QOCO: A Quadratic Objective Conic Optimizer with Custom Solver Generation (2503.12658v2)

Abstract: Second-order cone programs (SOCPs) with quadratic objective functions are common in optimal control and other fields. Most SOCP solvers which use interior-point methods are designed for linear objectives and convert quadratic objectives into linear ones via slack variables and extra constraints, despite the computational advantages of handling quadratic objectives directly. In applications like model-predictive control and online trajectory optimization, these SOCPs have known sparsity structures and require rapid solutions. When solving these problems, most solvers use sparse linear algebra routines, which introduce computational overhead and hinder performance. In contrast, custom linear algebra routines can exploit the known sparsity structure of problem data and be significantly faster. This work makes two key contributions: (1) the development of QOCO, an open-source C-based solver for quadratic objective SOCPs, and (2) the introduction of QOCOGEN, an open-source custom solver generator for quadratic objective SOCPs, which generates a solver written in C that leverages custom linear algebra. Both implement a primal-dual interior-point method with Mehrotra's predictor-corrector. Our benchmarks show that QOCO is faster and more robust than many commonly used solvers, and solvers generated by QOCOGEN are significantly faster than QOCO and are free of dynamic memory allocation making them an attractive option for real-time optimization on resource-constrained embedded systems.

Custom Solver Generation for Quadratic Objective Second-Order Cone Programs

The paper presents innovative contributions in the domain of optimization, specifically focusing on quadratic objective second-order cone programs (SOCPs). This class of problems plays a crucial role in various applications, including optimal control, model predictive control, and real-time trajectory optimization, where rapid and efficient solutions are imperative. The authors introduce two novel tools: QOCO and QOCOGEN, both of which substantially enhance the efficiency of solving SOCPs with quadratic objectives.

Core Contributions

1. QOCO: This is an open-source C-based solver explicitly designed for quadratic objective SOCPs. It employs a primal-dual interior-point method augmented with Mehrotra's predictor-corrector approach. The design leverages custom linear algebra routines to exploit problem-specific sparsity structures, ensuring superior performance over traditional solvers relying on sparse linear algebra methods. The solver has demonstrated remarkable robustness and speed, outperforming several well-known commercial and open-source solvers in benchmarking tests.
2. QOCOGEN: An open-source custom solver generator that produces a tailored solver in C for quadratic objective SOCPs. The generator takes advantage of known sparsity patterns within problem data to eliminate the computational overhead typically associated with generic sparse linear algebra routines. Solvers generated by QOCOGEN (referred to as QOCO in the runtime environment) eschew dynamic memory allocation, thereby optimizing for execution on resource-constrained embedded systems where memory determinism is crucial.

Implementation Details

The authors meticulously detail the implementation strategy, focusing on efficiently solving the KKT system arising in the interior-point method. Regularization techniques — both static and dynamic — are employed to guarantee stability and invertibility of the KKT matrix. Furthermore, the introduction and rigorous implementation of iterative refinement ensure the accuracy of the computed solutions despite numerical perturbations induced during matrix factorization. Static memory allocation in QOCO minimizes execution-time memory management overhead, further enhancing real-time application viability.

Numerical Performance

The paper showcases robust numerical evidence of the proposed tools' superiority through extensive benchmarking. Problems from domains such as controlled trajectory planning and financial optimization demonstrate QOCO's and QOCOGEN's significant runtime advantages. Additionally, comparisons against the Maros–Mészáros test set — a standard benchmark in quadratic programming — underscore QOCO's robustness and efficacy, often presenting it as the fastest available solver for both typical and challenging problem instances.

Practical and Theoretical Implications

The introduction of QOCO and QOCOGEN has compelling implications:

• Theoretical Impact: The tailored approach to linear algebra and the dedication to exploiting problem structure mark a progressive shift in how optimization problems, specifically SOCPs with quadratic objectives, are approached. This method not only enhances theoretical solutions but also paves the way for future work that can leverage problem-specific structures in linear programming and more complex convex programs.
• Practical Applications: The elimination of dynamic memory allocation and reduction in computational overhead position the tools as ideal candidates for real-time systems requiring swift computation, such as in aerospace applications, real-time control systems, and embedded devices in robotics.

Future Work

Prospective developments in this field could explore extending the custom solver generation philosophy to other classes of optimization problems, such as those involving different cone types or mixed-integer quadratic programming. Furthermore, integrating machine learning approaches to predict and prepare problem-specific solver configurations may offer even deeper performance enhancements.

In conclusion, this research presents a significant step forward in optimizing the efficiency and speed of solving quadratic objective SOCPs. The authors' focus on exploiting problem-specific characteristics and eliminating common computational bottlenecks poses substantial benefits across both practical applications and theoretical advances in optimization methodology.