Questions tagged [qclp]
A quadratically constrained linear program (QCLP) is an optimization problem in which the objective function is linear and the constraints are quadratic.
54 questions
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Convex optimization problem with quadratic constraint [closed]
$$ \min_{x} c^Tx $$ such that $$||Ax||^2_2 \le 1 $$ where $$A \succ 0$$
I feel like this should be easy but I have been struggling for hours. Is this a second-order cone constraint? How do I begin to ...
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Analytical solution for equality-constrained QCLP
I'm looking for a way to derive a closed form expression for $\omega$, which is the solution of the following maximization problem:
$$\max_{w} \omega^{T}\mu$$
$$\textrm{st.}\hspace{0.5cm} \omega^{T}\...
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Find infimum and supremum of the function $f$ defined by the formula $f(x, y, z) = 2x+ 2y-3z$ on the set $A$
Let $A = \{(x, y, z) \in \mathbb R^3 : x^2 + y^2 + z^2 = 2, xy + yz +zx +1 = 0\}$. Find infimum and supremum of the function $f$defined by the formula $f(x, y, z) = 2x+ 2y-3z$ on the set $A$.
I want ...
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Linear program plus sphere constraint
I have the following optimization problem:
\begin{align}
\min_{x} \quad & x^\top \hat{x} \\
\text{s.t.} \quad & x^\top a_i \leq 0, \quad \forall i \in \{ 1,\ldots,N \} \\
& \|x\|_2 = 1,
\...
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Dual formulation of a QCLP
I'm working on a quadratic constrained linear program (QCLP) which I'm trying to obtain its dual formulation. I would like this formulation to be linear, or at least convex. Here are my results so far....
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Solving a QCLP with both equality and inequality quadratic constraints
I have the following quadratically-constrained linear program (QCLP)
$$ \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 \...
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Gradient of Lagrangian can not be set to 0
I am trying to solve the following constrained optimization problem:
$$ \begin{array}{rl} \underset{V \in \Bbb R^{D \times d}}{\operatorname{minimize}} & \operatorname{trace} \left( W V^\top \...
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How to deal with the constraint $a \leq \|x\|_2 \leq b$ in an optimization problem?
How to solve the following optimization problem?
$$\begin{array}{c l} \underset {x} {\text{minimize}} & c^{\top} x\\ \text{subject to}~& a \leq \|x\|_2 \leq b
\end{array} $$
Can we convert it ...
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Maximizing linear objective subject to quadratic equality constraint
I've been trying to get through some practice questions on the Karush-Kuhn-Tucker (KKT) theorem but I can't seem to answer the following.
Given $f, g : \mathbb{R}^2 \to \mathbb{R}$ defined by $f(x) :=...
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Explicit solution of an QCLP
Give an explicit solution of the following QCLP $$\begin{split} & c^\top x \to \min\limits_{x \in \mathbb{R}^n }\\ \text{s.t. } & x^\top A x \leq 1\end{split}$$ where $A \in \mathbb{S}^n_{++}$ ...
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Sketch the graph of $ f (x_1, \: x_2) = x_2 $ over $ x_1 ^ 2 + x_2 ^ 2 \leq 2 $ and find the maximum value of $ f $.
Sketch the graph of $ f (x_1, \: x_2) = x_2 $ over $ x_1 ^ 2 + x_2 ^ 2 \leq 2 $ and find the maximum value of $ f $.
The graph would be what's inside the cylinder, I think. My affirmation is that the ...
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Maximize $f(x,y) = x+y$ subject to $x^2+xy+y^2+y=1$
I'm stuck trying to find the maximum value and at which point it occurs of the function
$$f(x,y) = x + y,$$
subject to the constraint $x^2 +xy +y^2 + y = 1$.
my working so far is
$f_x = \lambda g_x \...
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How can I solve this maximization problem with inequality constraint on the Frobenius norm?
$$
\max_{C\in \mathbb{S}^n} x^TCx\\
s.t. \|C\|_{tr}^2\leq b
$$
where $\|C\|_{tr}^2=tr(C^TC)=tr(CC)$, $C$ is a symmetric matrix.
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Solve with KKT min $x_1-4x_2+x_3$
Given min $x_1-4x_2+x_3$
s.t $x_1+2x_2+2x_3=-2$
$||x||^2\leq1$
(i)Given a KKT point of the problem above,must it be an optimal solution?
(ii) Find the optimal solution of the problem using KKT ...
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Linear objective with quadratic equality constraint
I've been given a problem:
min $f(x) = c^Tx$
s.t.
$\sum_{i}x_i = 0$
$\sum_{i}x^2_i$ = 1
I've created the Lagrangian function
$L(x,\lambda) = c^Tx - \lambda_1\sum_{i}x_i - \lambda_2\left(\left(\sum_{i}...
