[論文レビュー] Clarabel: An interior-point solver for conic programs with quadratic objectives

Paul J. Goulart, Yuwen Chen|arXiv (Cornell University)|May 21, 2024

Advanced Optimization Algorithms Research被引用数 13

ひとこと要約

Clarabelは、2次目的を持つ凸円錐計画の一般目的 primal-dual 内点ソルバーで、均質埋め込みと対称・非対称錐の特殊な扱い、SDPのChordal分解を活用します。

ABSTRACT

We present a general-purpose interior-point solver for convex optimization problems with conic constraints. Our method is based on a homogeneous embedding method originally developed for general monotone complementarity problems and more recently applied to operator splitting methods, and here specialized to an interior-point method for problems with quadratic objectives. We allow for a variety of standard symmetric and non-symmetric cones, and provide support for chordal decomposition methods in the case of semidefinite cones. We describe the implementation of this method in the open-source solver Clarabel, and provide a detailed numerical evaluation of its performance versus several state-of-the-art solvers on a wide range of standard benchmarks problems. Clarabel is faster and more robust than competing commercial and open-source solvers across a range of test sets, with a particularly large performance advantage for problems with quadratic objectives. Clarabel is currently distributed as a standard solver for the Python CVXPY optimization suite.

研究の動機と目的

• Provide a general-purpose interior-point method for convex conic optimization with quadratic objectives.
• Leverage homogeneous embedding to unify feasibility, optimality, and infeasibility certificates.
• Support a broad set of cones (symmetric and nonsymmetric) and enable chordal decomposition for SDPs.
• Demonstrate performance improvements over state-of-the-art solvers on benchmarks.
• Integrate the solver into the Python CVXPY optimization suite.

提案手法

• Formulate the cone program as a convex problem with quadratic objective and conic constraints.
• Adopt a homogeneous embedding approach adapted to convex problems with quadratic objectives to obtain a single solvable feasibility problem.
• Use a primal-dual interior-point method with central-path equations and Newton-like step directions.
• Apply different scaling and barrier strategies for symmetric (NT scaling) and nonsymmetric cones (BFGS-based scaling).
• Incorporate chordal decomposition for SDPs via compact/range-space reformulations and clique merging strategies.
• Provide solver initialization, equilibration (Ruiz), and robust termination/infeasibility detection criteria.

実験結果

リサーチクエスチョン

• RQ1 How can the homogeneous embedding approach be specialized to efficiently solve convex conic programs with quadratic objectives?
• RQ2 What are the performance and robustness benefits of Clarabel across symmetric and nonsymmetric cones compared to existing solvers?
• RQ3 How does chordal decomposition affect the solvability and speed of large-scale semidefinite problems within this framework?
• RQ4 Can the HSDE-based approach be adapted to produce certificates of primal/dual infeasibility in addition to optimal solutions?
• RQ5 How does integration with CVXPY impact usability and practical applicability?

主な発見

• Clarabel is reported to be faster and more robust than competing commercial and open-source solvers across several test sets, with notable gains for problems having quadratic objectives.
• The estimator incorporates duality-gap-based formulations and a robust initialization to handle a broad class of cones.
• Chordal decomposition is implemented for SDPs to enable scalable handling of large semidefinite constraints.
• The solver supports both symmetric and nonsymmetric cones, including exponential and power cones, as well as the zero cone for equalities.
• Empirical evaluation includes comparisons on standard benchmarks, highlighting performance advantages.