Questions tagged [quadratic-programming]
Questions on quadratic programming, the optimization of a quadratic objective function subject to affine constraints.
0
votes
0answers
34 views
Why is positive definiteness required for a global minimum to exist?
When we optimize
$$\min x'Ax$$
why does matrix $A$ need to be positive definite in order to have a global minimum?
0
votes
1answer
33 views
Solving weighted least squares with non-negative constraints
I have the optimization problem
$$
\begin{align}
\min_{\mathbf{P} \geq 0} \|\mathbf{A\odot(X-PQ^\top)}\|^2 + \frac{\|\mathbf{P}\|^2}{2}
\end{align}
$$
$\odot$ is the Hadamard product, $\mathbf{A,X,P,...
1
vote
2answers
57 views
Explanation of a convex quadratic program
I want to find the minimum of the following optimization problem but don't even understand the problem in the first place.
$$\min\limits_{AX=b}\frac{1}{2}X^{T}QX+C^TX+\alpha$$
where $Q, A \in M_{n \...
0
votes
0answers
27 views
Why would sequential quadratic programming fail to find global minimum?
I have a data set. A matrix $X$, $1300 \times 20$ and output vector $\mathbf{y} \in \Bbb R^{20}$ $$\mathbf{y} = \begin{bmatrix} 100\\100\\\vdots\\100\end{bmatrix}$$
I am trying to run OLS on this ...
1
vote
1answer
49 views
Best approximation of an arbitrary matrix by a distance matrix
A distance matrix $D$ is a $n\times n$ matrix obtained by computing the pair-wise distances $D_{ij} = \text{dist}(x_i, x_j)$ between a collection of points $\{x_1,\ldots, x_n\}$ in some metric space. ...
0
votes
0answers
34 views
Minimum of sum of six squares over a polytope
$$\begin{array}{ll} \text{minimize} & x_1^2 + x_2^2 + \cdots + x_6^2\\ \text{subject to} & x_1 + x_2 + \cdots + x_6 = K\\ & x_1 + x_2 \ge k_1\\ & x_1 + x_2 + x_3 + x_4 \ge k_2\\ & ...
2
votes
0answers
42 views
Minimum of $\langle Ax,x \rangle - 2 \langle b,x \rangle$.
In exercise 5 of section 0 of Fundamentals of convex analysis by Hiriart-Urrut, Lemaréchal, we're supposed to prove that if a self-adjoint linear operator $A:\mathbb{R}^n \rightarrow \mathbb{R}^n$ is ...
0
votes
0answers
56 views
How to solve this constrained maximization problem?
Does anyone have an idea of how to tackle the following maximization problem?
Maximize the function $ f(x,y,z) = x - y - \alpha z^2 $, $ \alpha > 0 $, under the following constraints:
C1:...
0
votes
1answer
22 views
Total Unimodularity in Integer QP
I have a Quadratic Programming problem with linear constraints. My objective is Quadratic-Convex, the constraint matrix is Totally Unimodular (similar to assignment or network flow problems), and the ...
0
votes
0answers
12 views
Quadratic programming terms
Help me to understand the terms of this quadratic programming problem?
$Z$ and $U$, $x$ and $α$ are fixed. We have to minimize $w$. I would like to put it into terms of cvxopt to solve it.
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0answers
38 views
Are step sizes in quadratic programming solvers analytically exact?
In this paper (DOI link), Goldfarb and Idnani describe an algorithm for solving a certain subset of quadratic programs. This algorithm (or a very similar one) is implemented in the quadprogpp package ...
0
votes
0answers
29 views
Quadratic programming: Approximate Solution
Let $A$ be a $p \times p$, positive definite and symmetric matrix and $t \in \mathbb{R}^p$, such that $t_i>0$ for at least one $i \in \{1,\dots,p\}$. Let $x^*$ be the unique minimizer of
$$min \{x'...
0
votes
2answers
14 views
Does this special case of convex quadratic programming have a partially-unique solution?
I know that a strictly convex quadratic programming problem has a unique solution, but I'm curious about the following situation:
If $Q$ is positive definite, does the following problem:
$$\min\...