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Solve Mean-Variance Problem by Expected Return Maximization
One period market with n securities with return R~ $N(\mu,\Sigma)$. An expected return on each security is $\mu_1,\dots,\mu_n$
Investor has 1 unit of wealth and invests into n assets, with weights $...
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Maximize $a^T x$ with constraint on $2$-norm of $x$
In a textbook i'm reading, a maximization statement is given to maximize $a^Tx$ where the 2-norm of $x$ is smaller or equal than $c$, where $a$ and $x$ are vectors and $c$ is a scalar. The answer is ...
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Linear program with quadratic (L2 norm) constraint
I would like to solve an optimization problem of the type
$$ \min_x x^T c, \quad s.t. \|x\|_2 \leq k \; \land \; a \leq x_i \leq b \; \forall i$$
efficiently (note the quadratic norm constraint), ...
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If $a^2+b^2+c^2 =3$, find the minimum value of $a+b+c$
Given that their are 3 positive real numbers $a,b,c$, such that $a^2+b^2+c^2=3$, find the minimum value of $a+b+c$
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Lagrange Multiplier Find Maximum and Minimum
Use Lagrange multiplier to find maximum and minimum of $f(x,y) = 3x-4y$ subject to $x^2+3y^2=129$.
So I start by getting the partial with respect to both f(x, y) and $x^2+3y^2=129$
Ultimately, I get ...
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Optimizing Trace$(Q^TZ)$ subject to $Q^TQ=I$
Let $Z \in \mathbb{R}^{m \times n}$ be a tall matrix ($m > n$). Solve the following optimization problem in $Q \in \mathbb{R}^{m \times n}$
$$\begin{array}{ll} \text{maximize} & \mbox{Tr} \left(...
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Constrained convex optimization [duplicate]
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 ?
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Minimize with Lagrange multiplier
$$\begin{array}{ll} \text{minimize} & f(x,y,z) := x+2y+z\\ \text{subject to} & g(x,y,z) := x^2+y^2+z^2-1=0\end{array}$$
I get the following $\nabla f=(1,2,1)$ and $\nabla g=(2x,2y,2z)$. Now, ...
user732574
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Minimize $x + z$ subject to $x^2 + y^2 = 1$ and $y^2+z^2 = 4$
I'm trying to solve this problem by KKT's condition:
$$\begin{align*}
\text{min} & \quad x + z \\
\text{s.t} & \quad x^2 + y^2 = 1 \\
& \quad y^2+z^2 = 4
\end{align*}$$
One ...
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Minimise $2x+y$ subject to $x^2+y^2=1$ using KKT
I'm doing this exercise in preparing for the final exam in optimization:
$$\begin{align*}
\text{min} &\quad 2x+y \\
\text{s.t} & \quad x^2+y^2=1
\end{align*}$$
Could you please verify if I ...
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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 $\...
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Change of variables in QCLP
Is there any change of variables that makes the following optimization problem easier to solve?
\begin{align}
\max_{x\in\mathbb{R}^n,t\in\mathbb{R}}\quad & c^\top x,\\
\mbox{s.t.}\quad\quad & ...
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Constrained optimization problem exam question
Consider the following constrained optimization problem:
$$\min_{x} f = x_1 \\ \text{such that} \quad g_1=x_1^2+x_2^2-9 \leq0 \\ \qquad \qquad \ \ \quad \ g_2=-x_1^2-x_2^2+4 \leq0 \\ \qquad \ \ \ \ \ ...
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"Convert" quadratic constraint to quadratic objective
I have a large sparse quadratic optimization problem with a single quadratic constraint:
$$\begin{array}{ll} \text{maximize} & c'x\\ \text{subject to} & l \leq Ax \leq u\\ & x'Qx + b'x \...
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Finding maximum of $y-z$ given quadratic constraint
Problem
According to my friend, this is a problem that seems to be solvable by observation.
Try to find the maximum of $(y-z)$ given following two constraints
$$
\begin{aligned}
x+y+z=3\\
x^2+y^2+z^...
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4
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Difficulty finding Lagrange multiplier because of $\leq$
Let $f: \mathbb R^3 \to \mathbb R$ be defined by $$f(x,y,z)=x-y+z$$
and $$E:=\{(x,y,z)\in \mathbb R^{3} \mid x^2+2y^2+2z^2\leq1\}$$ Find the extrema of $f$ on $E$.
Path: I have already proven that ...
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Find a minimum and a maximum of expression $E(x,y,z)=x-2y+z$
Find minimum and maximum of $E(x,y,z)=x-2y+z$ where $2x^2+y^2+z^2=3$.
For finding the maximum of the expression I first thought using the inequality of Cauchy-Buniakowski-Schwarz: $$\boxed{(ax+by+cz)^...
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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. ...
