Questions tagged [convex-optimization]
Convex Optimization is a special case of mathematical optimization where the feasible region is convex and the objective is to either minimize a convex function or maximize a concave function.
215
questions
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CVXOPT intermediate step valuation stepping out of function domain of defintion
I am using CVXOPT, particularly to solve a nonlinear convex optimization problem. Either the objective function or the constraints involve some functions that are only defined in a strict subset of $\...
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votes
0answers
75 views
Reformulate a maximization into a minimization problem
I have the following maximization problem:
$$\max_{\mathbf{w}, t, \Theta} t$$
$$\text{s.t. } ||\mathbf{w}||^{2} \leq P$$
$$\mathbf{S} \succ 0$$
$$||\mathbf{u}^{(n)}||+2\sum_{i=1}^{2} \mathbf{u}^{(n)...
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0answers
40 views
L2 norm optimization problem
I have an optimization problem where i need to find an image x, that is very close to x' such that:
monitor(x') is valid but monitor(x) is invalid. (output is valid
when the neural network output is ...
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0answers
19 views
Separating hyperplane [closed]
Let C be finitely generated cone and let x∉C.
How can we show that there can only be finitely many separating hyperplanes between C and x?
4
votes
1answer
58 views
SCP (Sequential Convex Programming) vs SQP (Sequential Quadratic Programming)
Can someone explain me at a high level the difference between an SCP and an SQP to solve a nonlinear (nonconvex) program?
Assume my problem is something like
$$\min\limits_x. \quad f(x)$$
$$s.t. \...
3
votes
0answers
49 views
What is this QR-factorization-based preconditioning called?
I have recently started to delve into someone else's code, and there is a part in there I don't quite understand. The authors of the code use some form of pre-conditioning to speed up the optimization....
2
votes
0answers
103 views
Understanding Illumination Optimisation Problem
I am a newbie to convex optimisation and I am learning with the aid of CVXPY. I am requesting for clarity on the illumination problem as described in Boyd & Vandenberghe lecture 1 slides here. I ...
2
votes
1answer
51 views
Why is CVXPY throwing a DCP error with cp.sqrt but no error with cp.norm
I am trying to use CVXPY to optimise signal-to-noise-plus interference ratio (SINR) for a visible light communication (VLC) system. I have one of my SINR constraints stated as
\begin{equation}
\...
2
votes
1answer
68 views
How to best code a problem with scipy, cvxpy or Convex.jl with given generated data
I have a curve fitting problem of the form:
$$
\textbf{y} = f(\textbf{x}, a,b,c,d) + \varepsilon
$$
$$
f(x, a,b,c,d) = \frac{b}{e^{x\cdot a}+c}+d
$$
with the constraint
\begin{equation}
\begin{aligned}...
2
votes
2answers
129 views
Solving a specific sparse linear system without dense materialization
I need to (computationally) solve a system of equations, for the purposes of an interior point method, of the form
$$
\left[\begin{array}{cc}B & A^T \\ A & 0\end{array}\right] \left[\begin{...
3
votes
1answer
69 views
Reformulating a convex optimization problem with $x \mapsto \max(x,0)$ in the constraint
I am wondering if there is a well-known transformation allowing one to solve convex optimization problems of the form
$$\begin{array}{ll} \underset{x}{\text{maximize}} & r^T x\\ \text{subject to} &...
5
votes
0answers
94 views
Generally quadratic convex problem with one non-convex term
How would you approach a standard convex quadratic problem with convex constraints but one non-convex term ? Say $|x|^{0.4}$.
$$\min_x \frac{1}{2} x^{T}Qx + g^Tx + c^T \mathrm{sign}(x) |x|^{0.4} $$
...
2
votes
1answer
75 views
Project to nearest point on convex polyhedron
I have a point $y \in \mathbb{R}^d$ and a convex polyhedron $\mathcal{P}$ given as the intersection of half-spaces:
$$\mathcal{P} = \{x \in \mathbb{R}^d \mid a_1 \cdot x \le b_1, \dots, a_n \cdot x \...
0
votes
1answer
122 views
Norm constraint in CVXPY
I'm trying to implement the algorithm outlined in https://arxiv.org/abs/1211.5608 on a small scale. I have a linear operator $\mathcal{A}$ which is defined as $$\text{trace}(A^*_l(hm^*))$$ where $$A_l ...
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vote
1answer
82 views
Solving a linear program with an active set method
Is it possible to solve a linear program with an active set method? If so what would be the similarities and differences to the simplex method?
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0answers
259 views
Finding points inside cells of power (generalized Voronoi) diagram
Suppose we have a set of points $p_1,\ldots,p_n\in\mathbb R^d$ as well as a set of weights $w_1,\ldots,w_n\in\mathbb R$. Recall that the power cell associated to the pair $(p_k,w_k)$ is given by:
$$\...
