Questions tagged [convexity]
For questions related to convex functions and convex sets, especially as they relate to optimization problems.
48
questions
1
vote
1
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98
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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 ...
- 6,206
2
votes
0
answers
45
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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 $x$ ...
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2
votes
0
answers
86
views
How to figure out integer variables in the relaxation set?
Suppose, there is mixed-integer programming as follows:
$(1)$
$$\begin{aligned}
\min&\quad c^{\top} x\\
\text{s.t.}& \quad A x \geq b \\
&\quad B x \geq d \\
&\quad x \geq 0 \\
&...
- 6,206
4
votes
1
answer
73
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How to know if a combinatorial optimization problem is linear or not?
I want to know if a combinatorial problem like the knapsack problem is linear or not. And how do we know if a given problem is convex or not?
- 335
3
votes
2
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254
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How to make following constraint a convex one?
I would like to write a constraint as follows, where $x,y>0$ are optimization variables, and $a,b,c,d,A$ are positive constants. How to make it a convex constraint?
\begin{equation}
\frac{{ax}}{{\...
- 177
2
votes
3
answers
135
views
Convex optimization on the unit hypercube with gradients and a bounded minimum
I'd like to find the minimum of a smooth, continuous function inside the unit hypercube (the dimensionality of which could go into the hundreds or even thousands). The function is convex (Hessian $\...
- 123
1
vote
1
answer
222
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How to convexify log(convex) function?
I have the following optimization problem: \begin{align}\max_x&\quad\log_2(1+|a+bx|^2+cx^2)\\\text{s.t.}&\quad0\le x\le1\\&\quad(1-x^2)\ge\text{constant}\end{align} where $a$ and $b$ are ...
7
votes
2
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137
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Is $\min \ x^3 \ \mathrm{s.t.}\ x \geq 0$ a convex problem?
The problem $$\min \ x^3 \ \mathrm{s.t.} \ x \geq 0$$
is sometimes said to be a convex optimization problem. $f(x) = x^3$ is not a convex function. However, in the domain of $x\geq 0$ it is convex. So ...
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3
votes
0
answers
91
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Proving convexity for a function with summation and integer variable
I would like to show that the function $f$ is convex in $\rho\in [0,1)$ under $s\in \mathbb{Z}^+$. When I use Sympy packages of Python to find $\displaystyle\frac{\partial^2 f(\rho)}{d\rho^2}$. I get ...
- 1,229
1
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0
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71
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Decomposition of Polyhedra
There is no doubt that clear examples consolidate the understanding of concepts being learnt. I am new to finding the structure and decomposition of a polyhedra. Suppose that we have the system
$$ \...
- 111
1
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1
answer
76
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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
...
- 395
4
votes
1
answer
101
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:
$$f(y) \geq ...
- 89
3
votes
1
answer
86
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How to evaluate the convexity of an optimal control problem?
Can we consider an optimal control problem, a convex optimization problem like static optimization problems? If it is possible, under what conditions, will this problem be a convex problem? For ...
- 2,083
1
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0
answers
54
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$\nabla_y\nabla_vf(x^*)\geq0$ for any concave $f$ if and only if $y=-v$
$f:\mathbb R^3\to\mathbb R$ is an arbitrary concave function.
$H$ is a plane. $v$ is a given vector on $H$.
$x^*=\max_{x\in H} f(x)$
Prove that $\nabla_y\nabla_vf(x^*)\geq 0$ if and only if $y=-v$.
I ...
- 179
2
votes
1
answer
75
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Quasi-convex function must be "partially monotonic"?
$f(x)$ is quasi-convex,
$$x^*\in\arg\min_{x\in C}f(x).$$
How to prove that, for any $a\in C$, $f(x) $ is weakly monotonic in the direction of $(x^*-a)$?
Is this simple result a part of an ancient ...
- 179
6
votes
1
answer
162
views
Convexity of the variance of a mixture distribution
$X$ is a random variable that is sampled from the mixture of uniform distributions. In other words:
$$X \sim \sum_{i=1}^N w_i \cdot \mathbb{U}(x_i, x_{i+1}),$$
where $\mathbb{U}(x_i, x_{i+1})$ denotes ...
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4
votes
1
answer
180
views
Can we get closed form solution for such a problem?
\begin{align}\min&\quad\sum_{i=1}^N\frac{A_i}{x_i}\\\text{s.t.}&\quad\sum x_i \le X\\&\quad x_i \ge 0\end{align}
wherein $A_i>0, (i\in\{1,\dots,N\})$ is constant, $x_i, (i\in\{1,\dots,N\...
- 177
1
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0
answers
87
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Question on quadratically constrained quadratic program
If the constrained optimization problem is a quadratically constrained quadratic program of the form \begin{align}\min&\quad x^HQx-a(x+x^H)+b|z^Hx|^2\\\text{s.t.}&\quad\|x\|^2\le1\end{align} ...
- 19
5
votes
3
answers
937
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Linear programming convexity
Is it possible for a linear programming model to be non-convex ? If it is, please, provide a simple 2 variables example and explain why it is non-convex.
