I only know of linear programming (LP) optimization by itself, and quadratic programming (QP) optimization by itself, but I have never seen them mixed within the same objective function. For example, a model that concurrently solves an outer maximization problem as well as an inner minimization one.
Say I have a "bigger picture" model that must be solved with linear programming as an outer maximization, which at the same time requires an inner minimization model to derive an optimal decision variable using quadratic programming.
Am I allowed to mix LP with QP like this in the same objective function? Could you give examples of actual models from any field that mix LP with QP? I want to know what the objective function looks like and how the notation is done
(To avoid only receiving min-max or minimax as an answer, the example I mentioned could even be outer and inner maximization.)
You seen to be referring to what is called Bilevel Optimization https://en.wikipedia.org/wiki/Bilevel_optimization .
If your inner problem is a QP, you would include the KKT conditions for the QP as additional constraints to the outer problem. This overall problem would no longer be an LP due to the complementarity constraints in the KKT conditions, and is called a Mathematical Program with Equilibrium Constraints (MPEC) https://neos-guide.org/content/mathematical-programs-equilibrium-constraints or Mathematical Program with Complementarity Constraints (MPCC).