views
Compute the distance from a point inside a convex set to the boundary of the set
Problem
Let $\mathcal C = \{ X \in \mathbb{R}^n \mid g(X) \leq 0\}$ where $g$ is convex, and let $X_c \in \mathcal{C}$. Is there any algorithm to compute the distance from $X_c$ to the boundary of $\...
6
votes
2answers
228 views
Existence of Optimal Solution
Assume we are solving $\min\{f(x) \ | \ x \in S \}$.
If $f: \mathbb{R}^n \mapsto \mathbb{R}$ is a proper closed convex function, and $S$ is a non-empty closed convex set, does this imply that the ...
3
votes
2answers
243 views
Convexity of a function
I would like to show that this function
$$2x^2 + 8y^2$$
is convex or concave by using the definition
$$f(θx+(1−θ)y) \le θf(x)+(1−θ)f(y)$$
From my understanding, using the Hessian matrix, I believe ...
5
votes
0answers
660 views
Convexity of the projection of a convex set
Question:
A set $S \subset \mathbb{R}^m \times \mathbb{R}^n$ is convex. Using the fact that affine maps preserves convexity prove that $S(y) = \{x \in \mathbb{R}^m\mid (x,y)\in S \}$ and $\hat{S} = ...
8
votes
2answers
177 views
Convex Optimization: Separation of Cones
I am trying to solve Exercise 2.39 at Boyd and Vandenberghe's Convex Optimization book. In one source, the answer is given as:
2.39 Separation of cones. Let $K$ and $\tilde K$ be two convex cones ...
4
votes
1answer
216 views
How to express this constraint?
I have the constraint \begin{align}\max&\quad\gamma\\\text{s.t.}&\quad a\ge\gamma b\\&\quad\gamma\le 1\end{align} where $\gamma$ is an optimization variable and $a$ is a function of some ...
9
votes
1answer
89 views
Problem solvable $\Rightarrow$ subproblems solvable if feasible region closed, convex?
Let $c \in \mathbb{R}^n$, $M \subseteq \mathbb{R}^n$ such that the problem
\begin{align}P:\quad\min_{x \in \mathbb{R}^n}&\quad c^\intercal x\\\textrm{s.t.}&\quad x \in M\end{align} is solvable....
8
votes
1answer
250 views
Convexity/Concavity of Average Number of Jobs in M/M/1 Queue?
I am working on a problem involving the average number of jobs $L$ in an M/M/1 queue with arrival rate $\lambda$, service rate $\mu$. For traffic intensity $\rho = \frac{\lambda}{\mu}$,
$$
L = \frac{\...
10
votes
1answer
184 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 ...
13
votes
2answers
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 ...
9
votes
2answers
333 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 ...
9
votes
3answers
583 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} & ...
11
votes
2answers
485 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 ...
17
votes
3answers
3k 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 ...
15
votes
1answer
340 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 ...
17
votes
5answers
3k views
Linear Programming with additional "if-then"/"Default to zero" constraints?
What approaches can I use for a Linear Programming problem with the additional constraint that if a decision variable falls below a certain threshold, then it should just be forced to 0.
I'm ...
12
votes
2answers
340 views
Convex vs Strictly Quasiconvex Functions in Optimization
I have read/heard quite a few time that in the old days, it was considered that linear programs constitute the class of optimization problems that can be solved efficiently in practice (as a rule of ...
13
votes
1answer
470 views
How to formulate a problem to prove/disprove convexity?
Given a general non-linear problem:
\begin{align}P:\qquad&\min_{x\in X} f(x)\\\text{s.t.}\qquad&g(x)\leq 0\end{align}
where $f$ is a non-linear function, $g$ is a vector of non-linear ...
9
votes
2answers
249 views
How to determine the convexity of my problem and categorize it?
My problem is:
\begin{align}\min\limits_{x_{ij}}\qquad&{\sum_{i\in N}\sum_{j\in M}\frac{x_{ij}}{C_j-\sum\limits_{i\in N} x_{ij}a_i}}\\\text{s.t.}\qquad&0<C_j-\sum_{i\in N} x_{ij}a_i\\\qquad&...
12
votes
1answer
216 views
Recovering primal optimal solutions from dual sub gradient ascent using ergodic primal sequences
My question concerns recovering a primal optimal solution while performing dual sub gradient ascent. Denoting by $y_i$ the dual multiplier in the $i^{\text{th}}$ iteration, let
\begin{equation}
x_i = ...
7
votes
2answers
379 views
How can I linearize or convexify this binary quadratic optimization problem?
I have an optimization problem as below. I am having a hard time with the last constraint.
$\max \eta$
subject to
${\bf U}(:,m)^T{\bf A}{\bf U}(:,m)=0,m=1,2,\cdots,M$
here
$\bf{A}$ is a Binary ...
15
votes
2answers
148 views
Bound on the number of constraints to be generated (lazy constraints)
I am working on a very large optimisation problem. All variables are continuous, the objective is linear and the constraints convex, but I have many such constraints (on the order of $2^n$ — actually, ...
7
votes
1answer
377 views
KKT inequality conditions
Let's say I have an objective function
$$f(x_1,x_2, \cdots, x_n)$$
and $N$ constraints
$$x_i \ge 0. $$
I am trying to solve it with KKT conditions. Now the objective function becomes
$$f(x_1,x_2,...
12
votes
2answers
552 views
Convex Maximization with Linear Constraints
I am doing active research in convex maximization w.r.t. linear constraints. There are many cases which can be efficiently approximately solved, e.g., convex quadratic maximization, log-sum-exp ...
20
votes
2answers
1k views
Reference for "expectation preserves convexity"
It is well known that expectation preserves convexity: If $f(x)$ is convex and $Y$ is a random variable, then $\mathbb E[f(x-Y)]$ is convex. This property arises in, for example, inventory theory.
I ...
