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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### Minimize $ {x}^{T} Q y $ subject to $ \left\| x \right\|^{2} + \left\| y \right\|^{2} = 1 $

Let $ Q $ be an $ n \times n $ diagonal matrix with positive diagonal entries $\lambda_{1}<\cdots<\lambda_{n}$.
Find local minimizer(s) for the function $f : \mathbb{R}^{n} \times \mathbb{R}^{n} \rightarrow \mathbb{R} $ and show which 1st and 2nd…

jackson5

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### Mixed Binary Quadratic Programming Quadratically Constrained

I would like to find an algorithm to find the optimal (or a close approximation) of the minimum of $f(X,Y)$:
$$
f(X,Y)=\sum^N_{i=1}{x_it_i}+\sum^N_{i=1}{x_iy_ir_i}\\
\begin{align}
s.t &&x_i \in \{0,1\} \tag 1\\
&&t_i \ge 0 \tag 2\\
&&A.X = b \tag…

chkone

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### Implement QCQP in CVXOPT

I'm struggling to formulate a simple QCQP in the correct format to solve with CVXOPT.
I'm trying to implement max-margin Inverse Reinforcement Learning from the paper Apprenticeship Learning via Inverse Reinforcement Learning (§3, p3), which is…

aaronsnoswell

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### Maximimization problem with quadratic equality constraint

Given this optimization problem,
$Max \quad Q(x)$
$s.t., \quad x \in X$
$\quad \sum_{i=1}^{n} x_i^2 = k$
where $x_i$ are integers, $x \in X$ is a set of linear inequalities, k is a parameter, and Q(x) is a quadratic function of x.
I'm interested…

user2512443

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### QCQP Formulation

My goal is to find a vector $c \subset \mathbb{R}^4$ that is as close to the vector $A \subset \mathbb{R}^4$ as possible while maintaining a constraint regarding ratios of elements of $c$, specifically $\frac{3}{4} \leq \frac{c[1]+c[2]}{c[3]+c[4]}…

Jerald Thomas

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### dual variable of dual problem is a solution for primal problem in SDP?

Hello I have a question on the dual variable in SDP problem
Actually, I'm reading the papers about QCR ( reformulating the 0-1 QCQP problem into a convex problem with tightest bound), but I will skip the detail and ask you the one that I haven't…

Jaeyoon Yoo

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### Proving tightness of a semidefinite relaxation of a nonconvex quadratic program

The Lagrangian of my semidefinite relaxation of a nonconvex QCQP (quadratically-constrained quadratic program)…

HD Lang

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### How to solve this quadratically constrained quadratic programming problem?

Could you please shed some lights on this? (Not a homework problem)
I am looking for solutions of the following problem in $b$:
$$\max \| X b \|_2 \quad\text{subject to} \quad \| b - b_0 \|_2 < a, \| b \|_2 = 1$$
where matrix $X$, $a$ and $b_0$ are…

Luna

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### How to obtain the optimal Lagrange multiplier vectors if the globally optimal solution for a nonconvex QCQP is found?

I am using a black-box solver to solve the following non-convex QCQP to global optimality.
$$ \min_x x^TQ_0x + c^T x \\
s.t. \quad x^TQ_1x+c_1^Tx=b_1 \\
Ax=b \\
l\leq x\leq u
$$
where $Q_0$ is indefinite diagonal matrix and $Q_1$ is positive…

Apurv

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### Prove that maximum value is the largest eigenvalue

Given a symmetric positive definite matrix $A \in \mathbb{R}^{n \times n}$ and a full rank matrix $B \in \mathbb{R}^{n \times n}$. Prove that the maximum value the following optimization problem is the largest eigenvalue of…

Andre Jackson

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### Distance between a point and a conic curve

I have a point $r=(100,0)$ and want to find the closest point to it from this set:
$$k = \{(a,b) : b^2=1+a/4\}$$
where $a$ belongs to $[-4,0]$.
I thought about defining function $h(x)=|r-x|$, and using Lagrange multipliers to locate minimum point.…

Alex11

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### Find Minimum + Maximum of function with constraints

homework assignment ask to find Max/Min for $$U(x,y,z) = x^2 + 2y^2 + 3z^2$$
with these constraints:
$x^2 + y^2 + z^2 = 1$
$x + 2y + 3z = 0$
Thank you.
First i tried to isolate x from the second constraint and then to put it in the first one .

Eyal Klemm

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