Conic Optimization via Operator Splitting and Homogeneous Self-Dual Embedding

Abstract

We introduce a first order method for solving very large convex cone programs. The method uses an operator splitting method, the alternating directions method of multipliers, to solve the homogeneous self-dual embedding, an equivalent feasibility problem involving finding a nonzero point in the intersection of a subspace and a cone.This approach has several favorable properties. Compared to interior-point methods, first-order methods scale to very large problems, at the cost of requiring more time to reach very high accuracy. Compared to other first-order methods for cone programs, our approach finds both primal and dual solutions when available or a certificate of infeasibility or unboundedness otherwise, is parameter-free, and the per-iteration cost of the method is the same as applying a splitting method to the primal or dual alone.We discuss efficient implementation of the method in detail, including direct and indirect methods for computing projection onto the subspace, scaling the original problem data, and stopping criteria. We describe an open-source implementation, which handles the usual (symmetric) non-negative, second-order, and semidefinite cones as well as the (non-self-dual) exponential and power cones and their duals. We report numerical results that show speedups over interior-point cone solvers for large problems, and scaling to very large general cone programs.

Embedded Control in Wearable Medical Devices: Application to the Artificial Pancreas

“…The solvers called by CVXPY are CVXOPT and ECOS for small-medium-scale problems and SCS (splitting conic solver) [117] for large-scale problems.…”

“…CVXPY [116] is an open source software package that converts the optimization problem, specified by the user, into a standard, conic form and interfaces with a solver to obtain the optimal solution. The solvers called by CVXPY are CVXOPT and ECOS for small-medium-scale problems and SCS (splitting conic solver) [117] for large-scale problems.…”

A New Architecture for Optimization Modeling Frameworks

“…In some problem domains the memory requirements even for sparse A matrices can be prohibitive (e.g., 2D convolution in large-scale image reconstruction), while at the same time efficient procedural evaluations of the matrix-vector computations with A and A T exist (e.g., FFT-based convolution). A possible solution is a first-order method, such as SCS [OCPB16], which only requires solving linear systems to moderate accuracy. This approach can be implemented with either a direct or indirect method for the linear solver subroutine.…”

“…Next we solve a convex problem with SCS [OCPB16]. In this case, the canonicalization step produces a problem in the standard cone form (1) and solver generation produces a graph implementing the SCS iterations.…”

“…In particular, the computation graph abstraction naturally represents the composition of structured linear operators which can be significantly more efficient than the standard sparse or dense matrix representation. We develop such a method in this paper and demonstrate that it outperforms the existing state of the art for solving large convex optimization problems, a GPU-enabled version of SCS [OCPB16], which itself is one of the few GPU-optimized solvers available, POGS [FB15] being another example.…”