0
votes
1answer
28 views

Reentrant constraints in active set algorithm?

Problem definition Supposing you're trying to solve a quadratic program: $$ \min_x f(x) = \frac{1}{2}x^T Q x + c^T x \\ \mbox{s.t} \, \; A x \ge b$$ Where Q is square ($n$x$n$), positive semi ...
3
votes
2answers
117 views

What numerical methods are known to solve $L_1$ regularized quadratic programming problems?

What numerical methods are suitable to solve the following problem $$\min_x \tfrac{1}{2}x^T A x + b^Tx + \lambda ||x||_1$$ where $x,b\in\mathbf{R}^n$, and $A\in \mathbf{R}^{n\times n}$ is positive ...
1
vote
1answer
51 views

A particular quadratic minimization problem

Given $n^2$ constants $a_{11},a_{12},\ldots,a_{1n},a_{21},\ldots,a_{nn}$ and $n^2$ non-negative variables $x_{11},x_{12},\ldots,x_{1n},x_{21},\ldots,x_{nn}$. Find the minimum value of $$\sum_{i=1}^n ...
0
votes
0answers
5 views

Perturbation of Polyhedral Projection

I am interested in understanding the behavior of the Euclidean projection $\pi_K(x)$ as the polyhedral set $K$ varies. I know there are different approaches to this, but for what I am doing it would ...
0
votes
1answer
46 views

Homogeneous non-negative least-squares

I would like to least-squares-"solve" a set of linear equations ($\underset{\mathbf{x}}{\mathrm{argmin}}\; \|\mathbf{Ax-b}\|_2$). In my case, $\mathbf{b=0}$, e.g. the system is homogeneous. I also ...
1
vote
1answer
29 views

Equality constrained Quadratic Program

Consider the QP $$ x^* = \arg \min_{\displaystyle x \in \mathbb{R}^n{\geq 0}} \ \frac{1}{2} x^\top P x + q^\top x \ \text{ sub. to: } A x = b, $$ where $P \succ 0$. Without the non-negativity ...
1
vote
0answers
17 views

Uniqueness of the solution to a quadratic opt problem

Consider a positive definite matrix $\boldsymbol H$, the known vectors ${\boldsymbol b}$ and ${\boldsymbol a}_i$. Now the minimization problem is casted with respect to the vector ${\boldsymbol x} $ ...
0
votes
0answers
31 views

Implementing SVM: Help converting equation into form of another

I'm currently programming a simple linear SVM (Support Vector Machine). For the optimization involved, I need to find a way to convert the equation $\sum\limits_{i=1}^L a_i ...
1
vote
2answers
57 views

Trace of quadratic function with 2 PSD matrices - convex?

If A & B are positive semi-definite, is this always convex: $$ trace(XAX^TB) $$ There was a similar question asked here: Trace of a quadratic function, Convexity and here: Confusion related to ...
0
votes
1answer
60 views

Minimization problem convex set

I'm trying to minimize the function: $$f(w)=w^T\mu+k\sqrt{w^T\Sigma w}$$ where $w$ is a vector in $W=\{x \in \mathbb{R}^n|x_1+...+x_n=1 , x_i \geq 0 \forall i\}$. The vector $\mu \in \mathbb{R}^n$, ...
2
votes
0answers
57 views

Quadratic Integer Programming

Would anyone mind helping me solve this problem $$ \min\space f(x) = \frac12 x^\mathrm TQx + bx + c \qquad \text{s.t. } \sum_i x_i=\lambda $$ where $x$ is a vector whose entries are positive ...
0
votes
1answer
60 views

How to re-parametrize for quadratic minimization?

Given a real-rectangular matrix $S$ and inorder to solve this simple quadratic programming problem: Minimize $w'S'Sw = \|S w\|^2$ over $w$ subject to $e^Tw = 1$ and $w \geq 0$ using a solver I ...
2
votes
1answer
170 views

Convex optimization and linear programming please help! :)

How would I write the following as a standard form LP? Minimizing $\sum_{i=1}^n x_i + c\max(a_i-x_i)$ for $a_i \ge 0$ and what is the optimal value for when $c=n$ How to express minimize $\frac{1}{2} ...
3
votes
2answers
317 views

A standard quadratic minimization problem

Consider the "Complex" Quadratic minimization problem \begin{align} \min_{\mathbb{x}\in \mathbb{C}^{N \times 1}}~\mathbf{{x}}^H\mathbf{Q}\mathbf{x}-2~\Re{(\mathbf{x}^H\mathbf{b})}+1 \end{align} ...
1
vote
0answers
90 views

Minimize a complex quadratic subject to two convex quadratic constraints

I have the following the optimization problem \begin{align} \min_{\mathbb{x}\in \mathbb{C}^{N \times 1}}~&||\mathbb{x}^H\mathbb{u}||_2^2-2*Real\{ \mathbb{x^Hu}\} \\\ ...
0
votes
1answer
156 views

Convex optimization problem to quadratic programming problem

Briefly, have the following problem: \begin{equation} \sum_{i = 0}^n a_i \ (max [ F_i( \bar x ), 0 ] )^2 \rightarrow min, \\\\ s.t.\\\\ A \bar x \leq b \end{equation} where $ F( \bar x ) $ is a ...