A Graphical Global Optimization Framework for Parameter Estimation of Statistical Models with Nonconvex Regularization Functions

TL;DR

This paper introduces a graph-based global optimization framework for solving complex parameter estimation problems with nonconvex regularization, improving over existing methods by directly constructing convex relaxations without auxiliary variables.

Contribution

It proposes a novel decision diagram-based approach integrated into a branch-and-cut framework to globally solve nonconvex regularized optimization problems efficiently.

Findings

Successfully solves benchmark sparse linear regression problems with nonconvex penalties.

Eliminates the need for auxiliary variables and artificial bounds in the optimization process.

Demonstrates effectiveness over existing global optimization techniques.

Abstract

Optimization problems with norm-bounding constraints arise in a variety of applications, including portfolio optimization, machine learning, and feature selection. A common approach to these problems involves relaxing the norm constraint via Lagrangian relaxation, transforming it into a regularization term in the objective function. A particularly challenging class includes the zero-norm function, which promotes sparsity in statistical parameter estimation. Most existing exact methods for solving these problems introduce binary variables and artificial bounds to reformulate them as higher-dimensional mixed-integer programs, solvable by standard solvers. Other exact approaches exploit specific structural properties of the objective, making them difficult to generalize across different problem types. Alternative methods employ nonconvex penalties with favorable statistical characteristics, but these are typically addressed using heuristic or local optimization techniques due to their structural complexity. In this paper, we propose a novel graph-based method to globally solve optimization problems involving generalized norm-bounding constraints. Our approach encompasses standard $\ell_p$-norms for $p \in [0, \infty)$ and nonconvex penalties such as SCAD and MCP. We leverage decision diagrams to construct strong convex relaxations directly in the original variable space, eliminating the need for auxiliary variables or artificial bounds. Integrated into a spatial branch-and-cut framework, our method guarantees convergence to the global optimum. We demonstrate its effectiveness through preliminary computational experiments on benchmark sparse linear regression problems involving complex nonconvex penalties, which are not tractable using existing global optimization techniques.