Evaluating the solution performance of the augmented Lagrangian function on Ising machines

We apply the augmented Lagrangian function (ALF) as a formulation for Ising machines and evaluate its performance by time-to-epsilon (\mathrm{TT\varepsilon}). The ALF has been well studied in continuous optimization for its numerical stability and convergence, and its advantage over the penalty function formulation is demonstrated here through the following results. Using the quadratic knapsack problem as a benchmark, we examine the dependence of \mathrm{TT\varepsilon} on the hyperparameters \mu and \lambda. The augmented Lagrangian formulation reduces \mathrm{TT\varepsilon} by roughly an order of magnitude compared with the penalty function formulation, keeping \mu small while obtaining feasible solutions and, for representative parameter settings, reaching high-precision solutions earlier in the search. These findings indicate that the augmented Lagrangian function is a promising formulation for improving the solution performance of Ising machines.