0
votes
0answers
13 views
How Sequential Quadratic Programming versus Quadratic programming and Iterative QP are related?
What is the difference between Sequential Quadratic Programming (SQP) versus Quadratic Programming (QP)? Is it the same as Iterative Quadratic Programming (IQP)? For example, BFGS, DFP are types of QP/...
0
votes
1answer
28 views
Methods for solving nonlinear constraints quadratic programming
Are there any other methods to solve nonlinear constraints quadratic programming? I have known that some effective numerical methods, i.e, SQP and Gauss pseudospectral method and some heuristic ...
1
vote
1answer
27 views
quadratic programming /symmetric matrix
I have a quadratic program with $ F: \mathbb{R^n} \rightarrow \mathbb{R}, F(x)=x^TQx$ I want to find a symmetric matrix M for Q, such that $F(x)=x^TMx$ holds for all x.
I can write Q as sum of ...
0
votes
2answers
86 views
Determining the coefficients of $y = a x^2 + b x + c$ so that the graph contains three given points
Determine coefficients of the equation for a second order polynomial in the format: $y = a x^2 + b x + c$
Here are the $x$-$y$ coordinates of three points:
$(0.8143,0.3500)$
$(0.2435,0.1966)$
$(0....
0
votes
1answer
51 views
Unconstrained convex quadratic integer programing
Let $N_i=\{0,1,\dots,\bar{n}_i\}$ and define $N=N_1\times \dots \times N_I$. I have a minimization problem of the form
$$
\min_{n\in N} \sum_i A_i n_i +\sum_i \sum_{j\neq i} B_{ij} (n_i-n_j)^2
$$
...
0
votes
1answer
44 views
Approximate function by stacking building blocks
I need some help with a 'generalised Lego problem':
Given a function $f(x)\geq 0$ on an interval $[a,b]$, and a 'building block' function $p(x)\geq 0$ with compact support. The maximum of f shall be ...
0
votes
1answer
38 views
Linearly constrained quadratic program
I have the following quadratic program
$$\min_x x^TAx \qquad \text{s.t} \quad Ax \in [a,b]^m$$
where matrix $A$ is positive semidefinite, and is similar both the objective function and in the ...
0
votes
0answers
20 views
Is there a time polynomial algorithm to solve a box-constrained quadratic convex program?
I am working on the following quadratic convex problem
$min_{x}x^{T}Qx+f^{T}x$
subject to
$a\leq x\leq b$
where $x,a,b,f\in\mathbb{R}^{n}$ and $Q$ is positive definite. Is there any
algorithm that ...
0
votes
1answer
49 views
Measuring infeasibility in convex optimization, relations with dual problem
A question regarding convex optimization and (maybe) duality.
I have a problem in the form
\begin{align}
x^* = \mathrm{arg} \min_x f(x) \quad \text{s.t.} \quad A x \leq b, C x = d,
\end{align}
where $...
1
vote
0answers
43 views
Derive LCP from KKT conditions of a QP
I'm working through this tutorial on LCPs and interior point methods. In it, the authors claim that the following quadratic program
$$
\begin{aligned}
\min \quad& \frac{1}{2}u^TQu - c^Tu\\
\text{...
0
votes
1answer
54 views
Is there only one unique minimizer for this scalar complex quadratic form?
Consider $c,b \in \mathbb{C}$ and $f: \mathbb{C} \mapsto \mathbb{R}$,
$$f(c) = bc' + b'c + cc'$$
is there only one extrema of $f$ corresponds to $$c^* = -b$$?
where, $'$ means complex conjugate.
...
0
votes
1answer
55 views
Local optimality of non-convex quadratic minimization with linear constraints
We are interested in minimizing a quadratic function, which is not convex. The feasible set is a polyhedron. We know that the global minimum is at a vertex when the function is concave.
Also, there ...
