# 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.

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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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### 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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### 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 ...
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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 ...
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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 ...
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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 ...
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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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### 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 $$\max\{c^Ty:y\in \mathcal{E} (A,x)\}$$ ...
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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 $y_3(... 1answer 417 views ### Linear programming with non-convex quadratic constraint Could anyone let me know if the following linear programming problem can be solved in polynomial time or should be NP-hard?$\min c^Tx$s.t.$x^TQx\geq C^2, x\in [0,1]^n,c\in \mathbb{R}_+^n,Q\in\...
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 $n \times n$ ...