SAT solvers can be used to solve the traveling salesman problem, where the sum of costs between chosen edges is important. I understand that you then ask the solver to go again finding a lesser cost. How is that maximum represented in conjunctive normal form?
How are objective functions represented in SAT solvers?
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The idea is to translate pseudo-Boolean constraints into CNF and many different encodings are available. A naive encoding is to enumerate all partial models that would result in a higher cost (which is exponential).
Note: I am co-author of the following publication: You may find the PBLib useful as it provides different encodings.