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Optimization of linear objective with quadratic integer constraint
Is there any way to solve an linear optimization problem with quadratic integer constraint?
E.g.
$\max a^Tx$, $x=\langle x_1,x_2,\cdots,x_n \rangle$
s.t. $x_ix_j<b_{ij}$, $\forall x_i,x_j \in x$
...
2
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1
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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 ...
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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?
$$\begin{array}{ll} \text{minimize} &...
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Boolean quadratically constrained linear program (QCLP)
1) I have the following problem that I would like to first solve optimally but I have not been able to express it in a way that can be accepted by Matlab optimization functions.
$$\begin{array}{ll}
...
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Solving linear program with 1 quadratic constraint complexity
Consider the following linear program,
$$\min y \\
xc_1 \leq c_2 + yz,\\
x = x_1 + \dots + x_n,\\
z \leq x_1 + x_2, \\
z \leq x_2 + x_3, \\
\vdots\\
z \leq x_{n-1} + x_n, \\
x,x_1, \dots, x_n,y,z \...
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If $a,b$ are positive integers and $x^2+y^2\leq 1$ then find the maximum of $ax+by$ without differentiation.
If $x^2+y^2\leq 1$ then maximum of $ax+by$
Here what I have done so far.
Let $ax+by=k$ . Thus $by=k-ax$.
So we can have that $$b^2x^2+(k-ax)^2 \leq b^2$$
$$b^2x^2+k^2-2akx +a^2x^2-b^2\leq 0 $$
By ...
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2
answers
224
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How to prove the maximum of n-dimensional linear function $a^Tx$ is $||a||_2$ for $||x||_2 \le 1$
I can get it when $x \in \mathbb R$, but I cannot understand why
$$\sup\{a^Tx \mid \|x\|_2 \le 1 \} = \| a \|_2$$ when $x \in \mathbb R^n$. To my understanding, this problem can be transformed into ...
- 1,243
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3
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Determining the minimum of a linear function subject to a quadratic inequality constraint
What is the minimum value of $x+4z$, a function defined on $\mathbb{R^3}$, subject to the constraint $x^2 + y^2 +z^2 \leq 2$?
I know how to solve this if the constraint is an equality, but what shall ...
user426277
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Maximize $\mbox{tr} (AX)$ subject to $\mbox{tr} (BX'CX) = 1$
Suppose $A,B,C$ are given square matrices and $X$ is a matrix variable. Is there a nice way to express the following?
$$\begin{array}{ll} \text{maximize} & \mbox{tr} (AX)\\ \text{subject to} &...
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Python solver for a linear objective function with quadratic constraints
I need to optimize a problem of the following form within Python
$ minimize$
$\qquad abs(\Delta w) \;\epsilon^T$
$s.t.$
$\qquad(w+\Delta w)C(w+\Delta w)^T \; \le 9500 (some\; finite\; number) \\\\[...
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How to solve this convex optimization problem (with absolute and linear objective function)?
I have the following problem:
\begin{align*}
\sup_y&\quad \big | \langle u,y \rangle\big |\\
\mbox{s.t.}&\quad \frac{1}{2}\langle y,y \rangle\ + \langle b,y \rangle\ \geq \gamma.
\end{align*...
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votes
2
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2k
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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 (...
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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
$$\sum_{i=1}^m x_i\phi_i - \left(\sum_{...
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1
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How to maximize $\langle X, Y \rangle$ over $X$ under constraint s.t $\mbox{Tr} (X^TAX)=1$?
Maximize $\langle X, Y \rangle$ over $X$ subject to the constraint $\mbox{Tr} (X^TAX) = 1$.
Here $A$ is positive semi-definite and also all matrices have real entries and $Y$ is a known matrix given ...
- 301
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4
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Maximize $\langle \mathrm A , \mathrm X \rangle$ subject to $\| \mathrm X \|_F = 1$
Given $\mathrm A \in \mathbb R^{m \times n}$,
$$\begin{array}{ll} \text{maximize} & \langle \mathrm A , \mathrm X \rangle\\ \text{subject to} & \| \mathrm X \|_F = 1\end{array}$$
I can think ...
2
votes
2
answers
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Minimize linear objective subject to unit norm constraint
I have a linear function with unit norm constraint that I need to minimize.
$$\begin{array}{ll} \underset{w}{\text{minimize}} & w^\top x\\ \text{subject to} & \|w\| = 1 \end{array}$$
Is ...
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votes
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Maximizing a linear function over an ellipsoid
Let $A \in \mathbb{R}^{n\times n}$ be a positive definite matrix, $x \in \mathbb{R}^n$ and $c \in \mathbb{R} \setminus \{0\}$. Determine the maximum $$ \max_{y \in \mathbb{R}^n} \left\{ c^T y : y \in \...
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