Questions tagged [quadratic-programming]
A quadratic program (QP) is an optimization problem in which the objective function is quadratic and the feasible region is a convex polytope.
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Positive definite quadratic form algorithm
Let $f(x,y)= ax^2+bxy+cy^2$, or similarly denote it by $(a,b,c)$. This question is about the case $(1,0,p)$ where $p$ is prime. Suppose I have one solution $\bar{x}_1=(x_0,y_0)$ for $f(x,y)=m$ for ...
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Does the value function of a quadratic program stay convex when adding constraints?
I am interested in the value function of a quadratic program of the form
$$
v(y)=\min_x \frac{1}{2} x^\top Q(y) x,
$$
subject to a linear equality constraint
$$
E(y)x=d(y),
$$
and a linear inequality ...
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Can we get the exact solution of large-scale quadratic programming problems (quadratic objective, linear inequality constraints) using KKT condition?
Crossposted at Computational Science SE
Consider a quadratic programming problem with the following format:
$$
\text{min} Q(x) = c^Tx+\frac{1}{2}x^TDx \\
$$
$$
\text{s.t.} Ax\leq b, \\
x\geq 0
$$
...
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Sensitivity of the solution of QP with respect to parameters
Given a quadratic program,
$$\begin{array}{ll} \text{minimize} & \displaystyle \frac12 x^TAx + b^Tx \\ \text{subject to} & Cx \le d \end{array}$$
Suppose $A \succ 0$, so the program strongly ...
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On optimizing a multivariate quadratic function subject to certain conditions
The problem is to maximize $f(x_1,x_2,\cdots,x_n)=\sum\limits_{i=1}^{n}\Big(x_i-k_i\Big)^2$ for $n\ge 3$ subject to the conditions (1) $\sum\limits_{i=1}^{n}x_i=\sum\limits_{i=1}^{n}k_i\le n(n-1)$ ...
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A property of quadratic optimization solution map
Define for a given $\mathbf{b}$ the quadratic optimization solution map by
$$
\mathcal{S} \ \colon \ U \mapsto \mathbf{x} \ \colon \ \mathbf{x}^\top \, U^{-1} \, \mathbf{x} = \min_{\mathbf{y} \geq \...
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Can I solve this quadratic program "fast"?
We are given a matrix $D \in \mathbb{Z}^{C \times C}$ of non-negative entries, an integer $k \geq 1$ and we need to maximize the quadratic form $x^T D x$ under some simple constraints. For all ...
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Drawing a 3D object in a 3D environment, and converting to math [closed]
So I have been granted a free time and I want to work on a project but first I had to research.
As we know, lines have infinite points, and with lines, we can create infinite shapes. I want to let ...
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Complexity of Quadratic Programming where the symmetric matrix Q is positive semidefinite only in the feasible directions
playing around with stuff for my dissertation, I derived a quadratic problem in the general form
\begin{equation}
\begin{aligned}
\min_{x} \quad & x^TQx + c^Tx \\
\textrm{s.t.} \quad & Ax \leq ...
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Convexity of a positive definite objective with min(x,y)-nonlinearity
I have derived an optimization objective of the form
$$
f(x) = \sum_{i,j} C_{ij}\min(x_i, x_j), s.t. g(x) \geq 0
$$
where $C \in \mathcal{R}^{N \times N}$ is a positive definite matrix, and $x \in \...
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(Iterative?) Solutions to a certain quadratic program with non-convex constraints
Let $y\in\mathbb{R}^m$, $\tau\in\mathbb{R}$ and $X\in\mathbb{R}^{m\times n}$, with $\tau>0$
I would like to efficiently solve the following problem:
Problem 1
Choose $\alpha,z\in\mathbb{R}^m,\beta\...
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Positivity of quadratic form minus linear form on the simplex
Let us $a_{ij}$ be the elements of a $n$-dimensional covariance matrix. Can we prove the following?
$$ 1-\sum_{k=1}^n a_{ik} \lambda_k + \sum_{j=1}^n \sum_{k=1}^n \lambda_j a_{jk} \lambda_k > 0, \...
