A quadratically constrained quadratic program (QCQP) is an optimization problem in which both the objective function and the constraints are quadratic.
Questions tagged [qcqp]
204 questions
19
votes
7 answers
Maximize the value of $v^{T}Av$
Let $A$ be a symmetric, real matrix. The goal is to find a unit vector $v$ such that the value $v^{T}Av$ is
maximized, and
minimized.
The answer is that $v$ should be the eigenvector of $A$ with
largest eigenvalue, and
smallest eigenvalue.…
user4205580
- 2,213
14
votes
3 answers
Solve least-squares minimization from overdetermined system with orthonormal constraint
I would like to find the rectangular matrix $X \in \mathbb{R}^{n \times k}$ that solves the following minimization problem:
$$
\mathop{\text{minimize }}_{X \in \mathbb{R}^{n \times k}} \left\| A X - B \right\|_F^2 \quad \text{ subject to } X^T X =…
Alec Jacobson
- 573
13
votes
8 answers
Distance of ellipse to the origin
Calculate the minimum distance from the origin to the curve
$$3x^2+4xy+3y^2=20$$
The only method I know of is Lagrange multipliers. Is there any other method for questions of such type? Any help appreciated.
Shobhit
- 6,960
12
votes
3 answers
Trace minimization with constraints
For positive semi-definite matrices $A,B$, how can I find an $X$ that minimizes $\text{Trace}(AX^TBX$) under 'either' one of these constraints:
a) Sum of squares of Euclidean-distances between pairs of rows in $X$ is a constant $\nu$.
or
b) $X$ is…
qlinck
- 1,181
11
votes
4 answers
Minimizing a quadratic function subject to quadratic constraints
Okay, so I am attempting to minimize the function
$$f(x,y, z) = x^2 + y^2 + z^2$$
subject to the constraint of
$$4x^2 + 2y^2 +z^2 = 4$$
I attempted to solve using Lagrange multiplier method, but was unable to find a $\lambda$ that made the system…
user345
- 791
10
votes
2 answers
Eigenvalue bound for quadratic maximization with linear constraint
This builds on my earlier questions here and here.
Let $B$ be a symmetric positive definite matrix in $\mathbb{R}^{k\times k}$ and consider the problem
$$\begin{array}{ll} \text{maximize} & x^\top B x\\ \text{subject to} & \|x\|=1 \\ & b^\top x =…
sven svenson
- 1,440
8
votes
1 answer
Minimizing Quadratic Form with Norm and Positive Orthant Constraints
Let $ M $ be a positive semi definite matrix.
I want to solve
$$ \arg \min_{x} {x}^{T} M x \quad \mathrm{s.t.} \quad \left\| x \right\| = 1, \ x \succeq 0 $$
where $ x \succeq 0 $ means each coordinate of $x$ is nonnegative.
Is there a standard…
user7530
- 50,625
7
votes
1 answer
How can I experiment with Lagrange multiplier in QCQP?
Suppose we want to solve following optimization problem (it is a PCA problem in this post)
$$
\underset{\mathbf w}{\text{maximize}}~~ \mathbf w^\top \mathbf{Cw} \\
\text{s.t.}~~~~~~ \mathbf w^\top \mathbf w=1
$$
As mentioned the the post, using the…
hxd1011
- 493
6
votes
2 answers
Quadratic equality constraints via SDP
I want to know if it is possible to solve a QCQP problem with quadratic equality constraints in SDP. I know it is possible to convert a QCQP to an SDP by using the Shur complement. The following worked for me thus…
Kirillvh
- 193
6
votes
1 answer
SDP relaxation of non-convex QCQP and duality gap
Short version
Is there a duality gap between a QCQP problem and the SDP problem obtained through Lagrangian relaxation?
A paper I'm studying is using this fact, but I cannot achieve the authors' results.
Longer version
I've been trying to reproduce…
user73053
- 61
6
votes
2 answers
Least squares problem with constraint on the unit sphere
It is easy to find the minimum of $\|Ax-b\|_2$, when $A$ has full column rank. But how is the case when we add the constraint $\|x\|_2=1$? Or, to be explicit,
$$\min_{\|x\|_2=1}\|Ax-b\|_2=?$$
My idea is to construct the corresponding Lagrange…
Gabriel
- 61
6
votes
4 answers
Quadratic optimisation with quadratic equality constraints
I would like to solve the following optimisation problem:
$$\text{minimize} \quad x'Ax \qquad \qquad \text{subject to} \quad x'Bx = x'Cx = 1$$
Where $A$ is symmetric and $B$ and $C$ are diagonal.
Does anyone have a suggestion for an efficient way…
user111950
- 547
5
votes
1 answer
Minimizing quadratic function subject to quadratic equality constraint
Given $N \times N$ positive (semi)definite matrix $\mathbf{A}$, vector $\mathbf{b} \in \Bbb C^N$ and $c > 0$,
$$\begin{array}{ll} \underset{\mathbf{x} \in \mathbb{C}^N}{\text{minimize}} & \mathbf{x}^H\mathbf{A}\mathbf{x} + 2 \Re\left\{ \mathbf{b}^H…
dineshdileep
- 9,113
5
votes
1 answer
Minimize $x^*(A+A^*)x$ such that $x^*A^*Ax=1$ and $x^*x=1$
Given $A\in\mathbb{C}^{n\times n}$, such that it has singular values larger than $1$ and smaller than $1$,
\begin{array}{ll} \underset{x\in\mathbb{C^n}}{\text{minimize}} & x^*(A+A^*)x.\\ \text{subject to} & x^*A^*Ax=1,\\&x^*x=1\end{array}
My…
Lee
- 1,978
5
votes
2 answers
What is the range of $\vec{z}^{ \mathrm{ T } }A\vec{z} $?
Let A be a 3 by 3 matrix
$$\begin{pmatrix}
1 & -2 & -1\\
-2 & 1 & 1 \\
-1 & 1 & 4
\end{pmatrix}$$
Then we have a real-number vector $\vec{ z }= \left(
\begin{array}{c}
z_1 \\
z_2 \\
z_3
\end{array}
\right)$ such that
$$\vec{z}^{…
ohisamadaigaku
- 317