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

**14**

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

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**13**

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

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**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

- 6,642
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**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

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**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

- 1,247
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**10**

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

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**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

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**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

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- 11

**5**

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

- 51
- 2

**5**

votes

**1**answer

### Convert a non-convex QCQP into a convex counterpart

Problem
We consider a possibly non-convex QCQP, with nonnegative variable $x\in \mathbb{R}^n$,
\begin{equation*}
\begin{aligned}
& \underset{x}{\text{minimize}}
& & f_0(x) \\
& \text{subject to}
& & f_i(x) \leq 0, \; i = 1, \ldots, m\\
& & & x \geq…

Mr.Robot

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**5**

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

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**5**

votes

**1**answer

### Minimize $x^T A y$, subject to $ x^Ty\geq 0$, where $A=\Phi^T\Phi$ is symmtric and semi-positive definite.

I try to solve it by KKT conditions.
The Lagrangian is
$L=x^TAy-\lambda x^Ty$.
Its KKT conditions are given by
$$
\begin{align}
Ay-\lambda y=&0\quad (1)\\
A^Tx-\lambda x=&0\quad (2)\\
\lambda\geq &0\quad (3)\\
x^Ty\geq &0 \quad (4)\\
\lambda…

Hao WANG

- 75
- 4

**5**

votes

**1**answer

### Maximizing a quadratic function subject to $\| x \|_2 \le 1$

Consider the $n$-dimensional quadratically constrained quadratic optimization problem
$$\begin{array}{ll} \text{maximize} & \frac12 x^T A x + b^T x\\ \text{subject to} & \| x \|_2 \le 1\end{array}$$
where $A$ is a symmetric $n\times n$ matrix that…

user856

**5**

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

- 51
- 4

**5**

votes

**3**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

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