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DDS 2.2 Users' Guide

DDS (Domain-Driven Solver) Version 2.2
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)

\begin{array}{r} F_{0}^{i} + x_{1} F_{1}^{i} + \dots + x_{n} F_{n}^{i} ⪰ 0, i = 1, \dots, ℓ . \end{array}

F_{j}^{i}

n_{i}

n_{i}

Generalized Power Cone
Quadratic Constraints
Direct Sum of 2-dim Convex Sets (including geometric programming and entropy programming)

\begin{array}{r} \sum_{i = 1}^{ℓ} α_{i} f_{i} (a_{i}^{⊤} x + β_{i}) + g^{⊤} x + γ \leq 0, a_{i}, g \in R^{n}, β_{i}, γ \in R, i \in {1, \dots, ℓ}, \end{array}

α_{i} \geq 0

f_{i} (x)

i \in {1, \dots, ℓ}

e^{x}

\ln (x)

Epigraph of Matrix Norm (including nuclear norm minimization)

\begin{array}{rcl} X - U U^{⊤} ⪰ 0, \\ X = A_{0} + \sum_{i = 1}^{ℓ_{1}} x_{i} A_{i}, \\ U = B_{0} + \sum_{i = 1}^{ℓ_{2}} u_{i} B_{i}, \end{array}

A_{i}

i \in {0, \dots, ℓ_{1}}

m

m

B_{i}

i \in {0, \dots, ℓ_{2}}

m

n

Vector Relative Entropy

f : R_{+ +}^{ℓ} \oplus R_{+ +}^{ℓ} \to R

f (u, z) := \sum_{i = 1}^{ℓ} u_{i} \ln (u_{i}) - u_{i} \ln (z_{i}) .

Quantum Entropy

\begin{array}{r} q e (A_{0}^{i} + x_{1} A_{1}^{i} + \dots + x_{n} A_{n}^{i}) \leq g_{i}^{⊤} x + d_{i}, i \in {1, \dots, ℓ}, \end{array}

q e (X) := TR (X \ln (X))

F_{j}^{i}

n_{i}

n_{i}

Quantum Relative Entropy

\begin{array}{r} q r e (A_{0}^{i} + x_{1} A_{1}^{i} + \dots + x_{n} A_{n}^{i}, B_{0}^{i} + x_{1} B_{1}^{i} + \dots + x_{n} B_{n}^{i}) \leq g_{i}^{⊤} x + d_{i}, i \in {1, \dots, ℓ}, \end{array}

q r e (X) := TR (X \ln (X) - X \ln (Y))

A_{j}^{i}

B_{j}^{i}

n_{i}

n_{i}

Hyperbolic Optimization

p (A x + b) \geq 0,

p (x)