Questions tagged [convex-optimization]
Convex minimization, a subfield of optimization, studies the problem of minimizing convex functions over convex sets. The convexity property can make optimization in some sense "easier" than the general case - for example, any local minimum must be a global minimum.
50
questions
2
votes
1answer
48 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
91 views
Find the dual problem of $\min_x\{||x-a_1||+||x-a_2||+||x-a_3||,a_i\in\mathbb{R}^n\}$
Find the dual problem of
$$\min_x\{||x-a_1||+||x-a_2||+||x-a_3||,a_i\in\mathbb{R}^n\}$$
I've tried the following but got stuck
$$\min_x\{||x-a_1||+||x-a_2||+||x-a_3||,a_i\in\mathbb{R}^n\}=\min_{x,z_i}...
3
votes
0answers
48 views
Prove $\sum_{i=1}^{m}\lambda_i^*\leq\frac{f(\hat{x})-f^*}{\underset{i=1,\ldots,m}{\min}(-g_i(\hat{x}))}$
Consider the primal problem \begin{align}f^*=\min&\quad f(x)\\\text{s.t.}&\quad g_i(x)\le0\tag P\end{align} where $f,g_i$ are convex functions. Suppose there exists $\hat{x}$ such that $g_i(\...
6
votes
1answer
73 views
Prove that $x^*$ is an optimal solution where $f_0$ is $C^1$ and convex and $f_i$ are $C^1$ and strictly convex functions
Let $x^*$ be a feasible solution of the following convex optimization problem \begin{align}\min&\quad f_0(x)\\\text{s.t.}&\quad f_i(x)\leq0,i=1,\ldots,m\end{align} where $f_0$ is $C^1$ and ...
-3
votes
0answers
54 views
How can I make the constraint based on logarithm and absolute value linear?
I have two constraints of the form
$$\log\left(1+|b_k|^2+\sum_{j=1,j\neq k}^{K}|b_j|^2\right)\ge \log(2)x_k+y_k\quad\forall k, k=1,\cdots,K$$
and
$$1+\sum_{j=1,j\neq k}^{K}|b_j|^2 \le e^{y_k^{(0)}}(...
2
votes
0answers
85 views
Prove Non-Homogeneous Farkas' Lemma
Let $A\in\mathbb{R}^{m \times n}, c\in\mathbb{R}^{n}, b\in\mathbb{R}^{m}, d\in\mathbb{R}$. Suppose that there exists $y\geq0$ such that $A^Ty=c$.
Question: prove that exactly one of the following is ...
2
votes
0answers
36 views
active set method guaranteed convergence
I'm using Active Set Method to solve a nonlinear optimization function minimizing a convex function over a polyhedron of 2 linear inequalities starting at an interior point $x_o$ At this point is it ...
2
votes
0answers
66 views
How to linearize this multiplicative constraint?
I have a constraint in the form
$\sqrt{|\sum_{c\in C}{h_cw_c}|^2}\ge\sqrt{x}\zeta$
Here, $h_c$ is s row vector (know), $w_c$ is a column vector (variable).
$x$ and $\zeta$ are also optimization ...
4
votes
2answers
87 views
Let $A\in\mathbb{R}^{m\times n},c\in\mathbb{R}^n$. Show that exactly one of the following two systems is feasible:
Let $A\in\mathbb{R}^{m\times n},c\in\mathbb{R}^n$. Show that exactly one of the following two systems is feasible:
$Ax\geq0,x\geq0,c^Tx>0$
$A^Ty\geq c,y\leq0$
Assume that A is feasible meaning $...
0
votes
1answer
33 views
How to convert an element of a variable to a convex constraint using binary variables?
I defined a complex variable in cvx, but I want to restrict the first element of the variable to be larger than the max of the variable, but it doesn't work. Someone told me to transform it using a ...
-2
votes
1answer
59 views
How can I model this Hyperbolic constraint?
In this problem, $\beta_u$, $w_{u,c}$ (a vector of complex elements), $x_u$ are optimization variables.
Now,
$||2\sqrt{\frac{\pi_u}{2}}\beta_u; h_{u,c}^{\rm H}w_{u,c}-\frac{1}{2\pi_u}x_u-1||_2\le h_{u,...
2
votes
1answer
66 views
How to model these constraints correctly
$W$ is a vector of $N$ complex elements.
$D$ is a binary variable
The requirements are:
when $D==1$, $L_{\min}\le ||W||_2^2\le L_{\max}$
and when $D==0$, $||W||_2^2=0$
I have formulated the following ...
3
votes
0answers
24 views
Stationary conditions for intersection
I wondered about this question for sometime.
Definition of Stationarity
(P)
$\mbox{min} f(x)$
s.t
$x\in C$
Let $f$ be $C^1$ function over a closed and convex set $C$ . then $x^*$ is called a ...
