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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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 ...
2 votes
0 answers
62 views

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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2 votes
1 answer
49 views

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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1 vote
0 answers
46 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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0 votes
1 answer
167 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 \...
2 votes
0 answers
465 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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3 votes
0 answers
130 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 ...
1 vote
1 answer
232 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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2 votes
1 answer
359 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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5 votes
2 answers
577 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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0 votes
1 answer
185 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 ...
1 vote
0 answers
89 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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1 vote
0 answers
89 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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0 votes
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115 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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1 vote
0 answers
36 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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1 vote
1 answer
288 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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2 votes
2 answers
101 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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2 votes
1 answer
66 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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2 votes
1 answer
167 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? $$\min_{x \in \mathbb{R}^n} \frac{1}{2} x^\top AA^\top x + b^\top x$$
1 vote
1 answer
377 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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3 votes
0 answers
232 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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1 vote
1 answer
228 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
307 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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3 votes
0 answers
92 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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1 vote
0 answers
83 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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3 votes
1 answer
522 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: $$ \...
5 votes
2 answers
434 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 "...
2 votes
0 answers
142 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 votes
1 answer
143 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 vote
1 answer
182 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 $\|a+Qb\|_2$?
2 votes
3 answers
935 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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1 vote
1 answer
160 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 vote
0 answers
49 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 ...
1 vote
0 answers
124 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
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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2 votes
0 answers
101 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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1 vote
0 answers
123 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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1 vote
0 answers
80 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 votes
0 answers
144 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 ...
4 votes
0 answers
161 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{...
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4 votes
0 answers
491 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}. \...
0 votes
2 answers
115 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\...
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3 votes
1 answer
684 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
345 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 votes
2 answers
204 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 vote
1 answer
891 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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33 votes
4 answers
4k views

Why are optimization problems often called "programs"?

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