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]
56 questions
11
votes
2 answers
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
- 111
7
votes
4 answers
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
- 6,226
7
votes
1 answer
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 $c \in \mathbb{R} \setminus \{0\}$. Determine the maximum $$ \max_{y \in \mathbb{R}^n} \left\{ c^T y : y \in \mathcal{E} (A,x) \right\} $$ where $\mathcal{E}…
Thesinus
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- 9
- 17
3
votes
1 answer
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
- 29
3
votes
2 answers
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
- 319
3
votes
2 answers
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
- 49
3
votes
1 answer
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
$$\sum_{i=1}^m x_i\phi_i - \left(\sum_{i=1}^m x_i\right) \left(\sum_{j=m+1}^n x_j\right)…
user402078
- 31
3
votes
3 answers
Linear programming with quadratic constraints
I have a given set of variables:
$x_1,y_1,x_2,y_2,x_3,y_3$
The objective function is to minimize the sum of these with quadratic equality constraints:
$y_1(x_1+x_2+x_3)$=0
$y_2(x_2+x_3)$=0
$y_3(x_3)$=0
There are other inequality constraints which…
user1234
- 71
- 1
- 9
3
votes
1 answer
How to solve a quadratically constrained linear program (QCLP)?
Can anybody suggest some techniques to solve a quadratically constrained linear program (QCLP)? Any references on standard techniques would be helpful.
Priyanka
- 31
2
votes
2 answers
Find infimum and supremum of the function $f$ defined by the formula $f(x, y, z) = 2x+ 2y-3z$ on the set $A$
Let $A = \{(x, y, z) \in \mathbb R^3 : x^2 + y^2 + z^2 = 2, xy + yz +zx +1 = 0\}$. Find infimum and supremum of the function $f$defined by the formula $f(x, y, z) = 2x+ 2y-3z$ on the set $A$.
I want to use Lagrange multipliers so I got the…
qerty149
- 106
2
votes
1 answer
Gradient of Lagrangian can not be set to 0
I am trying to solve the following constrained optimization problem:
$$ \begin{array}{rl} \underset{V \in \Bbb R^{D \times d}}{\operatorname{minimize}} & \operatorname{trace} \left( W V^\top \right) \\ \text{such that} & \frac12 x^\top VFV^\top x =…
Jiachun Jin
- 21
2
votes
3 answers
How to deal with the constraint $a \leq \|x\|_2 \leq b$ in an optimization problem?
How to solve the following optimization problem?
$$\begin{array}{c l} \underset {x} {\text{minimize}} & c^{\top} x\\ \text{subject to}~& a \leq \|x\|_2 \leq b
\end{array} $$
Can we convert it into a convex optimization problem?
Ryan
- 725
2
votes
1 answer
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 $g(x) :=…
ji zhi
- 21
- 3
2
votes
1 answer
Optimizing Trace$(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 $Q \in \mathbb{R}^{m \times n}$
$$\begin{array}{ll} \text{maximize} & \mbox{Tr} \left(Q^T Z \right)\\ \text{subject to} & Q^T Q = I_{n…
kkcocoqq
- 316
2
votes
2 answers
Maximisation of a piecewise affine function over an ellipsoid
Given vectors $\mathrm a, \bar{\mathrm x} \in \mathbb R^n$ and matrix $\mathrm P \in \mathbb S^n_{++}$, how to deal with the absolute value in the objective function of this optimization problem in $\mathrm x \in \mathbb R^n$?
$$\begin{array}{ll}
…
fire-bee
- 330