Lower Bounds on Syntactic Logic Expressions for Optimization Problems and Duality using Lagrangian Dual to characterize optimality conditions

In this paper, we prove lower bounds on the logical expressibility of optimization problems. There is a significant difference between the expressibilities of decision problems and optimization problems. This is similar to the difference in computation times for the two classes of problems; for example, a 2SAT Horn formula can be satisfied in polynomial time, whereas the optimization version in NP-hard. Grädel [E. 91] proved that all polynomially solvable decision problems can be expressed as universal (Π1) Horn sentences. We show here that, on the other hand, optimization problems defy such a simple characterization, by demonstrating that even a simple Π0 formula is unable to guarantee polynomial time solvability. See Section 4 for Duality using Lagrangian Dual to characterize optimality conditions — that section also describes using a single call to a “decision machine” (a Turing machine that solves decision problems) to obtain optimal solutions.