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

# Questions tagged [qclp]

46 questions

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### Linear programming with one quadratic equality constraint

I have a problem that can be formulated as a linear program with one quadratic equality constraint:
where variable $x$ is an $n$-dimensional vector and $H$ is a positive semidefinite $n \times n$ matrix.
I know this optimization problem can always…

user123346

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### If $a,b$ are positive integers and $x^2+y^2\leq 1$ then find the maximum of $ax+by$ without differentiation.

If $x^2+y^2\leq 1$ then maximum of $ax+by$
Here what I have done so far.
Let $ax+by=k$ . Thus $by=k-ax$.
So we can have that $$b^2x^2+(k-ax)^2 \leq b^2$$
$$b^2x^2+k^2-2akx +a^2x^2-b^2\leq 0 $$
By re-writing as a quadratic of $x$…

Angelo Mark

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### Maximizing a linear function over an ellipsoid

Let $A \in \mathbb{R}^{n\times n}$ be a positive definite matrix, $x \in \mathbb{R}^n$ and $c \in \mathbb{R} \setminus \{0\}$. I got to determine the maximum
$$\max\{c^Ty:y\in \mathcal{E} (A,x)\}$$
where $\mathcal{E} (A,x)$ is an ellipsoid defined…

Thesinus

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### Lagrange multiplier when decisions variables are not in the same set

Find the maximum of $2x+y$ over the constraint set $$S = \left\{ (x,y) \in \mathbb R^2 : 2x^2 + y^2 \leq 1, \; x \leq 0 \right\}$$
I want to use Lagrange multipliers to find the optimal solution. However, Lagrange requires $\vec x \in A$. In our…

Mario Zelic

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### Linear objective function with quadratic constraints

The context is ordinary multivariate regression with $k$ (>1) regressors, i.e. $Y = X\beta + \epsilon$, where
$Y \in \mathbb{R}^{n \times 1}$ vector of predicted variable,
$X \in \mathbb{R}^{n \times (k+1)}$ matrix of regressor variables(including…

Preetam Pal

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### Indefinite quadratic constraint

I'm trying to solve an optimization problem with a linear objective function and mostly linear constraints. However, I do have several constraints of the form
$$\sum_{i=1}^m x_i\phi_i - \left(\sum_{i=1}^m x_i\right) \left(\sum_{j=m+1}^n x_j\right)…

user402078

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### How to solve a quadratically constrained linear program (QCLP)?

Can anybody suggest some techniques to solve a quadratically constrained linear program (QCLP)? Any references on standard techniques would be helpful.

Priyanka

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### Maximizing linear objective subject to quadratic equality constraint

I've been trying to get through some practice questions on the Karush-Kuhn-Tucker (KKT) theorem but I can't seem to answer the following.
Given $f, g : \mathbb{R}^2 \to \mathbb{R}$ defined by $f(x) := x_1 + x_2$ and $g(x) :=…

ji zhi

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### Optimizing Trace$(Q^TZ)$ subject to $Q^TQ=I$

Let $Z \in \mathbb{R}^{m \times n}$ be a tall matrix ($m > n$). Solve the following optimization problem in $Q \in \mathbb{R}^{m \times n}$
$$\begin{array}{ll} \text{maximize} & \mbox{Tr} \left(Q^T Z \right)\\ \text{subject to} & Q^T Q = I_{n…

kkcocoqq

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### Constrained convex optimization

Solve
Maximize $f(x)=c^Tx$
subject to $x^TQx \leq 1$
where $Q$ is a positive definite matrix.
what is the solution if the objective function is to be minimized ?

rgk

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### Maximisation of a piecewise affine function over an ellipsoid

Given vectors $\mathrm a, \bar{\mathrm x} \in \mathbb R^n$ and matrix $\mathrm P \in \mathbb S^n_{++}$, how to deal with the absolute value in the objective function of this optimization problem in $\mathrm x \in \mathbb R^n$?
$$\begin{array}{ll}
…

fire-bee

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### Optimality condition for convex QCLP

I am trying to get the optimality condition for the following problem
$$\begin{array}{ll} \text{minimize} & c^T y\\ \text{subject to} & Ay = 0\\ & y^T B y \le 1\end{array}$$
where $B$ is positive semidefinite.
I observed that $y^TBy$ must be equal…

david

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### Optimization of linear objective with non-convex quadratic constraint

Is there any technique to deal with a problem where we have a linear objective function and one or many quadratic non-convex function(s) like the problem below?
$$\begin{array}{ll} \text{minimize} & c_1 x + c_2 y\\ \text{subject to} & xy =…

Sourav Mondal

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### How to prove the maximum of n-dimensional linear function $a^Tx$ is $||a||_2$ for $||x||_2 \le 1$

I can get it when $x \in \mathbb R$, but I cannot understand why
$$\sup\{a^Tx \mid \|x\|_2 \le 1 \} = \| a \|_2$$ when $x \in \mathbb R^n$. To my understanding, this problem can be transformed into a Second Order Cone Programming (SOCP)…

Finley

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### Determining the minimum of a linear function subject to a quadratic inequality constraint

What is the minimum value of $x+4z$, a function defined on $\mathbb{R^3}$, subject to the constraint $x^2 + y^2 +z^2 \leq 2$?
I know how to solve this if the constraint is an equality, but what shall I do if it's an inequality? Could anyone help me,…

user426277