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Can this quadratic program be solved analyticaly?
I have a convex quadratic program wich is structured as follows :
\begin{align*}
\operatorname{argmin}_{p} &\hspace{0.5em} p' A p - 2 p' b \\
\mathrm{s.t.} &\hspace{0.5em} Ep=f \\
...
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Necessary optimality condition for quadratic programming: a solution of a constrained QAP is a solution of a LP
I have a concern about a result given by Murty in [1] and also written by Floudas and Visweswaran in [2]
They consider a QP:
\begin{array}{ll}{\min _{x} Q(x)} & {=c^{T} x+\frac{1}{2} x^{T} D ...
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A quadratic program with non-negativity constraints
Is there any closed form solution for the optimal value of the folowing optimization problem?
$$\begin{array}{ll} \text{minimize} & (\mathbf{x} - \mathbf{y})^{\mathrm{T}}\mathbf{B}(\mathbf{x} - \...
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Does the Perron vector maximize $x^TAx$ in the simplex?
Let $\mathbf{A}$ be any $n\times n$ symmetric positive matrix ($A_{ij}>0$). It is easy to show that the solution to the following optimization problem
\begin{align}
\max_{\mathbf{x}}~~\mathbf{x^...
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Symmetric linear least-squares solution
Given tall matrices $A$ and $Y$ and the following overdetermined linear system in square matrix $X$
$$AX=Y$$
is there an explicit formula for the least-squares solution if $X$ is constrained to be ...
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KKT conditions for min-cost flow QP [closed]
I'm working on a convex quadratic separable min-cost flow problem with the following structure:
$P = \{\min \frac{1}{2}x^tQx + qx : Ex = b, 0 \leq x \leq u\}$
But I'm stuck on deriving the KKT ...
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Mixed integer formulation of union of polytopes?
Given $t$ different unbounded polyhedra $P_1:A^{(1)}x^{(1)}\leq b^{(1)},\dots,P_t:A^{(t)}x^{(t)}\leq b^{(t)}$ we are looking for the representation of $\bigcup_{i=1}^tP_i$ (not their convex hull) with ...
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Finding a point on a convex set
Given a compact bounded convex set $\mathcal C\subseteq\mathbb R^n$ given by $t$ hyperplane inequalities I want to find a point $u\in\mathcal C$ such that for all $v\in\mathcal C$
a convex relation $...
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Does positivstellensatz and SOS proof system help here?
I have a system of $m$ homogeneous degree $2$ polynomial equations in $\mathbb Z[x_1,\dots,x_n]$ where $m=poly(n)$. Take
$$f_1(x_1,\dots,x_n)=0$$
$$\dots$$
$$f_m(x_1,\dots,x_n)=0$$
to be the system.
...
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How Sequential Quadratic Programming versus Quadratic programming and Iterative QP are related? [closed]
What is the difference between Sequential Quadratic Programming (SQP) versus Quadratic Programming (QP)? Is it the same as Iterative Quadratic Programming (IQP)? For example, BFGS, DFP are types of QP/...
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A "nice" (but non-definite) quadratic programme
For integers $n\geq k>0$, let $f$ be the following quadratic form:
$$f(x_1,\ldots,x_n)=\sum_{i=1}^n\sum_{j=0}^{k-1}x_ix_{i+j\bmod n}.$$
Is it true that the minimum of $f$ over the unit simplex is ...
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Correlation between the first and a random position of an ergodic bit sequence
Edit: Since the geometric approach did not work, I try now another approach: phrasing the problem as a quadratic programme.
Probabilistic version.
Let $x=(x_1,x_2, \ldots) $ be an ergodic random ...
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answer
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Clarification on FPTAS optimization in a paper
In the abstract of this paper by Hildebrand, Weismantel & Zemmer it is stated that they provide an FPTAS for $$\min x'Qx$$ over a fixed dimension polyhedron when $Q$ has at most one negative or ...
