DDS (Domain-Driven Solver) Version 2.1
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.