SDPJSolver

SDPJ is a native Julia semidefinite program (SDP) solver.

• Motivated by the eft/modular bootstrap programs, SDPJ is a parallelized, arbitrary precision SDP solver based on the primal-dual interior-point method.
• SDPJ is largely inspired by SDPA and SDPB, with slightly different parallelization architecture.
• The solver is still in a development stage, which is far from fully optimized and might contain bugs. Corrections and suggestions are welcome and will get serious attention : )

The optimization problem

The function

sdp(c, A, C, B, b; β = 0.1, Ωp = 1, Ωd = 1, ϵ_gap = 1e-10, ϵ_primal = 1e-10, ϵ_dual = 1e-10, iterMax = 200, prec = 300)

solves the following SDP:

Primal

$$ \begin{aligned} \text{Minimize } \quad & c^T x \\ \text{subject to } \quad & X^{(l)} = \sum_i x_i A_i^{(l)} - C^{(l)} \geq 0, \quad l = 1, 2, ..., L \\ & B^T x = b \end{aligned} $$

Dual

$$ \begin{aligned} \text{Maximize } \quad & \sum_l tr(C^{(l)} Y^{(l)}) + b^T y \\ \text{subject to } \quad & \sum_l tr(A_{\star}^{(l)} Y^{(l)}) + B y - c = 0 \\ & Y^{(l)} \geq 0, \quad l = 1, 2, ..., L \end{aligned} $$

Domain

$$ \begin{aligned} x & \in \mathbb{R}^m \\ B & \in \mathbb{R}^{m \times n} \\ b, y & \in \mathbb{R}^n \\ A_i^{(l)}, C^{(l)}, X^{(l)}, Y^{(l)} & \in \mathbb{S}^{k^{(l)}} \\ \end{aligned} $$

Interior-point method

In each iteration, the program solves the following deformed KKT conditions to determine the Newton step:

• Primal feasibility

$$ X^{(l)} = \sum_i x_i A_i^{(l)} - C^{(l)} $$

• Dual feasibility

$$ \sum_l tr(A_{\star}^{(l)} Y^{(l)}) + B y - c = 0 $$

• Complementarity

$$ X^{(l)} Y^{(l)} = \mu^{(l)} I $$

Mehrotra's predictor-corrector method is used to accelerate convergence.

After a search direction is obtained, the step size is determined by requiring that $X$ and $Y$ remain positive.

The feasibility problem

The function

findFeasible(A, C, B, b; β = 0.1, Ωp = 1, Ωd = 1, ϵ_gap = 1e-10, ϵ_primal = 1e-10, ϵ_dual = 1e-10, iterMax = 200, prec = 300)

determines whether the SDP above is feasible. Note that the arguments are basically the same as sdp() except no vector c for the objective function is needed. The function converts the feasibility problem to the following optimization problem:

$$ \begin{aligned} \text{Minimize } \quad & t \\ \text{subject to } \quad & X^{(l)} = \sum_i x_i A_i^{(l)} - C^{(l)} + t I\geq 0, \quad l = 1, 2, ..., L \\ & B^T x = b \end{aligned} $$

If $t^* \geq 0$, the problem is infeasible; otherwise, the problem is feasible.

⚠️ Known issue: It seems that findFeasible() will fail if the feasible set is unbounded but works fine when the problem is infeasible.

Inputs

prec: arithemetic precision in base-10, which is equivalent to

setprecision(prec, base = 10)

The default value of the global variable T is BigFloat, which supports arbitrary precision arithmetic.

If accuracy is not a concern, the user can manually set T to other arithmetic types for improved performance, say Float64:

setArithmeticType(Float64)

c: $m$-element Vector{T}

A: $L$-element Vector{Array{T, 3}}

C: $L$-element Vector{Matrix{T}}

B: $m$ x $n$ Matrix{T}

b: $n$-element Vector{T}

β: factor of reduction in μ in each step

Ωp and Ωd are initial values for the matrices X and Y: $X = Ω_p I, Y = Ω_d I$

mode: can be either "opt" or "feas".

The iteration terminates if any of the following occurs:

• The function sdp() is used, and the duality gap, primal infeasibility, and dual infeasibility are below ϵ_gap, ϵ_primal, and ϵ_dual, respectively.
• The function findFeasible() is used, and the primal/dual infeasibilities reach their thresholds, with a certificate of $t^* > 0$ or $t^* < 0$ found.
• The number of iteration exceeds iterMax.

Outputs

The function sdp() returns a dictionary with the following keys:

• "x": value of the variable x
• "X": value of the variable X
• "y": value of the variable y
• "Y": value of the variable Y
• "pObj": value of the primal objective function
• "dObj": value of the dual objective fucntion
• "status": reports the status of optimization. Can be either
  • "Optimal"
  • "Feasible"
  • "Infeasible"
  • "Cannot reach optimality (feasibility) within iterMax iterations."