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Efficient algorithm for solving a convex quadratic program [duplicate]
Let $A \in \mathbb{R}^{n \times m}$ and $b \in \mathbb{R}^n$. Suppose $m \ll n$. How to solve this quadratic program efficiently?
$$\min_{x \in \mathbb{R}^n} \frac{1}{2} x^\top AA^\top x + b^\top x$$
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Maximizing quadratic form subject to inequality constraints [closed]
Given a $n \times n$ symmetric matrix $\rm S$, solve the optimization problem in $n \times k$ (where $n \geq k$) matrix $\rm X$
$$\begin{array}{ll} \text{maximize} & \mbox{tr} \left( \mathrm X^\...
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Solve Huber's M-Estimation using quadratic programming
The Huber's M-estimate is to solve the problem
$$\underset{\mathbf x}{\rm{minimize}} \rho(\mathbf b-\mathbf A\mathbf x) + \alpha|\mathbf x|$$
where
$$ \rho(t) = \left\{\begin{array}{c}\frac{t^2}{2\...
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1
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Norm of solution of quadratic program
In a quadratic program (QP), do linear equality constraints always reduce the norm of the minimizer? Specifically, let $P \succ 0$, $A \in \mathsf{M}_{m\times n}$ and $q\in\mathbb{R}^n$. Define
$$x^* ...
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Least square solution to $AXB+CXD=E$
I am trying to find the least-squares solution $X$ of the following matrix equation
$$AXB+CXD=E$$
Of course, I know that this equation can be written in the form
$$(B^T \otimes A+D^T \otimes C) \...
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Are there scenarios under which feasibility bilinear programming is easy?
Given $c\in\Bbb R^{n_1},d\in\Bbb R^{n_2}$, $E\in\Bbb R^{n_1\times n_2}$, $A\in\Bbb R^{m_1\times n_1}$, $B\in\Bbb R^{m_2\times n_2}$ $a\in\Bbb R^{m_1}$, $b\in\Bbb R^{m_2}$ and $t\in\Bbb R$ we know ...
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On convex quadratic programming clarification
We know convex quadratic programming is in $P$.
Is it also in $P$ if the function of interest is only convex in the domain of interest?
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Is this result on an unconstrained inverse quadratic programming problem new or known already?
Is this problem and solution actually new, or has someone done this earlier?
The details can be found in the preprint: arxiv:1701.01477.
Let us consider a direct quadratic programming problem:
$$
\...
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Minimizing $x_1^2+x_2^2+x_3^2+x_1x_2+x_2x_3+x_3x_1$
Look at the expression
$$
f(x_1,x_2,x_3) = x_1^2+x_2^2+x_3^2+x_1x_2+x_2x_3+x_3x_1.
$$
The numbers $x_1,x_2,x_3$ are non-negative, and I assume that $x_1+x_2+x_3=3$. This is a sum of squares and "...
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Optimization of quadratic form with band matrices
Let $A_1$ be the $N \times N$-matrix for which $a_{i,j} = 1$ for $i=j$ and 0 otherwise. Let $A_2$ be the matrix for which $a_{i,j}=1$ for $|i-j| \leq 1$ and 0 otherwise. Similarly define $A_3$ (which ...
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Convex Decomposition of matrix
For a matrix $\mathbf{X} \in \mathbb{R}^{n\times l}$, we have the following problem of representing vectors in $\mathbf{X}$ as a convex combination of other vectors excluding the vector itself:
$\min\...
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Least squares problem with constrained solution [closed]
If $a_{m\times 1}$ and $Q_{m\times n}$ ($m<n $) are known, and we know every element of $b$ is between $[-1\ \ 1]$, how to determine $b$ to minimize $\|a+Qb\|_2$?
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Quadratic Programming With Piecewise Linear Term
The problem I have can be defined as:
$$
\min \frac{1}{2}\mathbf{x}^T\mathbf{Q}\mathbf{x} + \mathbf{c}^T\mathbf{x}
$$
s.t. linear equality constraints:
$$
\mathbf{Ax=b}
$$
and linear inequality ...
