SDP Benchmark Problems

In the following a set of benchmark problems is presented. These convex optimization problems all involve positive semidefinite constraints and occur in various fields, such as portfolio optimization, graph theory, or robust control. These problems were collected to benchmark conic solvers against each other and to study the performance gains of various extensions to conic solvers. The problems are given in the following format:

min 1/2x'Px+q'x+r
s.t. Ax+s=b,
     s in K

with decision variables x ϵ R^n, s ϵ R^m and data matrices P=P'>=0, q ϵ R^n, A ϵ R^(m×n), and b ϵ R^m. The convex cone K is a composition of the zero cone, the non-negative orthant, a set of second order cones, and a set of positive semidefinite cones. The following list gives an overview over the considered problems (problem types with nonzero P in italic):

1. Random SDP with Quadratic Objective
2. Nearest Correlation Matrix
3. Smallest Sphere around multiple Ellipsoids
4. LMI-based Robust Control Problems
5. Semidefinite Relaxation of MIQO Problems
6. Lovasz Theta Function in Graph Theory

For more information about the problems check the Documentation. The problems are available as Julia JLD files and Matlab MAT files (see respective directories). Each file describes one problem and contains at least the following variables (some contain extra problem information):

Variable	Description	Type
m,n	problem dimension	Int
A,b	constraint data	SparseMatrixCSC, Vector
P,q,r	objective function data	SparseMatrixCSC, Vector, Float
objTrue	optimal obj value from MOSEK with standard tolerance	Float
problemType	descriptive name of the problem	String
problemName	short problem tag with number	String
Kf	Number of variables in zero-cone	Int
Kl	Number of variables in nonnegative orthant	Int
Kq	Dimensions of second-order cone variables	Array{Int}
Ks	Dimensions of positive semidefinite cone variables	Array{Int}

The Codefolder contains the Julia scripts used to generate the .jld-files. Furthermore, the script convertToMAT.jlcan be used to convert .jld-files into Matlab .mat-files.

Installation / Usage

• In Julia the .jld files can be loaded by using the JLD package. An example for a random SDP with quadratic objective is given here:

using JLD

filePath = #insert filepath here, e.g. "/Users/User1/DataFiles/Julia/SDPQuad/SDPQuad01.jld"
data = JLD.load(filePath)
P = data["P"]
q = data["q"]
r = data["r"]

A = data["A"]
b = data["b"]

m = data["m"]
n = data["n"]

Kf = data["Kf"]
Kl = data["Kl"]
Kq = data["Kq"]
Ks = data["Ks"]

objTrue = data["objTrue"]

Licence

This project is licensed under the Apache License - see the LICENSE.md file for details.

Contact

For questions or suggestions for other psd-problems (especially with quadratic cost), send an email 📧 to Michael Garstka 🚀!