# 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.

41
questions

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107 views

### 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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votes

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108 views

### 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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**1**answer

83 views

### 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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56 views

### 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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72 views

### 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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41 views

### When does nuclear norm mininimization give rank $1$ solution when we have at most one solution?

We have a quadratic system of equations
$$A(X\otimes Y)=b$$ where $A\in\mathbb Z^{m\times n^2}$ and $b\in\mathbb Z^m$ are known and $X,Y\in\mathbb Z^{n}$ are unknown. $A$ and $b$ have at most $b$ bit ...

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70 views

### 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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16 views

### 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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121 views

### 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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46 views

### Convex Relaxation in Affine Subspace

I was reading an article about solving a correspondence problem. There, the original non-convex problem is formulated like that:
$$
min_X E(x) = x^t W x \\
s.t. \ X \in \Pi_n
$$
Where $\Pi_n$ is a ...

**2**

votes

**2**answers

90 views

### 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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**1**answer

53 views

### 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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**1**answer

123 views

### 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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197 views

### 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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**0**answers

183 views

### 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\...

**1**

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**1**answer

77 views

### 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^* ...

**4**

votes

**1**answer

200 views

### 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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81 views

### 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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78 views

### 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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**1**answer

457 views

### 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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**2**answers

369 views

### 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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**0**answers

123 views

### 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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**1**answer

94 views

### 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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**1**answer

144 views

### 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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565 views

### 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 ...

**1**

vote

**1**answer

150 views

### 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....

**1**

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**0**answers

40 views

### 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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99 views

### 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{...

**2**

votes

**1**answer

141 views

### 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 $...

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**0**answers

95 views

### 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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112 views

### 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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**0**answers

73 views

### 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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139 views

### 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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157 views

### 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{...

**3**

votes

**0**answers

447 views

### 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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votes

**2**answers

112 views

### 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\...

**3**

votes

**1**answer

542 views

### 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 ...

**4**

votes

**0**answers

218 views

### Efficiently factorize a KKT system with block diagonal upper corner

I have a system resulting from a quadratic energy minimization with linear equality constraints enforced with Lagrange multipliers which has the form:
\begin{equation}
A =
\left[\begin{array}{c|c}
\...

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**2**answers

187 views

### combinatorial and linear duality

Let $S$ be a finite set, and let $W$ be a nonempty set of subsets of $S$; we will identify every subset of $S$ with its characteristic function, a 0-1 vector in $\mathbb R^S$. The combinatorial dual $\...

**1**

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**1**answer

821 views

### Block Covariance Matrix - Positive Definite? (Quadratic Optimization) [closed]

I have a covariance matrix C. I have then formulated an quadratic optimization problem that involves the following matrix in the quadratic form:
[ C C ]
[ C C ]
However, the quadratic solver ...

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**4**answers

3k views

### Why are optimization problems often called “programs”?

Why are optimization problems often called programs?
linear programming
geometric programming
convex programming
Integer programming
...