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.
  • Download and Install DDS software package
  • 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
    • Quadratic Constraints
    • Direct Sum of 2-dim Convex Sets (including geometric programming and entropy programming)
    • 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 \(\ln(x)\).
    • Epigraph of Matrix Norm (including nuclear norm minimization)
    • 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.
    • Vector Relative Entropy
    • 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
    • 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
    • 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.
    • Hyperbolic Optimization
    • Constraints of the form \[ p(Ax+b) \geq 0, \] where \(p(x)\) is a hyperbolic polynomial.