Questions tagged [quadratic-programming]
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22 questions
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Adam gradients for least squares approaches?
Least squares leads to an n-dimensional "parabola" in the parameters. I assume the same is valid for other constrained least squares like non-negative least-squares.
This may be a wrong ...
- 1
1
vote
1
answer
51
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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
$$
\forall i_1,j_1,i_2,j_2 : \\
i_1 \leq i_2 \land ...
- 456
2
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2
answers
278
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Approximation algorithms for integer convex quadratic programs over a linear subspace
Consider the problem
$$
\min_x \frac{1}{2} x^\top Q x + c^\top x \qquad \text{s.t.}\\
Ax=b\\
x \in \mathbb{Z}^n
$$
where $Q$ is a positive (semi)definite matrix.
Clearly, feasibility can be decided in ...
- 555
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1
answer
136
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What is the complexity of minimising a convex quadratic function over the integers?
The problem of interest is
$$
\min_{x\in\mathbb{Z}^n} \frac{1}{2}x^\top Q x + c^\top x
$$
where $Q$ is a positive definite matrix. I am reasonably sure this can't be solved in poly-time, since the ...
- 387
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1
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277
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What's the difference between "convex/non-convex optimization" and "quadratic programming"?
I'd like to start by clarifying I'm by no means an expert in any of this, so take everything I say with a grain of salt.
Convex and Non-convex Optimization are subfields of mathematical optimization ...
2
votes
1
answer
202
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Is the following binary quadratic integer programming NP-Hard？
I'am trying to prove the following binary quadratic integer programming problem NP hard.
$$
\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. \...
- 23
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73
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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 ...
- 121
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38
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Polynomial Deterministic Algorithm for MAX-QP problem with multiplicative errror $1/n$
MAX-QP problem (quadratic programming on boolean cube) statement:
Given symmetrical matrix $A$ with 0 on diagonal
$$A = (a_{ij}) : A^T = A, a_{ii} = 0$$
Find maximum $x^TAx$ where $x \in \{0, 1\}^n$, ...
- 158
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154
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SVMs - Fat (Margin) Boundary: Why is $\max\frac1{||\theta||}=\min \frac{ ||\theta||^2}{2}$?
I am trying to understand SVMs in depth watching lectures from MIT.
The professor to reduces the classification problem into an optimization problem. To do that, he first defines the decision and ...
- 167
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281
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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 ...
- 405
4
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answers
134
views
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 ...
- 168
5
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1
answer
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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 ...
- 271
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2
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282
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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'...
- 653
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131
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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, ...
- 955
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How to prove QUADPROG is NP-hard using 3COLOR? [duplicate]
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?
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NP Reduction from 3Color to QuadProg [duplicate]
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 ...
MRX
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1
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78
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Max-min of a nonconvex quadratic problem
I am trying to solve the following problem, which is a simplification of our original question:
$\max\limits_{x,y}\min \{x_iy_i-b_i \mbox{ for } i=1,\ldots, n: x,y\in \Delta_n\}$
where $\Delta_n$ is ...
- 1
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2
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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 + \...
- 195
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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 matrix):
https://...
- 121
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Qudratic Equations - Promise Problem
Is the follwing promise problem NP-hard?
Input: A system of quadratic equations.
Promise: The system has either one or zero solutions.
Question: Does the system have a solution?
- 231
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331
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Complexity of active set method for Quadratic Programming
The Quadratic Programming problem is as follows:
$$\min_x \{\frac12x^THx+x^Tg\}$$
$$Ax\le b$$
where $H$ is symmetric and positive semi-definite.
What is the complexity of the active set method for ...
- 131
2
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1
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239
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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 ...
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