[Paper Review] Towards practical generic conic optimization

Chris Coey, Lea Kapelevich+1 more|arXiv (Cornell University)|May 3, 2020

Advanced Optimization Algorithms Research5 references7 citations

TL;DR

This paper introduces Hypatia, a generic conic primal-dual interior point solver in Julia that natively supports exotic cones via efficient self-concordant barrier oracles. It demonstrates that solving natural formulations with Hypatia outperforms solving extended formulations with Hypatia or MOSEK in solve time, memory usage, and numerical robustness across six test problems.

ABSTRACT

Many convex problems can be represented through conic extended formulations with auxiliary variables and constraints using only the small number of standard cones recognized by advanced conic solvers such as MOSEK 9. Such extended formulations are often significantly larger and more complex than equivalent conic natural formulations, which can use a much larger class of exotic cones. We define an exotic cone as a proper cone for which we can implement efficient logarithmically homogeneous self-concordant barrier oracles for the cone or its dual. Our goal is to establish whether a generic conic interior point method supporting natural formulations can outperform an advanced conic solver specialized for standard cones. We introduce Hypatia, a highly-configurable open-source conic primal-dual interior point solver with a generic interface for exotic cones. Hypatia is written in Julia and accessible through a native interface or the modeling language JuMP, and currently implements over two dozen predefined cones of interest. Using six example problems, we demonstrate significant advantages in terms of solve time, memory usage, and numerical robustness from solving natural formulations with Hypatia, compared to solving extended formulations with Hypatia or MOSEK.

Motivation & Objective

• To evaluate whether a generic conic interior point method supporting natural formulations can outperform specialized solvers for standard cones.
• To develop a highly-configurable open-source conic solver that natively supports exotic cones through efficient barrier oracles.
• To demonstrate the advantages of using natural conic formulations over extended formulations in terms of computational efficiency and numerical stability.
• To provide a flexible, accessible interface for exotic cones via Julia and JuMP, enabling broader adoption in conic optimization.

Proposed method

• Design and implement Hypatia, a primal-dual interior point solver in Julia with a generic interface for exotic cones.
• Implement efficient logarithmically homogeneous self-concordant barrier oracles for over two dozen predefined exotic cones and their duals.
• Support both native modeling through a Julia API and integration with the JuMP modeling language for ease of use.
• Formulate problems using natural conic formulations involving exotic cones, avoiding the need for extended formulations with standard cones.
• Benchmark performance across six test problems using solve time, memory usage, and numerical robustness as metrics.
• Compare Hypatia’s performance on natural formulations against both Hypatia and MOSEK on equivalent extended formulations.

Experimental results

Research questions

• RQ1 Can a generic conic interior point method that natively supports exotic cones outperform specialized solvers for standard cones?
• RQ2 Does using natural conic formulations with exotic cones lead to better computational performance than extended formulations with standard cones?
• RQ3 To what extent do exotic cones improve memory efficiency and numerical robustness in conic optimization?
• RQ4 How does the performance of Hypatia compare to MOSEK 9 when solving equivalent problems in different formulations?

Key findings

• Hypatia significantly reduces solve time when solving natural formulations with exotic cones compared to extended formulations with standard cones.
• Memory usage is substantially lower when using natural formulations with Hypatia than when solving extended formulations with Hypatia or MOSEK.
• Numerical robustness is improved when using natural formulations with Hypatia, as evidenced by fewer convergence failures and better conditioning.
• The performance advantage of Hypatia on natural formulations is consistent across six diverse test problems, demonstrating broad applicability.