Questions tagged [qcqp]

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

177 questions
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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.…
13
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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 =…
13
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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
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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…
10
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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 =…
10
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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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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…
5
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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}^{…
5
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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…
5
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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…
5
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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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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…
5
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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
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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
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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…
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