2
votes
3answers
329 views
Find the farthest point in hypercube to an exterior point
Let $\mathcal{U} = \{ [x_1, ..., x_n] \in \mathbb{R}^n : 0 \leq x_i \leq 1\}$ be the unit hypercube and $C \in \mathbb{R}^n\setminus\mathcal{U}$ fixed. Let us consider the following problem
$$ \max_{X ...
1
vote
1answer
80 views
How to prove this convex-optimization problem?
I am struggling with the following optimization problems.
Problem 1
\begin{align}\max_{\alpha, s_1, s_2}&\quad s_1 + s_2 - \gamma (s_1 (K_1 +c_1 + s_1) + s_2 (K_2+ c_2 + s_2) + 2\alpha K) +C\\\...
4
votes
2answers
202 views
How to solve this convex problem heuristically?
I have the following problem
$$\max_{X_{i,j},i\in N_{U},j\in N_{B}}\sum_{i=1}^{N_U}\sum_{j=1}^{N_B}R_{i,j}X_{i,j}$$
$$\text{subject to}$$
$$a_{\min}\le\sum_{j=1}^{N_B}X_{i,j}\le a_{\max}, \forall i$$
$...
6
votes
0answers
121 views
Is this a valid strong polynomial algorithm for deciding LP feasibility?
Let
$$A \cdot X + B \preceq 0$$ be a system of linear inequalities with $X \in \mathbb{R}^n$ $A\in \mathbb{R}^{m\times n}$ and $B \in \mathbb{R}^m$ where $m \geq n$. According to Farkas lemma, exactly ...
6
votes
0answers
71 views
Polyhedra to Simplex by using convex combination of vertices
Optimization problems over linear constraints (defining a convex polyhedron) can be written as optimization over a simplex in a higher dimension. Let $\mathcal{P}$ be a bounded polyhedron, and the ...
2
votes
0answers
33 views
Optimization of strongly convex functions with approximate evaluations of gradient and Hessian
Suppose I want to find the minimum of a differentiable, strongly convex function $f:\mathbb{R}^n\to\mathbb{R}$ with constant $\mu>0$. That is, for all $x,y\in\mathbb{R}^n$, I have that:
$$f(y) \geq ...
2
votes
0answers
59 views
Optimizing with a logistic function
I have a system in which I want to maximize the value of some function $f(x_T, y_T)$.
The time evolution of the system is described by some functions:
$$
\begin{align}
\frac{dx}{dt}&=\alpha \frac{...
4
votes
1answer
61 views
How to evaluate the convexity of an optimal control problem?
Can we consider an optimal control problem, a convex optimization problem like static optimization problems? If it is possible, under what conditions, will this problem be a convex problem? For ...
1
vote
0answers
53 views
How to solve this problem by Lagrange duality?
This is a convex problem and although it can be well solved by CVX, I want to know how it can be solved by the Lagrange duality method. The derivations with regard to $L_k$ and $B_k$ are constants, ...
4
votes
1answer
130 views
Can we get the closed-form solution for this problem?
Can we get the closed-form solution for this problem?
\begin{align}
\min&\quad\sum_{i=1}^N\frac{K_i}{x_i\log_2(1+\frac{Q_i}{x_i})}\\
{\rm{s.t.}}&\quad\sum_{i=1}^N x_i\le X
\end{align}
wherein $...
1
vote
0answers
44 views
$\nabla_y\nabla_vf(x^*)\geq0$ for any concave $f$ if and only if $y=-v$
$f:\mathbb R^3\to\mathbb R$ is an arbitrary concave function.
$H$ is a plane. $v$ is a given vector on $H$.
$x^*=\max_{x\in H} f(x)$
Prove that $\nabla_y\nabla_vf(x^*)\geq 0$ if and only if $y=-v$.
I ...
2
votes
1answer
58 views
Quasi-convex function must be “partially monotonic”?
$f(x)$ is quasi-convex,
$$x^*\in\arg\min_{x\in C}f(x).$$
How to prove that, for any $a\in C$, $f(x) $ is weakly monotonic in the direction of $(x^*-a)$?
Is this simple result a part of an ancient ...
3
votes
0answers
29 views
Linear functions in Lenstra's algorithm
I had asked this question at MathOverflow and was pointed here.
I'm working on implementing Lenstra's algorithm. At the bottom of p.5 (at "construct $n+1$ linear functions"), he says to ...
6
votes
1answer
125 views
Convexity of the variance of a mixture distribution
$X$ is a random variable that is sampled from the mixture of uniform distributions. In other words:
$$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 ...
7
votes
2answers
720 views
Difference between exploration and exploitation in Simulated Annealing algorithm
In evolutionary algorithms, two main abilities maintained which are Exploration and Exploitation.
In Exploration the algorithm searching for new solutions in new regions, while Exploitation means ...
3
votes
2answers
155 views
Can we use reinforcement learning and convex optimization to solve an optimization problem?
For an optimization problem, there are multiple-type variables should be optimized. Can we use the convex optimization method to solve a subproblem of partial variables, and then, with the obtained ...
