# QSDPNAL

#### QSDPNAL — a MATLAB software for convex quadratic semidefinite programming

##### Xudong Li, Defeng Sun, Kim-Chuan Toh

This software is designed to solve primal convex quadratic semidefinite programming (QSDP) of the form:

$$\begin{eqnarray*} \begin{array}{rl} \min&\frac{1}{2}\langle X,\, \mathcal{Q}X \rangle +\langle C, \, X \rangle \\[5pt] {\rm s.t.}&{\cal A}_E(X)=b_E \\[5pt] &\mathcal{A}_I (X) \leq b_I \\[5pt] & X \in \mathbb{S}^n_+, \, X\in \mathcal{K} \end{array} \end{eqnarray*}$$

where $$\mathcal{Q}: \mathbb{S}^n \rightarrow \mathbb{S}^n$$ is a self-adjoint positive semidefinite linear operator, $$\mathcal{A}_E : \mathbb{S}^n\rightarrow \mathbb{R}^{m_E}$$, $$\mathcal{A}_I : \mathbb{S}^n\rightarrow \mathbb{R}^{m_I}$$ are linear maps, $$C\in \mathbb{S}^n$$, $$b_E\in \mathbb{R}^{m_E}$$, $$b_I\in \mathbb{R}^{m_I}$$ are given data, $$\mathcal{K}$$ is a simple closed convex polyhedral set defined by  $$\mathcal{K}=\{X\in\mathbb{S}^n \mid L\leq X\leq U\}$$ with $$L,U\in\mathbb{S}^n$$ being given bounds.

Important note.

• The software is still under development. Thus it will invariably be buggy. We would appreciate your feedback and bugs’ report.
• This is a research software. It is not intended nor designed to be a general purpose software at the moment.

##### Citation
• Xudong Li, Defeng Sun, and Kim-Chuan Toh, QSDPNAL: A two-phase augmented Lagrangian method for convex quadratic semidefinite programming, Mathematical Programming Computation, 10 (2018), pp. 703–743.