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Questions tagged [quadratic-programming]

22 questions
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2 answers

Approximate subset sum with two-dimensional vectors

Consider the following optimization problem: Given $n\leq 10^3$ vectors $v_i\in\mathbb{R}^2$, all of which are small, i.e., $\|v_i\| \leq 1$, find a subset $S$ of them that minimizes $ \| w + \sum_{i\in S} v_i \|^2$, where $w$ is a fixed known…
Kirill
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Is unconstrained quadratic programming NP-hard?

I could not find the answer on the Internet. The case of quadratic programming with constraints is already solved on this forum, see Transforming SAT to Quadratic Programming in polynomial time. But what happens if there are no constraints or if…
Daniel Porumbel
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Approximation algorithms for indefinite quadratic form maximization with linear constraints

Consider the following program: \begin{align} \max_x ~& x^TQx \\ \mbox{s.t.} ~& Ax \geq b \end{align} where $Q$ is a symmetric (possibly indefinite) matrix and the inequality is element-wise and constrains feasible solutions to a convex…
cangrejo
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What is the complexity of minimising a convex quadratic function over the integers?

The problem of interest is $$ \min_{x\in\mathbb{Z}^n} \frac{1}{2}x^\top Q x + c^\top x $$ where $Q$ is a positive definite matrix. I am reasonably sure this can't be solved in poly-time, since the closest lattice vector problem (in $\ell_2$ norm)…
Sriram
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Verifying a matrix is Copositive

A symmetric matrix $A\in \Bbb{R}^{n\times n}$ is copositive if for every vector $x\in\Bbb{R}^n$ with non-negative entries, we have $$x^TAx \ge 0.$$ What are known methods to check if a specific matrix $A$ is copositive that are efficient in…
Zach Hunter
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Can any Mixed Integer Quadratic Program be approximately solved with Semi Definite Programming relaxation?

I understand that we can approximate solutions to Integer Quadratic Programming optimization problems containing just a positive semi definite matrix, as outlined here (i.e. the Q…
Alexis Pribula
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1 answer

Modeling $(x > 0 \wedge y > 0) \Leftrightarrow z > 0$ in a linear program: impossible?

In this question, we see how to model boolean logic in $0 - 1$ ILPs. Moving to a relaxation, modelling $(x > 0 \vee y > 0) \Leftrightarrow z > 0$ with $x,y,z \in [0,1]$ with linear constraints is quite simple: $$z \geq x\\ z \geq y\\ x+y \geq…
G. Bach
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Approximation algorithms for integer convex quadratic programs over a linear subspace

Consider the problem $$ \min_x \frac{1}{2} x^\top Q x + c^\top x \qquad \text{s.t.}\\ Ax=b\\ x \in \mathbb{Z}^n $$ where $Q$ is a positive (semi)definite matrix. Clearly, feasibility can be decided in polynomial time. Given that the problem is…
Lisa E.
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Is the following binary quadratic integer programming NP-Hard？

I'am trying to prove the following binary quadratic integer programming problem NP hard. $$ \min \frac{\sum\limits_{i=1}^m(u_i-\bar u)^2}{m}\text{ , where }u=Q x,Q\in\mathbb{R}^{m\times n}\\ s.t. \begin{cases} Ax\leq b, \ A\in\mathbb{R}^{m\times…
OvinaSun
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PAVA for partially-ordered (lattice) x-axis

Given an MxN matrix A, I want to find the nearest matrix B (least squares fit: ie minimize$\displaystyle\sum_{i,j} (B_{ij} - A_{ij})^2$) such that $$ \forall i_1,j_1,i_2,j_2 : \\ i_1 \leq i_2 \land j_1 \leq j_2 \implies B_{i_1,j_1} \leq…
dspyz
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Finding $\text{argmin}_{x\in \Delta_n} ||A x -y||^2$

Let $\Delta_n :=${$x\in \mathbb{R}^n\colon \sum_{i=1}^n x_i = 1$} denote the probability simplex in $\mathbb{R}^n$ and $A^{n \times n}$ be diagonalizable matrix. I would like to know what is the fastest way (computationally-speaking) of solving the…
user6952886
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Can an optimization algorithm be "universal"?

I am wondering if a Bayesian Optimization framework (e.g. Google's Vizier) can be used in lieu of a traditional solver like Gurobi or CPLEX. In trying to answer this question, I realized that I don't know enough about optimization theory to answer…
Sasha the Noob
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Does Quadratically-Constrainted Quadratic Programming get easier if all constraints are equalities?

A Quadratically-Constrainted Quadratic Program consists of optimizing a quadratic objective function while imposing quadratic constraints, which can be inequalities or equalities. Obviously, describing the problem with inequalities alone can…
Alex Meiburg
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NP Reduction from 3Color to QuadProg

i just signed up here because im struggling very hard with a problem i gotta solve. What I wanna do is reducing an Instance of 3color to an instance of Quadprog to prove that quadprog is np-hard, and i have no idea how to do that. My first idea was…
MRX
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How to prove QUADPROG is NP-hard using 3COLOR?

I am given a task to prove using 3COLOR that Quadratic Programming is NP-hard. Does anyone have a clue on how this is meant to be done?
Mina Akram
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