# DDS 2.0 Users' Guide

DDS (Domain-Driven Solver) Version 2.0
Matlab-based software package for convex optimization problems given in the Domain-Driven form.

• Theory behind the Domain-Driven formulation
• DDS Users' Guide (pdf), arXiv link for citation.
• How to use DDS
• Types of function/set constraints that DDS solves:
• Linear Programming (LP) and Second Order Cone Programming (SOCP)
• Semidefinite Programming (SDP)

• Constraints of the form \begin{eqnarray} \label{SDP-1} F^i_0+x_1 F^i_1+ \ldots+x_n F^i_n \succeq 0, \ \ \ i=1,\ldots,\ell. \end{eqnarray} where $$F^i_j$$'s are $$n_i$$-by-$$n_i$$ symmetric matrices.
• Generalized Power Cone
• Every inequality of the form \begin{eqnarray} \label{intro-3} \sum_{i=1}^\ell \alpha_i f_i(a_i^\top x + \beta_i) + g^\top x + \gamma \leq 0, \ \ \ a_i, g \in \mathbb R^{n}, \ \ \beta_i, \gamma \in \mathbb R, \ \ i \in \{1,\ldots,\ell\}, \end{eqnarray} where $$\alpha_i \geq 0$$ and $$f_i(x)$$, $$i \in \{1,\ldots,\ell\}$$, can be a univariate convex function such as $$e^x$$ or $$\ln(x)$$.
• Matrix constraints of the form \begin{eqnarray} \label{EO2N-1} && X-UU^\top \succeq 0, \nonumber \\ && X=A_0+\sum_{i=1}^{\ell_1} x_i A_i, \nonumber \\ && U=B_0+\sum_{i=1}^{\ell_2} u_i B_i, \end{eqnarray} where $$A_i$$, $$i \in \{0,\ldots,\ell_1\}$$, are $$m$$-by-$$m$$ symmetric matrices, and $$B_i$$, $$i \in \{0,\ldots,\ell_2\}$$, are $$m$$-by-$$n$$ matrices.
• Constraints involving the vector relative entropy function $$f: \mathbb R_{++}^\ell \oplus \mathbb R_{++}^\ell \rightarrow \mathbb R$$ defined as $f(u,z):= \sum_{i=1}^{\ell} u_i\ln(u_i) - u_i\ln(z_i).$
• Quantum entropy constraints of the form \begin{eqnarray} \label{eq:QE-1} qe(A^i_0+x_1 A^i_1+ \cdots+x_n A^i_n) \leq g_i^\top x+d_i, \ \ \ i\in\{1,\ldots,\ell\}, \end{eqnarray} where $$qe(X):=\text{TR}(X\ln(X))$$, and $$F^i_j$$'s are $$n_i$$-by-$$n_i$$ symmetric matrices.
• Quantum relative entropy constraints of the form \begin{eqnarray} qre(A^i_0+x_1 A^i_1+ \cdots+x_n A^i_n, B^i_0+x_1 B^i_1+ \cdots+x_n B^i_n) \leq g_i^\top x+d_i, \ \ \ i\in\{1,\ldots,\ell\}, \end{eqnarray} where $$qre(X):=\text{TR}(X\ln(X)-X\ln(Y))$$, and $$A^i_j$$, $$B^i_j$$'s are $$n_i$$-by-$$n_i$$ symmetric matrices.
• Constraints of the form $p(Ax+b) \geq 0,$ where $$p(x)$$ is a hyperbolic polynomial.