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0answers
18 views
Convex performance measure of classification
In the context of binary classifcation methods, I am looking for a performance metric that can be optimized in MATLAB.
Since the data is not balanced, a good choice seems to be the so-called F1-...
0
votes
1answer
68 views
3
votes
2answers
355 views
log(det(X)) in Semidefinite Programming
I have been solving problems of the form $$max \ log(det(A)) \\ s.t. \ A = A^{T} \succeq 0, \\ p_{i}^{T}Ap_{i} \leq b_{i}$$ where $b_{i}$ and $p_{i}$ are input vectors (to be clear there is more than ...
4
votes
1answer
178 views
Underdetermined Minimum Volume Enclosing Ellipsoid
Given three vectors in $\mathbb{R}^{512}$, my task is to compute a Minimum Volume Enclosing Ellipsoid (MVEE). I have tried Kachiyan's algorithm, but it requires at least as many vectors as there are ...
1
vote
1answer
85 views
In which cases does the nonlinear conjugate gradient method take more than $n$ steps?
I have programmed a couple of Matlab implementations of nonlinear Conjugate Gradient methods (Fletcher Reeves and Polak Ribeire). However, I am concerned with how many steps it's taking to optimise ...
2
votes
0answers
102 views
Proving convexity of Frobenius norm and correlation function formulations of an optimization problem
I have been working on formulating my requirements in the form of an optimization problem in a multi-output regression setting.
Firstly, I would like to make the variables I used in the problem and ...
2
votes
0answers
49 views
Interior point of convex polytope
Suppose the convex polytope is the set of feasible solutions $\mathbf{x}\in\mathbb{R}^n$ for the linear system $\mathbf{A}\mathbf{x}=\mathbf{b}\,,\; \mathbf{A}\in\mathbb{R}^{m\times n}$ subject to ...
2
votes
1answer
77 views
Could the convex problem be tackled by CVX?
I want to solve the convex optimization as follows:
\begin{align}
\underset{X_1,X_2}{\min} &\ -\frac{1}{N}\sum_{i=1}^N\log\det\left(I+H_i^HX_2H_i\right)-\log\left[1+h^H(X_1+X_2)h\right]\\
&\...
1
vote
1answer
85 views
Disciplined convex programming expression of $x\sqrt{1-x}$
Anyone have an idea for a DCP (disciplined convex programming) representation of the concave function $x\sqrt{1-x}$, which is has domain $[0,1]$?
The Taylor series about $x=0$ is
$$x - \frac{x^2}{2}...
3
votes
1answer
162 views
Gradient descent in constrained optimization of barrier function
This question may be too basic, but I was wondering if it is possible to implement simple methods such as gradient descent or its variations to find the minimum of barrier functions in constrained ...
5
votes
0answers
104 views
Minimum of quadratic assignment (QAP) with convex objective
Suppose $A,B\succeq0$ and $C\in\mathbb R^{n\times n}$. I am hoping to solve an instance of the following optimization problem:
$$
\min_{\textrm{permutation matrices }P}
\mathrm{tr}(BP^\top AP+C^\top ...
6
votes
1answer
72 views
Sparsity-Promoting Convex Optimization Over Simplex
Say we want to find a sparse approximate minimizer to the function $f(x) : \mathbb{R}^d \to \mathbb{R}$. Then in line with the work in the field of compressed sensing, we can instead minimize $$f(x) + ...
3
votes
1answer
360 views
Convex optimization with constraints involving matrix inverse
I have the following convex optimization problem. I would like to ask is there any efficient way to solve it in Python? Can I use CVXOPT package? If so, any detailed instruction? Thanks a lot.
$$
\...
3
votes
0answers
108 views
A maximization problem, with motivation in machine learning
Consider the minimization problem described this paper. Let $f_{\lambda}$ be the minimizer. As a part of extending my work, I am able to show the following facts
$$\lim_\limits{\lambda \to 0}\|f_{\...
4
votes
1answer
84 views
Geometric Programming - symbolic version
I am interested in finding minimizers of functionals of the type $\sum x^ay^bz^c$ where the exponents are 1, 0 or -1. I have codes to find such minimizers when they exist up to machine precision, ...
3
votes
2answers
363 views
Convexity of Sum of $k$-smallest Eigenvalue
If I have a real positive definite matrix $A\in\mathbb{R}^{n\times n}$, and denote its eigenvalues as $\lambda_1\leq \lambda_2 \leq ... \leq \lambda_n $.
Define the function as $f(A)=\sum_{i=1}^{k} \...
4
votes
1answer
53 views
Determine image of hypercube under linear map
Let $A$ be an $3\times N$ matrix (where $N$ is large) with nonnegative real entries. I'd like an algorithm for determining when a vector $v\in\Bbb R^3$ can be written as $Aw$ for some vector $w\in\Bbb ...
