Questions on quadratic programming, the optimization of a quadratic objective function subject to affine constraints.

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### What is the fastest way to solve a large-scale quadratic programming problem with inequality constraint?

I am trying to solve problem as follows: \begin{equation} \begin{split} &\mathop{min} \limits_{\textbf{x}_1...\textbf{x}_k} \ \frac{1}{2}\textbf{x}_1^T\textbf{Q}_1\textbf{x}_1+\textbf{q}_1^T\...
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### Second Order Cone Program with Quadratic Objective Function

The standard form for a Second Order Cone Program (SOCP) is \begin{equation} \begin{array}{c} \min _{x} f^{T} x \\ \left\|A_{i} x+b_{i}\right\|_{2} \leq c_{i}^{T} x+d_{i}, i=1, \ldots, m \end{array} \...
1 vote
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### Rewriting quadratic optimization problem with negative definite matrix

Consider the following problem: Minimize $x(1-x) + y(1-y) + z(1-z)$ subject to: $$a. 0\leq x, y, z \leq 1$$ $$b. x+y+z = 1$$ If I plug in this problem in a standard quadratic optimizer, which ...
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### Explicit Solution of Quadratic Opt. Problem

I have the following optimization problem I am unsure whether I've got it correct: $$\text{min } x^Tx \\ \text{so that } a + c^Tx <= 0$$ I have introduced a slack variable $s$ to make it ...
1 vote
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1 vote
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### Quadratic Minimization Problem with positivity constraint

Let $A \in\mathbb{R}^{m\times n}$, $b,c\in\mathbb{R}^m$, $x\in \mathbb{R}^n$. Consider the following minimization problem: $$\min_{x\succeq 0} f(x):= \frac{1}{2}\|Ax-c\|^2 + b^\top Ax.$$ For the ...
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Let $A \in\mathbb{R}^{m\times n}$, $b,c\in\mathbb{R}^m$, $x\in \mathbb{R}^n$. Consider the following minimization problem: $$\min_{x} f(x):= \frac{1}{2}\|Ax-c\|^2 + b^\top Ax.$$ Since the function ...
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### The dual function of non-convex QP

I am trying to find the dual function of the follwoing non-convex QP \begin{equation*} \min \frac{1}{2}x^T Q x \\ Ax = b, 0\leq x \leq e \end{equation*} The Lagrangian function is given by \begin{...
1 vote
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### Minimizing $b^Tb$ subject to $Ax=b, x\geq 0, x\leq 1$

I have the following quadratic program: Minimize $b^Tb$ subject to $Ax=b$ where $A$ is a $n\times m$ matrix ($n\leq m$) of rank $n-1$. I also want $0\leq x\leq 1$. For my choice of $A$, I can prove ...
1 vote
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### Hessian of quadratic objective function

I have the quadratic function $$f(\boldsymbol{x}) = \frac{3}{2} \left (x_{1}^{2}+x_{2}^{2} \right) + (1+a) x_{1} x_{2} - \left(x_{1} + x_{2} \right) + b$$ where $a, b \in \Bbb R$ are unknown ...
1 vote