0
votes
0answers
41 views
Quadratic program reformulation maximum to minimum
I am newbie in optimization problem, I have the following optimization problem:
$$\max \quad \frac{1}{2}x^THx - q^THx$$
$$\text{s.t.} \;\;l\leq x \leq u$$
where $H, q$ are known constants, and H is ...
1
vote
2answers
52 views
Quadratic optimization problem with quadratic constraint
Given $A = A^T \in \mathbb{R}^{n \times n}, X = \begin{bmatrix}x & \eta\end{bmatrix} \text{with} \ x, \eta \in \mathbb{R}^n$ and $||x|| = ||\eta|| = 1$ (vectors in $\mathbb{R}^n$ are treated as ...
1
vote
0answers
52 views
Karmarkar's canonical form for convex quadratic programs??
I'm working through this paper A Simple Polynomial-Time Algorithm for Convex Quadratic Programming which gives an interior point algorithm for convex quadratic programming. I've reached the end of the ...
0
votes
1answer
21 views
confusion in proof of lemma for LCP
I'm looking at the proof of lemma 2 on page 6 of https://cowles.yale.edu/sites/default/files/files/pub/d05/d0549.pdf
Which claims that the LCP
$$
Mx+q \geq 0 \\
x \geq 0 \\
x^{T}(Mx+q) = 0
$$
has a ...
0
votes
1answer
56 views
minimum of $\frac{1}{2}x^tQx-c^tx$ subject to $Ax=b$ is local $\iff$ is global
$$\min \frac{1}{2}x^tQx-c^tx\\ Ax=b$$ where $Q\in \mathbb{R}^{n\times
n}$ is symmetric, $c\in \mathbb{R}^n$, $A\in \mathbb{R}^{m\times n}$
and $b\in\mathbb{R}^m$. Prove that if ${x}^*$ is a local
...
3
votes
2answers
229 views
How to minimize $\Vert Ax \Vert$ subject to $\hat{b}^Tx=0$ and $\Vert x \Vert = 1$?
Given $n\times 3$ matrix $A$ and unit vector $\hat{b}\in R^3$, is there a closed-form solution for $x\in R^3$ that minimizes the Euclidean norm $\Vert Ax \Vert$ subject to $\hat{b}^Tx=0$ and $\Vert x \...
0
votes
0answers
11 views
Two dimensional sorting: finding $\max_\sigma \langle M^\sigma,C\rangle$ for permutation function $\sigma$
Suppose that we have a non-negative matrices $C,M\in \mathbb{R}^{m\times m}_{\ge0}$. Also suppose $\sigma:[m] \rightarrow [m]$ is a permutation of $\{1,2,...,m\}$. We want to find the permutation that ...
0
votes
0answers
77 views
Large scale mixed integer (quadratic) programming
Here we have this optimization problem:
Given positive semi-definite matrix $A \in \mathbb{R}^{n \times n }$, and matrix $B \in \mathbb{R}^{n \times m} \text{ and vectors } d \in \mathbb{R}^n,e \in ...
5
votes
2answers
153 views
Transform an optimisation problem into a linearly-constrained quadratic program?
I would like your help with a minimisation problem. The minimisation problem would be a linearly-constrained quadratic program if a specific constraint was not included. I would like to know whether ...
0
votes
1answer
21 views
How to find uppper bound for the hyper parameters in linear model?
I have loss function of the following form:
$$argmin_w= ||Y-w^TX||_2^2 + \lambda||w||_1 + \gamma \sum_{g=1}^G||w_g||_1$$
where $w_g$ is group of coefficents.
Given $Y$ and $X$, I would like to find ...
0
votes
2answers
74 views
Minimise $||Ax - b||_2$ given $x \geq 0$ and $\textbf{1}^Tx = 1$
$$
\min_x ||Ax - b||_2\; \;\text{given }x \geq 0\;\;\text{and}\;\;\textbf{1}^Tx = 1
$$
I am trying to do the above optimization, I was using common Quadratic programming libraries but their speed is ...
0
votes
1answer
56 views
Silly Quadratic programming
Suppose that I want to minimize the function $x^2$ subject to the contraint
$$
ax\leq b,
$$
for some $b> 0$. I solved the problem if the contraint is an equality but I'm not sure how to go about ...
