A quadratically constrained quadratic program (QCQP) is an optimization problem in which both the objective function and the constraints are quadratic.

# Questions tagged [qcqp]

177 questions

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### Time complexity of a convex quadratically constrained quadratic program (QCQP)

Could someone tell me the time complexity of a convex quadratically constrained quadratic program (QCQP) problem? And any references?
Thank you very much.

Giantron

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### Recovering original matrix from its kernel matrix

For a given linear kernel matrix, $K [n \times n]$, we would like to recover the original matrix $X$.
In general it is not possible to recover the data matrix from a kernel matrix, as the projection of data matrix into possibly…

Amir

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### Optimizing non-convex functions on convex sets

I'm interested in the optimization problem
$$\begin{array}{ll} \text{maximize} & x^T M \, x\\ \text{subject to} & \| x \| \leq 1\\ & x \geq 0\end{array}$$
where $M$ is positive definite. As I understand it, this isn't a convex problem, but the…

bibliolytic

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### Finding the exterma of $x^2+y^2+z^2-yz-zx-xy$ s.t. $x^2+y^2+z^2-2x+2y+6z+9=0$ using Lagrange's multiplier,

Using Lagrange's multiplier method, obtain the maxima and minima of $$x^2+y^2+z^2-yz-zx-xy$$ subject to the condition $$x^2+y^2+z^2-2x+2y+6z+9=0$$
My attempt:
I formed the…

user467745

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### What's the shortest distance from a point inside of an ellipsoid to its surface?

In general in arbitrary dimension, what's the shortest distance from a point inside of an ellipsoid to its surface?
Any good resources on this topic would help greatly as well.
Edit: I know there are ways to do this with constrained optimization,…

td777

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### Maximizing a quadratic form on unit cube lattice points $\{ -1 , 1 \}^n$

Consider the problem: given a symmetric, positive-definite, quadractic form with matrix $A = A^T \in \mathbb R^{n \times n}$ compute the vector $y \in \{-1,1\}^n$ achieving the maximum
$$\max_{x \in \{-1,1\}^n } x^TAx$$
Is there an efficient…

spitespike

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### Find all the points on the cone $z^2=x^2+4y^2$ that are the closest to the point $(0,0,c)$

From all the points on the cone $z^2=x^2+4y^2$ find the closest to the point $(0,0,c)$. State explicitly the minimal distance. $c$ is a constant.
Lagrange multipliers can be used here.
Let the constraint function $g=x^2+4y^2-z^2$.
Let the…

Yos

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### Maximize $c^t x + x^T A x$ subject to $x^T x = 1$ where $A \succeq 0$

The vector Bingham-von Mises-Fisher distribution is defined on on the sphere $S^{p-1}$ and has density
$$p(x \vert c, A) \propto \text{exp}\{c^Tx + x^TAx\}$$ with respect to the uniform measure on $S^{p-1}.$ Assume $c\in\mathbb{R}^p\setminus \{0\}$…

stats_qs

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### QCQP as a SDP or SOCP?

I have a QCQP as shown below:
\begin{equation}
\begin{aligned} & \underset{\mathbf{x}}{\text{minimize}} & & \mathbf{x}^{T}\cdot\mathbf{P}\cdot\mathbf{x}\\
& \text{subject to} & & \mathbf{x}^{T}\cdot\mathbf{P}_{i}\cdot\mathbf{x}-r_{eq}^{i}=0, & …

TropE

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### Minimizing $\|Ax\|_2$ subject to $\|x\|_2 = 1$

I have a Matlab program to estimate a vector $x$ from noisy measurements. I use the singular value decomposition (SVD) to solve the linear equation $Ax=0$ (where the number of equations is greater than the number of variables). I have read before…

maruchan

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### Relaxation of non-convex QCQP with one quadratic and one linear constraint

According to Boyd we know that a non-convex QCQP problem with one quadratic constraint has strong duality with the relaxed SDP or Lagrange counterpart. (check "Convex Optimization" by Boyd, Appendix B)
My question is if I add a linear constraint to…

MChawa

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### Can a convex QCQP with an additional linear constraint be converted into a SOCP?

I have a quadratically constrained quadratic programming problem that I massaged into the form
$$
\begin{aligned}
& \underset{x}{\text{minimize}}
& & x^T Q x \\
& \text{subject to}
& & x^T P x < \sigma^2 \\
&&& 0 \leq x_i \leq x_i^* \\
&&&…

rhaskett

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### Minimize Frobenius norm with constraints

As a follow-up on my previous question, I would like to solve the following optimization problem:
$$\begin{array}{ll} \text{minimize} & \| \mathrm M \mathrm A - \mathrm B \|_F^2 - \mathrm x^H \mathrm M \mathrm y\\ \text{subject to} & \mathrm M^H…

ThP

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### Is Quadratically Constrained Quadratic Program (QCQP) in NP?

The general version of QCQP is NP-hard, but is it also NP-complete? That means, is there a non-deterministic algorithm, which solves QCQP in polynomial time complexity?
If the general version of QCQP is not in NP, are there restricted versions such…

i8r

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### Orthogonal Procrustes Problem

The classical Orthogonal Procrustes Problem is
$$\begin{array}{ll} \text{minimize} & \|A\Omega-B\|_{F}\\ \text{subject to} & \Omega'\Omega=I\end{array}$$
where $A$ and $B$ are known matrices.
Suppose $A$ is the identity matrix. I would like to…

Lindon

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