Thermodynamic optimization equalities in weakly driven processes

Equalities are generally more suitable for experimental verification than inequalities. In this work, I derive valid equalities from the Euler-Lagrange equation for the optimization of macroscopic thermodynamic averages in weakly driven classical open systems. These equalities show that optimization occurs when work and heat become path-independent. I illustrate their applicability by employing them as a convergence criterion in the global optimization technique of genetic programming. Moreover, due to fluctuation-dissipation relations for internal energy, work, and heat, analogous results hold for their variances.