0
votes
1answer
34 views
binary quadratic mixed integer nonlinear programming to inequalities
I have this unconstrained z=x.y where x,y are 0-1 integers. How to reformulate that into set of mixed integer linear inequalities with exactly the same feasible region?
0
votes
0answers
76 views
Non-positive-definite quadratic program minimum simplification
I was finding the minimum of a simple quadratic program with equality constraints:
$$
m = \min_x x^TQx\qquad\mathrm{s.t.}\quad Ax=b.
$$
Using a formulation with Lagrange multipliers from Wikipedia I ...
0
votes
0answers
29 views
Constraint breaking in constrained least squares
In a constrained least squares problem $\boldsymbol{y} = \boldsymbol{X}\boldsymbol{\beta},\ \boldsymbol{\beta} = \{\beta_1, \ldots, \beta_n\}$ solved with quadratic programming, suppose that I have ...
0
votes
1answer
187 views
How to map quadratic programming formulation to dual soft margin SVM
I am trying to use quadratic programming for SVM and I am confused about how to map SVM formulation to quadratic programming formulation given in CVXOPT (Python package).
This is what CVXOPT gives us
...
1
vote
0answers
30 views
different block size in Block Matrix
I am following a book and having trouble understanding.
so from Section 6.6.2 page 496, i have
A = NxN matrix
b = Nx1 vector
c = Nx1 vector
from Section 14.3.2 ...
1
vote
1answer
31 views
How to Reformulate an Image Constraint in the Dual of a Quadratic Program with Nonconvex Constraints?
I have the following nonconvex optimization problem, for which I want to formulate the dual:
$\mathcal{P}:\underset{x}{\text{min}} \quad x^\top A x + b^\top x \\
\quad \ \ \text{s.t.} \quad x\in\{0,1\...
2
votes
0answers
33 views
On approximation of maximization of quadratic function over a convex set
Given a convex function $f : \mathbb{R}^n \to [0,\infty)$. Let $L = \left\lbrace x \in \mathbb{R}^n \mid f(x) \leq 1 \right\rbrace$ be it's sub-level set and suppose that $L$ is not empty as well as ...
0
votes
0answers
44 views
If the relative value of the most negative eigenvalue is small, can we view the quadratic program as convex?
Consider symmetric matrix $A \in \mathbb{R}^{n \times n}$ with $n-1$ negative eigenvalues and a positive one such that $\lambda_1 > \lambda_2> \dots > \lambda_n$, where $\lambda_n$ is the ...
0
votes
0answers
32 views
Quadratic Matrix programming
I would like to find a numerical solution to the following quadratic matrix programming problem. Square matrix A has thousands of rows, so I need a algorithm which works fast. I looked into Python but ...
1
vote
1answer
144 views
What is the fastest algorithm to solve an equality-constrained convex quadratic program?
I am trying to solve the following convex problem
$$\begin{array}{ll} \text{minimize} & x^T Q x + px\\ \text{subject to} & Ax + b = 0\end{array}$$
which arises in the active set method for ...
2
votes
1answer
141 views
Is the optimal solution of a convex problem continuous with respect to parameters?
Given the convex problem
$$\begin{array}{ll} \text{minimize} & \displaystyle\sum_{i=1}^{N} f(x_i)\\ \text{subject to} & Ax = B\\ & 0\leq x_{\min} \leq x_i \leq x_{\max}\end{array}$$
...
0
votes
0answers
49 views
Support Vector Machines: question about the underlying math
I'm new to Support Vector Machines and I've been trying to get into the underlying math (instead of just using Scikit Learn or something like that).
I understand the math behind it up to the point ...
1
vote
1answer
84 views
Solving inequality-constrained least-norm problem non-iteratively
I am looking into the following constrained quadratic program in $x \in \mathcal{R}^4$
$$\begin{array}{ll} \text{minimize} & \| x \|_2^2\\ \text{subject to} & A x = b\\ & x_{\min} \le x_i ...
