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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24 views

Quadratic optimization under Frobenius norm constraint

Let $A,B \in \mathbb{R}^{p \times p}$ be positive semi-definite. I have troubles on the following optimization problem: \begin{equation} \max_{\|X\|^{2}_F=1} \text{tr}(X^{T}AXB) \end{equation} It ...
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20 views

Binary optimisation to maximise covariance or L2-error, is it even possible? Could it be solved by relaxation?

I have a matrix A and a vector y, given A invertible the solution to find x is simple, however if we restrict x to be a binary vector i.e. $$ x_{i} \in \{0,1\} \quad \forall i $$ then we are not ...
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40 views

(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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25 views

what kind of functions can be optimized over matroid polytope

We know that we can optimize linear functions over matroid polytope in polynomial time. Is this the only class of functions that can be optimized in polynomial time? Do we have any impossibility ...
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23 views

Most efficient way to solve against large sparse matrix with a few dense rows and columns?

I have a constrained optimization problem of the form: $$ \min_{Bx=g} \frac{1}{2} x^T A x - x^T f $$ $A \in \mathbb{R}^{n\times n}$ is positive semi-definite (with a tiny null space of dimension &...
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1answer
82 views

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 that: $ 1-\sum_{k=1}^n a_{ik} \lambda_k + \sum_{j=1}^n \sum_{k=1}^n \lambda_j a_{jk} \lambda_k >0$ for $i=1 \...
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189 views

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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105 views

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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44 views

Strict complementarity for quadratic programming

Consider the quadratic program \begin{align*} \text{min}_{x\in \mathbb{R}^n} \ &\tfrac{1}{2}x^THx + f^Tx\\ \text{st.} \ & Ax \leq a \\ & Cx = c \\ \end{align*} for matrices $A\in \mathbb{R}...
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1answer
145 views

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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1answer
180 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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2answers
255 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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1answer
113 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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69 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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79 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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83 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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23 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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1answer
192 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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2answers
98 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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1answer
61 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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1answer
128 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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1answer
271 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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208 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\...
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1answer
115 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^* ...
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1answer
236 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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0answers
86 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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80 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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1answer
474 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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2answers
392 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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0answers
128 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 ...
3
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1answer
107 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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1answer
159 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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3answers
692 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 ...
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1answer
153 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....
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0answers
45 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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0answers
103 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{...
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1answer
148 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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0answers
96 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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0answers
118 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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0answers
74 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}, \...
2
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0answers
140 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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0answers
158 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
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0answers
451 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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2answers
114 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
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1answer
588 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
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0answers
260 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} \...
4
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2answers
194 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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1answer
852 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 ...
31
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4answers
3k views

Why are optimization problems often called “programs”?

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