EDIT 1:
I have been wondering, maybe the ...
- 185
1
vote
1
answer
123
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Simple nonlinear programming using convexity analysis and KKT
I want to solve the following two-variate nonlinear programming using KKT conditions:
$$ \begin{align} \begin{split}
\max \quad & 15 \sqrt{x_{1}} + 16 \sqrt{x_{2}} \\
\text{s.t.} \quad &...
- 381
3
votes
1
answer
198
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Convexity Analysis
For the following function, I am testing the convexity in $\lambda$. All parameters are in $\mathbb{R}^+$.
$$\frac{\left(- \lambda \left(b + \frac{p}{\beta}\right) + 1\right) \left(\left(1 + \frac{p}{...
- 1,229
8
votes
2
answers
123
views
how to penalize a shortfall of a sum of absolute values
I have a model where there is a constraint on the sum of absolute values, and I would like to add a penalty on the shortfall from the constraint. More specifically:
\begin{align*}
\text{maximize}\ &...
- 542
5
votes
1
answer
108
views
Concavity: Definition similar to sub-modularity
We know that for $b \geq a$, and some $s \geq 0$, a concave function $f$ satisfies:
$f(a+s) - f(a) \geq f(b+s) - f(b)$.
This is not a frequent definition of concavity, but can be found, e.g., here....
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9
votes
1
answer
428
views
Compute the distance from a point inside a convex set to the boundary of the set
Problem
Let $\mathcal C = \{ X \in \mathbb{R}^n \mid g(X) \leq 0\}$ where $g$ is convex, and let $X_c \in \mathcal{C}$. Is there any algorithm to compute the distance from $X_c$ to the boundary of $\...
- 487
6
votes
2
answers
456
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Existence of Optimal Solution
Assume we are solving $\min\{f(x) \ | \ x \in S \}$.
If $f: \mathbb{R}^n \mapsto \mathbb{R}$ is a proper closed convex function, and $S$ is a non-empty closed convex set, does this imply that the ...
- 3,786
3
votes
2
answers
260
views
Convexity of a function
I would like to show that this function
$$2x^2 + 8y^2$$
is convex or concave by using the definition
$$f(θx+(1−θ)y) \le θf(x)+(1−θ)f(y)$$
From my understanding, using the Hessian matrix, I believe ...
- 135
5
votes
0
answers
911
views
Convexity of the projection of a convex set
Question:
A set $S \subset \mathbb{R}^m \times \mathbb{R}^n$ is convex. Using the fact that affine maps preserves convexity prove that $S(y) = \{x \in \mathbb{R}^m\mid (x,y)\in S \}$ and $\hat{S} = ...
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8
votes
2
answers
227
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Convex Optimization: Separation of Cones
I am trying to solve Exercise 2.39 at Boyd and Vandenberghe's Convex Optimization book. In one source, the answer is given as:
2.39 Separation of cones. Let $K$ and $\tilde K$ be two convex cones ...
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4
votes
1
answer
251
views
How to express this constraint?
I have the constraint \begin{align}\max&\quad\gamma\\\text{s.t.}&\quad a\ge\gamma b\\&\quad\gamma\le 1\end{align} where $\gamma$ is an optimization variable and $a$ is a function of some ...
- 2,209
9
votes
1
answer
93
views
Problem solvable $\Rightarrow$ subproblems solvable if feasible region closed, convex?
Let $c \in \mathbb{R}^n$, $M \subseteq \mathbb{R}^n$ such that the problem
\begin{align}P:\quad\min_{x \in \mathbb{R}^n}&\quad c^\intercal x\\\textrm{s.t.}&\quad x \in M\end{align} is solvable....
- 193
8
votes
1
answer
256
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Convexity/Concavity of Average Number of Jobs in M/M/1 Queue?
I am working on a problem involving the average number of jobs $L$ in an M/M/1 queue with arrival rate $\lambda$, service rate $\mu$. For traffic intensity $\rho = \frac{\lambda}{\mu}$,
$$
L = \frac{\...
- 333
10
votes
1
answer
241
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 ...
- 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 ...
- 2,259
9
votes
2
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396
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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 ...
- 705
9
votes
3
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653
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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} & ...
- 2,285
11
votes
2
answers
503
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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 ...
- 11.4k
18
votes
3
answers
4k
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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 ...
- 2,039
15
votes
1
answer
388
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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 ...
- 3,786
17
votes
5
answers
4k
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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 ...
- 2,039
12
votes
2
answers
517
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 ...
- 221
13
votes
1
answer
500
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 ...
- 11.4k
9
votes
2
answers
256
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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&...
- 101
12
votes
1
answer
241
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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 = ...
- 1,311
7
votes
2
answers
419
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 ...
- 2,209
15
votes
2
answers
164
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, ...
- 948
7
votes
1
answer
441
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,...
- 1,589
12
votes
2
answers
934
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 ...
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20
votes
2
answers
2k
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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 ...
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