$\gamma$-weakly $\theta$-up-concavity: A Unified Framework for Non-Convex Optimization Beyond DR-Submodular and OSS Functions

Researchers introduce γ-weakly θ-up-concavity, a mathematical framework that unifies optimization approaches for non-convex functions by generalizing DR-submodular and One-Sided Smooth functions. The framework proves these functions are upper-linearizable, enabling improved approximation guarantees for both offline and online optimization problems across various constraint structures.

This theoretical computer science paper addresses a fundamental challenge in optimization: handling non-convex functions that appear across machine learning and combinatorial problems. The introduction of γ-weakly θ-up-concavity represents a significant mathematical contribution by identifying common structural properties across previously disparate function classes, particularly DR-submodular and One-Sided Smooth functions. This unification matters because it allows researchers to apply a single analytical framework and derive approximation guarantees rather than developing separate algorithms for each problem type.

The key innovation—proving upper-linearizability for this function class—enables practical implications. By constructing linear surrogates that approximate non-linear objectives, the framework reduces complex non-convex problems to linear optimization, which is computationally tractable. The nonuniform upper-linearization argument provides explicit bounds tied to curvature parameters and feasible region geometry, offering concrete guarantees rather than asymptotic results.

From an industry perspective, this work strengthens the theoretical foundations underlying optimization algorithms used in machine learning, resource allocation, and combinatorial optimization. While primarily academic, improved approximation guarantees translate to better algorithm design for real-world applications in recommendation systems, portfolio optimization, and constraint satisfaction problems. The unified framework also reduces research fragmentation by consolidating scattered results across subfields.

Looking ahead, practitioners should monitor whether these theoretical guarantees stimulate new algorithm development or optimization libraries. The work particularly impacts researchers working with matroid constraints and submodular-adjacent functions, where the paper demonstrates concrete improvements over existing approaches.

• →γ-weakly θ-up-concavity unifies DR-submodular and OSS functions under a single analytical framework
• →Upper-linearizability enables reduction of non-convex problems to tractable linear optimization
• →Framework recovers optimal coefficients for DR-submodular maximization and improves OSS bounds on matroid constraints
• →Approximation guarantees depend explicitly on curvature parameters and feasible region geometry
• →Applicable to both offline optimization and dynamic online regret minimization settings