# Questions tagged [qcqp]

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

13
questions

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votes

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### How to solve a QCQP where constraints are balls?

I want to solve the following optimization problem in variables $\theta_1, \theta_2, \dots, \theta_K$
\begin{equation}
\begin{aligned}
& \underset{\theta}{\text{minimize}}
& & \...

**0**

votes

**0**answers

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### Dealing with degeneracy in nonlinear programming by “small” perturbations of constraints

CONTEXT: Suppose you have the nonlinear program
$$
\begin{aligned}
&\min f(x)\\
\text{subject to: }\quad & h_1(x) = 0 \\
&\quad\quad\vdots\\
&h_m(x) = 0
\end{aligned}
$$
where $x\in\...

**0**

votes

**0**answers

28 views

### Find a complex matrix on a unit sub-spheres

I am new to optimization theory. I have a following question. For a given $X = [x_1 x_2 \ldots x_N] \in \mathbb{C}^{N \times N}$, where $x_i \in \mathbb{C}^{N\times 1}$ for $i \in \{1,\ldots,N\}$, $U =...

**8**

votes

**5**answers

394 views

### Nearest matrix orthogonally similar to a given matrix

Given $A,B\in\Bbb R^{n\times n}$ is there technique find $$\min_{ T\in O(n,\Bbb R)}\|A-TBT^{-1}\|_F\mbox{ or }\min_{ T\in O(n,\Bbb R)}\|A-TBT^{-1}\|_2$$ within additive approximation error in $\...

**7**

votes

**2**answers

364 views

### Eigenvalue problem with two quadratic constraints

I would like to solve the following problem:
$$\begin{array}{ll} \text{minimize} & \mathbf{x}^T \mathbf{A} \mathbf{x}\\ \text{subject to} & \mathbf{x}^T\mathbf{B}\mathbf{x} = 0\\ & \...

**4**

votes

**3**answers

2k views

### Maximize the Euclidean norm of a matrix times a vector on unit sub-spheres

For $X = (x_1^T,\ldots,x_N^T)^T \in \mathbb{R}^{Nm \times 1}$, where $x_i \in \mathbb{R}^{m \times 1}$ for $i \in \{1,\ldots,N\}$, $A \in \mathbb{R}^{r \times Nm}$, and $r \geq Nm$, I want to obtain a ...

**1**

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**0**answers

243 views

### Is this QCQP convex or nonconvex?

\begin{equation}
\begin{split}
\min_{x\in \mathbb{R}^n}\:f(x)=(1/2)x^{T}Q_0x+c_0^T x
\end{split}
\end{equation}
s.t.
$$
g_i(x)=\frac{1}{2}x^T Q_ix-lmax_i\leq0,i\in\{1,...,m/2\}
$$
$$
g_i(x)=\frac{...

**12**

votes

**2**answers

4k views

### Linearly constrained eigenvalue problem

Suppose I'd like to:
\begin{align}
\mathop{\text{min}}_\mathbf{x} && \mathbf{x}^T\mathbf{A}\mathbf{x} \\
\text{subject to:} && \mathbf{x}^T \mathbf{M} \mathbf{x} = 1\\
&& \...

**1**

vote

**1**answer

611 views

### A non-convex quadratically constrained quadratic program

$$\begin{array}{ll} \text{minimize} & \beta^{T} A \beta\\ \text{subject to} & \beta^{T} C \beta=1\\ & \beta \geqslant 0\end{array}$$
where $A, C\in \mathbb{R}^{M\times M}$ and $\beta \in ...

**3**

votes

**2**answers

503 views

### A certain type of quadratic constrained quadratic program (QCQP)

Let $P_1$, $P_2$ be two Hermitian matrices. Can anyone comment on the following QCQP?
$$\begin{array}{ll} \text{minimize} & z^{H} z\\ \text{subject to} & z^{H} P_1 z +1 \leq 0\\ & z^{H} ...

**1**

vote

**1**answer

375 views

### Solving a QCQP problem with sparse regularization

I want to solve the follong QCQP problem:
$$
\mbox{Minimize}\quad\beta^TA\beta+\mu\Omega(\beta)$$
$$
\mbox{s.t.}\quad\beta^TB\beta=1 \quad\mbox{and}\quad\beta\ge0
$$
where $A$ and $B$ are both ...

**2**

votes

**3**answers

2k views

### Solving a non-convex quadratically-constrained quadratic program

I have the following quadratic optimization problem: $\min_{\vec{x}} |\vec{x}|^2$ subject to $\vec{x}^T G_j \vec{x} \geq 1$, $j = 1 \ldots m$, where the $G_j$ are positive semidefinite. $|\vec{x}|$ is ...

**2**

votes

**2**answers

2k views

### Complexity of convex quadratically constrained quadratic programming (QCQP)

Could someone tell me the time complexity of a convex quadratically constrained quadratic program (QCQP)? Any references?