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Mathematics

Questions tagged [semidefinite-programming]

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This tag is for questions regarding semidefinite programming (SDP) which is a subfield of convex optimization concerned with the optimization of a linear objective function (an objective function is a user-specified function that the user wants to minimize or maximize) over the intersection of the cone of positive semidefinite matrices with an affine space, i.e., a spectrahedron.

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I'm trying to understand how to express a simple LP in the standard semidefinite programming (SDP) form.In particular, consider the following linear program:$$\begin{aligned}\min_{x_1, x_2} \;&...
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I am interested in whether the following structured partial symmetric matrix can be completed to a positive semidefinite (PSD) matrix.The matrix $Z$ is a real symmetric $(n+1) \times (n+1)$ matrix....
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I am considering the following regularized binary quadratic optimization problem with a sparsity penalty$$ \min_{{\bf x} \in \{\pm 1\}^n} \; {\bf x}^\top {\bf C} \, {\bf x} + \| {\bf A} {\bf x} - {\...
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Let $V\subset \mathbb{R}^n$ be a set of points on the $n$ dimensional unit sphere. I want to find the maximum size of $V$ such that $d(x,y)=\|x-y\|_2^2$(the square of the usual Euclidean distance) is ...
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I've been trying to provide some examples of semidefinite programs (SDPs) with only equality constraints, but I'm having a surprising amount of trouble! In particular, I was hoping to find symmetric ...
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Given matrices $Q$ and $R \succ 0$, the Kalman filter problem is$$ \begin{array}{ll} \underset {P \succeq0, K} {\text{minimize}} & \operatorname{tr} (P) \\ \text{subject to} & (A - K C) P (A -...
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Can we formulate as a linear matrix inequality (LMI) one of the regions in which a concave quadratic polynomial is negative (without solving its roots)?Let the quadratic polynomial $f \in {\Bbb R} [x]...
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I have the following convex optimization problem:$$ \begin{array}{ll} \underset {{\bf A}, {\bf B}, {\bf X},{ \bf Y}} {\text{minimize}} & \sum\limits_{k \in \mathcal{K}} {\left\| {\bf x}_k \right\|...
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I want to find a low-rank symmetric positive semidefinite matrix, subject to some linear constraints, while maximizing a secondary cost that depends on the sum of the entries of this matrix.$$ \begin{...
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On the Wikipedia page on spectrahedron, it shows the following picture of a spectrahedron (I avoid using the term image since it seems to have special meaning in this context)I understand everything ...
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I have a Hermitian matrix $H \in \mathbb{C}^{n^2 \times n^2}$ and I want to solve the following optimization problem.$$\max_{v \in \mathbb{C}^n, \ v^\dagger v = 1} \ (v \otimes v)^\dagger H (v \...
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I am trying to solve the following optimization problem (with application in sensor placement)$$\begin{align}\max_{x,y,\eta} \quad & \eta\\\text{s.t.}\quad & y \leq a\\& x^2+\eta \leq ...
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In system theory, we often encounter Semi-Definite Programs (SDPs) with Linear Matrix Inequality (LMI) constraints, such as those presented in this paper. I have introduced new variables based on the ...
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I have the following convex program that is derived from the theory of distributionally-robust optimization.\begin{align*}&\inf \quad y_0 + \gamma (\rho^2 - \|\hat\mu\|^2 - \mathrm{tr}[\hat\...
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Let $X, Y \in \mathbb{R}^{n \times p}$ be generic. Let $M^*\in \mathbb{R}^{p\times p}$ minimize $\left\|Y - X M\right\|^2$ subject to the semidefinite constraint $M \geq 0$. Under what conditions is $...

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