2
votes
1answer
132 views
Optimize multivariable function with interdependent variables
I have a cost function with 2 parameters. The variables are dependent on each other. So, if I just take a partial derivative with respect to one variable the slope is in terms of the other variable ...
2
votes
0answers
46 views
Biconvex problem whose objective function depends on only one variable
I am solving the following biconvex problem:
$$\min_{x,y} f(y)$$
$$s.t. ~~ g(x) \leq 0$$
$$~~~~~h(x,y) = 0$$
$$x \in X, y \in Y$$
where $X$ and $Y$ are compact convex sets, $g(x)$ and $f(y)$ are ...
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votes
0answers
33 views
Minimizing the ratio of two specific non negative quadratic convex functions
$F$ is $m\times m$ diagonal, with real non negative elements
$D$ is $n \times m$ complex
$P$ is $n \times 1$ complex
$A$ is $m \times 1$ complex.
Minimize $\Gamma(A)$, with respect to $A$.
$$\...
6
votes
1answer
115 views
numerical solution of an under-determined linear equation in high dimensions
I need to solve a linear regression problem $$Ax=y$$ which is hugely underdetermined. I have around $10^6$ features but only $10^3$ equations. So $A$ is a $1,000\times 1,000,000$ matrix and $y$ a ...
0
votes
1answer
71 views
Formulate and solve a simple conic programs in cvxpy language [closed]
Let $r,\epsilon > 0$ and $a, b \in \mathbb R^n$ with $\|a\|_2 \le r$. Define $C(a) := \{x \in \mathbb R^p | \|x+a\|_2 \le r,\;\|x\|_\infty \le \epsilon\}$, and assume it is non-empty.
Question
(A)...
0
votes
1answer
30 views
Gradient ascent method with a constant step size?
I'm trying to use the gradient ascent method on a convex function like the multivariate-Normal density function with respect to its parameters (the original is a bit more complicated), something ...
2
votes
1answer
115 views
Question about strange outputs from the CVXPY solver
I am familiarizing myself with CVXPY, and encountered a strange problem. I have the following simple toy optimization problem:
...
3
votes
0answers
92 views
Why the MIRACLE of Lanczos/CG-like?
Lanczos/Arnoldi/Rietz/CG-like algorithm share the same core strategy...
In each, a little miracle appears, most of the Gram-Schmidt inner products are zeroes! In others words, new direction need only ...
-1
votes
1answer
25 views
Semi-Definite relaxation of non-linear constraint?
I am implementing an optimization problem using semi-definite approach. One of my constraints is of following form
$ trace(A∗X)−(k∗trace(A∗X))+(k∗\sqrt {(trace(B∗X)} )==0$
where k is a constant, A ...
2
votes
0answers
39 views
Where can I find sample data for large linear programming optimization problems?
I am doing a comparison of different algebraic modeling langues (AMPL, AIMMS, GAMS, Pyomo) in both theoretical and practical terms. As a practical experiment I am trying to measure problem model ...
1
vote
1answer
453 views
Why does Newton's method with Linear Equality Constraints use KKT condition?
Goal: Optimize convex function $f(\vec{x})$ subjected to constraint $A\vec{x} = \vec{b}$ starting at a point $\vec{x}_0$ that satisfies the constraint.
The problem only has equality constraint. Why ...
1
vote
0answers
47 views
Order of a principal term
In Yurii Nesterov's Introductory Lectures on Convex Optimization, there is a bound for the total number of iterations for some process. See page 109:
$$\left[\frac{1}{\ln(2(1-\kappa))} \ln\frac{t_0-t^...
2
votes
1answer
128 views
Convergence rate and complexity for convex minimization problem
In Yurii Nesterov's Introductory Lectures on Convex Optimization, there is a description of the rate of convergence and corresponding upper bound for the analytical complexity of a minimization ...
3
votes
0answers
96 views
First order methods for a large scale semidefinite program
I am interested in solving the following semidefinite optimization problem:
\begin{equation}
\begin{split}
\underset{X,\lambda}{\rm maximize} \;\;\;\;&\lambda^Tc \\
&-\mathbb{I} \le X \le \...
4
votes
1answer
1k views
How to transform this SOCP to the format required by cvxopt
I'm new to SOCP and want to try to get familiar with the format and how to solve it with cvxopt in python. However, for a simple toy example I'm struggling to get ...
0
votes
1answer
2k views
How to efficiently solve a QCQP with “dynamic” constraints in Python?
I want to solve a QCQP in Python. It is a problem from finance: maximise return (linear function) given some linear constraints and one quadratic constraint that turns it into a QCQP. Formally,
$$\...
1
vote
1answer
222 views
How to use CSDP to express a semidefinite program?
I am trying to use CSDP and am struggling with it. Consider, for example, the following semidefinite program
$$\begin{array}{ll} \text{minimize} & 0\\ \text{subject to} & Q - A' Q A - \...
