Branch-and-Bound Performance Estimation Programming: A Unified Methodology for Constructing Optimal Optimization Methods

Abstract: We present the Branch-and-Bound Performance Estimation Programming (BnB-PEP), a unified methodology for constructing optimal first-order methods for convex and nonconvex optimization. BnB-PEP poses the problem of finding the optimal optimization method as a nonconvex but practically tractable quadratically constrained quadratic optimization problem and solves it to certifiable global optimality using a customized branch-and-bound algorithm. By directly confronting the nonconvexity, BnB-PEP offers significantly more flexibility and removes the many limitations of the prior methodologies. Our customized branch-and-bound algorithm, through exploiting specific problem structures, outperforms the latest off-the-shelf implementations by orders of magnitude, accelerating the solution time from hours to seconds and weeks to minutes. We apply BnB-PEP to several setups for which the prior methodologies do not apply and obtain methods with bounds that improve upon prior state-of-the-art results. Finally, we use the BnB-PEP methodology to find proofs with potential function structures, thereby systematically generating analytical convergence proofs.

• The paper introduces BnB-PEP, a unified QCQP framework for constructing optimal first-order methods for both convex and nonconvex problems.
• It employs a specialized branch-and-bound algorithm to ensure global optimality and drastically reduce computation times.
• The method transcends traditional SDP-based approaches, offering improved performance and practical impact for large-scale optimization applications.

Branch-and-Bound Performance Estimation Programming: A Unified Methodology for Constructing Optimal Optimization Methods

The paper introduces Branch-and-Bound Performance Estimation Programming (BnB-PEP), an integrated approach for designing optimal first-order methods targeting both convex and nonconvex optimization landscapes. The core proposition of this research is the formulation of the optimization method discovery process as a nonconvex quadratically constrained quadratic program (QCQP). This transformation addresses the inherent nonconvexities directly, allowing the authors to eschew the conventional reliance on semidefinite programming (SDP) relaxations and other heuristic-based method derivations previously constrained by convexity requirements.

Methodology and Results

BnB-PEP leverages a branch-and-bound strategy tailored to solve these nonconvex QCQPs, ensuring global optimality of the solution within a tractable timeframe. By exploiting special structural properties intrinsic to the optimization problems, BnB-PEP significantly accelerates solution times compared to existing off-the-shelf implementations. The results are substantial—reducing runtimes from several hours or weeks to mere seconds or minutes in several benchmark cases.

The paper demonstrates the prowess of BnB-PEP across multiple scenarios where traditional methods falter, producing optimization strategies with performance bounds exceeding current state-of-the-art techniques. Notably, it generates first-order methods for various settings, such as smooth and nonsmooth, convex and nonconvex objectives, and delivers proven performance guarantees supported by potential function structures.

Technical Innovations

1. QCQP Framework: The research redefines the optimization method discovery process, framing it as a QCQP—a stark deviation from the convex SDP-based formulations traditionally employed. This nonconvex framework inherently supports a wider array of optimization problems that elude the capabilities of convex SDPs.
2. Branch-and-Bound Algorithm: A customized algorithm is central to the BnB-PEP methodology, strikingly outpacing current implementations by exploiting the unique properties of the optimization problems. This advancement significantly broadens the scope of solvable problems within practical timelines.
3. Generalization Beyond Convex Settings: By effectively dealing with the nonconvexity aspect, BnB-PEP presents an approach that can seamlessly apply to problems involving both convex and nonconvex challenges, an attribute that previous methods notably lack due to their intrinsic reliance on convexity.

Implications and Future Directions

The successful application of BnB-PEP opens several avenues for future research in optimization and algorithm design. Practically, the capability to efficiently derive optimal first-order methods for nonconvex scenarios is transformative, particularly in fields reliant on large-scale optimization, such as machine learning, data science, and operations research. Theoretically, the insights gained from nonconvex formulations and their successful resolution via tailored algorithms could inspire novel optimization frameworks and methodologies, further bridging gaps between theoretical optimality and empirical efficiency.

Future work could extend BnB-PEP to composite optimization, randomized algorithms, monotone operator settings, and beyond, potentially influencing the development of adaptive and stochastic optimization methods. Moreover, the continuous improvement of branch-and-bound techniques, including the integration of parallel computing resources, could further expedite solutions, cementing BnB-PEP's role in cutting-edge optimization research and applications.