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Quadratically constrained quadratic programming/optimization involving piece-wise function
I have a quadratically constrained quadratic programming/optimization problem involving kind-of piece-wise quadratic functions $f_n (x_m)=a_{n,m} (x_m-\theta_n)^2$, if $|x_m-\theta_n|<c$; $c^2a_{n....
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Which algorithm is most efficient for a specific QP problem
I have a QP problem of the following kind:
$\min_{\alpha\in\mathbb{R}^n}\frac{1}{2}\alpha^T M \alpha - p^T\alpha$
s.t. $l\leq \alpha \leq u$
The matrix $M$ is symmetric and positive definite and of ...
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Hessian matrix positive definiteness (concavity test) [closed]
I have a rather simple scenario based optimization problem:
Maximize
$$
Q_1{_s}(A_1{_s}-Q_1{_s}-bQ_2{_s})+ Q_2{_s}(A_2{_s}-Q_2{_s}-bQ_1{_s})-[(Q_1{_s}-K_1)^+ + (Q_2{_s}-K_2)^+]c
$$
subject to $Q_1{...
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votes
1
answer
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A graph assignment problem
Consider bipartite graph with vertex set $V_1\cup V_2$ where $|V_1|=\frac{n(n-1)}2$ and $|V_2|=n$. The vertices in $V_1$ all have degree $2$ and connected to two vertices in $V_2$. The vertices in $...
user76479
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A (non-convex) minimization quadratic programming problem with d constraints
Minimize $0<\omega_{dd}<2$ subject to
$$\sum_{j=1}^{d}(\omega_{dj} - \omega_{ij})^{2} \geq 4, i=0,1,...,d-1,$$
where $-2<\omega_{ij}<2$ is known for $0 \leq i \leq d-1$ and $1 \leq j \leq ...
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Optimization question: maximize quadratic objective with semidefinite constraints
I recently encountered the following optimization problem:
$\max \|AX\|_F^2$
subject to: $X\succeq0$ and $Xb_i\leq c_i$ for a collection of $T$ conditions: $i=1,\ldots,T$.
Matrices $A$ and $X$ are ...
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Frobenius nearest non-negative Gram matrix of balanced row-sums
Let $W \in \mathbb{R}^{n \times n}$ be any non-negative real symmetric matrix. For $k \leq n$, let $\mathcal{F} := \{X \in \mathbb{R}^{n \times k} \ | \ X \geq 0, X \mathbf{1} = \alpha \mathbf{1}, \...
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How to solve the following generalized quadratic programming problem [closed]
I want to solve a generalized form of a quadratic programming problem
$$\min_x \left(\sqrt{x^TPx}+\sqrt{x^TQx}\right)^2+c^Tx$$,
$$\textrm{ s.t. } Ax\le b.$$ Here, $P$ and $Q$ are both positive ...
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Two quadratic programming problems always same answer? [closed]
Was exploring quadratic programming optimization and for two types of problems the answers seemed to always equal.
Problem 1:
Minimize $\tfrac{1}{2} \mathbf{x}^T Q\mathbf{x}$
Subject to $ A \mathbf{...
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Solution of a linearly constrained quadratic programming problem [closed]
What is the solution of the following optimization problem:
\begin{align}
&\min{\mathbf{p}^\mathrm{T} \mathbf{B} \mathbf{p}}\\
&\text{subject to}: \mathbf{0}\leq{\mathbf{p}}\leq \mathbf{1}.
\...
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2
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Inner Product of Given Sum Positive Sequence
Let $$A = \Big\{(a_1,a_2,\dots)\ \Big|\ a_i\ge 0, \sum_{i=1}^\infty a_i=1\Big\},$$ $$v(x)=\sup\left(\bigg\{\sum_{i=1}^\infty a_ib_i\ \bigg|\ (a_i)_{i=1}^\infty,\, (b_i)_{i=1}^\infty \in A,\,\sup\...
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General method for under and over determined systems?
Suppose I have a system:
$$
Ax = b
$$
where $A$ is a $m$ by $n$ matrix which is less than full rank (neither full column nor row rank). In my particular case $m<n$.
I'd like a combination of a ...
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