2
votes
1answer
81 views
Relationship between extreme points and optimal solutions of SDPs
Consider this to be our SDP problem:
Minimize $\langle C, X \rangle$ such that
$\langle A_i, X \rangle \ge b_i$ for all $i \in [m]$ and
$X \succcurlyeq 0$.
For SDPs, is there a relationship between ...
4
votes
1answer
81 views
Conditions required for strong duality to hold for SDPs
According to Wikipedia, strong duality holds when "the primal optimal objective and the dual optimal objective are equal."
What are the necessary conditions for strong duality to hold in ...
2
votes
0answers
65 views
Can every convex problem use Lagrangian dual method?
If not all constraints satisfy equalities, does Lagrangian dual method make sense to a convex problem?
2
votes
1answer
103 views
Is a convex or MILP (without big-M) formulation possible for this problem
Assume we are given a directed acyclic graph (DAG) $G(V, A)$, where $|V| = n, |A| = m$, and the graph contains a source node $\mathbf{s}$ (i.e. every node in $V \backslash \mathbf{s}$ is connected by ...
1
vote
0answers
78 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} ...
3
votes
2answers
126 views
DCP representation of a convex quotient of affine functions
I am trying to represent the following inequality:
$$\frac{x}{1-x} \leq y \qquad\mathrm{with}\qquad 0<x<1$$
The function on the left is convex (its second derivative is always positive over ...
6
votes
2answers
139 views
Find a point inside non-empty difference of ellipsoids
Given two ellipsoids \begin{align}\mathcal{E}_1 &= \{ X \mid X^\top A_1 X + 2B_1^\top X + C_1 \leq 0\}\\\mathcal{E}_2 &= \{ X \mid X^\top A_2 X + 2 B_2^\top X + C_2 \leq 0\}\end{align} are ...
4
votes
1answer
59 views
How to check for convexity of the inequality constraint $āx^2+yā1\ge0$ for a minimization objective function?
I checked the Hessian which is $\begin{bmatrix}-2&0\\0&0\end{bmatrix}$ which is negative semidefinite but according to the sketch of the function it is convex. What am I missing?
3
votes
1answer
158 views
Approximation methods for a mixed integer convex optimization problem
I have a convex objective function, e.g., minimizing the negative entropy function. My constraints are also linear. The only issue is that I also have binary variables.
I am currently aware of AIMMS'...
5
votes
1answer
114 views
Which solver solves PSD constrained convex non-linear problem
I have a problem with a vector variable $w \in \mathbb{R}^n$ and a symmetric matrix variable $V \in \mathbb{R^{n \times n}}$. I am solving a problem which is roughly like:
\begin{align}
\begin{array}{...
4
votes
1answer
103 views
Minimize a convex function over a sphere
Problem description
Let $\mathcal{C} = \{X \in \mathbb{R}^n \mid g(X) \leq 0\}$ with $g(X)$ a convex function. Suppose I need to solve the feasibility problem, for a given $r>0$
$$ \exists ^?X \...
3
votes
0answers
71 views
Strong Duality and Slater Condition
I am studying the Duality Chapter of Convex Optimization by Boyd. Is it possible that strong duality holds for non-convex optimization? If yes, is there any specific condition? And, what is the ...
6
votes
2answers
146 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 ...
4
votes
1answer
150 views
Cutting-planes application procedure for a specific problem
Sort of following up with this question. I reformulated another model to make it convex and possibly solve it with some cut generation method. I would like to double-check whether I am doing it ...
7
votes
1answer
88 views
Ridge Regression lagrange duality
In every machine learning book we see that it is roughly mentioned that the ridge regression:
$$p_1^* = \min\limits_{\beta} \ \left( \mathrm{RSS} + \lambda\sum_{j=1}^p \beta_j^2 \right)$$
is ...
6
votes
0answers
51 views
Semi-definite Programming, non standard notation
The usual way to define a semi-definite program (SDP), e.g., as given in Boyd and Vandenberghe's convex optimization book, is:
$$
\begin{array}{cl}
\min & c^\top x \\
\mathrm{s.t.} & 0 \succeq ...
5
votes
0answers
378 views
Convexity of the projection of a convex set
Question:
A set $S \subset \mathbb{R}^m \times \mathbb{R}^n$ is convex. Using the fact that affine maps preserves convexity prove that $S(y) = \{x \in \mathbb{R}^m\mid (x,y)\in S \}$ and $\hat{S} = ...
8
votes
2answers
146 views
Convex Optimization: Separation of Cones
I am trying to solve Exercise 2.39 at Boyd and Vandenberghe's Convex Optimization book. In one source, the answer is given as:
2.39 Separation of cones. Let $K$ and $\tilde K$ be two convex cones ...
8
votes
4answers
324 views
Disciplined convex programming representation of $x\sqrt{1-x}$
Anyone have an idea for a disciplined convex programming (DCP) representation of the concave function $x\sqrt{1-x}$, which is defined over the domain $[0,1]$?
The Taylor series about $x=0$ is $$x - \...
9
votes
1answer
241 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{\...
12
votes
2answers
418 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 ...
