Overview
This research introduces a methodology utilizing subproblem objective value suboptimality bounds within inexact augmented Lagrangian methods and ADMM (Alternating Direction Method of Multipliers) algorithms. The central mechanism is a “partial strong convexity” lemma. This approach facilitates the development of two novel methods for computing Lagrangian bounds on the optimal values of stochastic mixed-integer programming problems.
Research Context
The field of optimization frequently encounters problems that require the solution of large, complex optimization problems, often involving stochastic elements and mixed-integer variables. Lagrangian relaxation is a common technique to address such problems, transforming a constrained optimization problem into an unconstrained one, or one with simpler constraints, by incorporating the constraints into the objective function. Augmented Lagrangian methods and ADMM are iterative algorithms used to solve such problems, often involving inexact solutions of subproblems within each iteration.
Prior work in this area has established various criteria for approximately solving subproblems within ADMM. This paper refines a long-standing criterion specific to ADMM applications.
Approach
The core of the methodology lies in the application of a “partial strong convexity” lemma. This lemma establishes how bounds on the suboptimality of subproblem objective values can be systematically integrated into inexact augmented Lagrangian methods and ADMM algorithms. For ADMM, the research incorporates a refinement to an existing criterion used for approximately solving subproblems, which is described as “small but important.”
The practical implementation of these methods involves solving the subproblems using variants of the classical Frank-Wolfe algorithm. This choice of algorithm for subproblem solution offers increased flexibility compared to prior methods of a similar type.
Findings
- The “partial strong convexity” lemma provides a mechanism for incorporating bounds on subproblem objective value suboptimality into inexact augmented Lagrangian methods and ADMM algorithms.
- The ADMM result includes a refinement of an existing criterion for approximately solving subproblems.
- This methodology enables two new approaches for calculating Lagrangian bounds on the optimal values of stochastic mixed-integer programming problems.
- These new approaches exhibit a simpler convergence analysis compared to the state-of-the-art methods that existed prior to this work.
- The subproblems within these new methods are solved using variants of the Frank-Wolfe algorithm.
- The choice of Frank-Wolfe variants for subproblem solution offers greater freedom compared to earlier methods of a similar nature.