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
12
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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
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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
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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 $c \in \mathbb{R} \setminus \{0\}$ . 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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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
- 41
3
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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
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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
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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 $$\sum_{i=1}^m x_i\phi_i - \left(\sum_{i=1}^m x_i\right) \left(\sum_{j=m+1}^n x_j\right)…
user402078
- 31
3
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3 answers
Linear programming with quadratic constraints
I have a given set of variables: $x_1,y_1,x_2,y_2,x_3,y_3$ The objective function is to minimize the sum of these with quadratic equality constraints: $y_1(x_1+x_2+x_3)$=0 $y_2(x_2+x_3)$=0 $y_3(x_3)$=0 There are other inequality constraints which…
user1234
- 71
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2 answers
Find infimum and supremum of the function $f$ defined by the formula $f(x, y, z) = 2x+ 2y-3z$ on the set $A$
Let $A = \{(x, y, z) \in \mathbb R^3 : x^2 + y^2 + z^2 = 2, xy + yz +zx +1 = 0\}$ . Find infimum and supremum of the function $f$ defined by the formula $f(x, y, z) = 2x+ 2y-3z$ on the set $A$ . I want to use Lagrange multipliers so I got the…
qerty149
- 106
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Solving a QCLP with both equality and inequality quadratic constraints
I have the following quadratically-constrained linear program (QCLP) $$ \begin{array}{ll} \underset {x} {\text{minimize}} & -c^T x \\ \text{subject to} & x^T Q x \leq a^2 \\ & x^T L x = 1 \end{array} $$ where $Q$ and $L$ are both symmetric positive…
2
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1 answer
Gradient of Lagrangian can not be set to 0
I am trying to solve the following constrained optimization problem: $$ \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
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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
- 705
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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 $g(x) := x_1^2+3x_1x_2+3x_2^2-3$…
ji zhi
- 21
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
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