A graph-based global optimization framework for problems with nonconvex norm constraints and penalty functions
Keywords: Norm Constraints, Sparse Parameter Estimation, Nonconvex Regularization, Global Optimization, Mixed-Integer Nonlinear Programs, Decision Diagrams
TL;DR: This paper introduces the first algorithm in the literature to globally solve mixed-integer nonlinear programs that contain complex nonconvex regularization terms arising in statistical parameter estimation models
Abstract: Optimization problems with norm-bounding constraints appear in various applications, from portfolio optimization to machine learning, feature selection, and beyond. A widely used variant of these problems relaxes the norm-bounding constraint through Lagrangian relaxation and moves it to the objective function as a form of penalty or regularization term. A challenging class of these models uses the zero-norm function to induce sparsity in statistical parameter estimation models. Most existing exact solution methods for these problems use additional binary variables together with artificial bounds on variables to formulate them as a mixed-integer program in a higher dimension, which is then solved by off-the-shelf solvers. Other exact methods utilize specific structural properties of the objective function to solve certain variants of these problems, making them non-generalizable to other problems with different structures. An alternative approach employs nonconvex penalties with desirable statistical properties, which are solved using heuristic or local methods due to the structural complexity of those terms. In this paper, we develop a novel graph-based method to globally solve optimization problems that contain a generalization of norm-bounding constraints. This includes standard $\ell_p$-norms for $p \in [0, \infty)$ as well as nonconvex penalty terms, such as SCAD and MCP, as special cases. Our method uses decision diagrams to build strong convex relaxations for these constraints in the original space of variables without the need to introduce additional auxiliary variables or impose artificial variable bounds. We show that the resulting convexification method, when incorporated into a spatial branch-and-cut framework, converges to the global optimal value of the problem under mild conditions. To demonstrate the capabilities of the proposed framework, we conduct preliminary computational experiments on benchmark sparse linear regression problems with complex nonconvex penalty terms that existing global solvers cannot model or solve. This establishes our framework as the first algorithm capable of globally solving such challenging mixed-integer nonlinear programs.
Primary Area: optimization
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Submission Number: 8500
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