Max-Min Optimization Task

A Max-Min Optimization Task is an online optimization task where the maximizing player moves first and the minimizing player has perfect information of the outcome of this move.

Context:
- It can range from being a Linear Max-Min Optimization Task to being a Nonlinear Max-Min Optimization Task.
See: Integer Linear Programming Task.

References

(Cea & Malanowski, 1998) ⇒ Jean Cea, and Kasimierz Malanowski. (1998). “An Example of a Max-Min Problem in Partial Differential Equations." California University Los Angeles
- ABSTRACT: The paper studies a 'Max-min' problem. Solving explicitly the 'min problem,' the original problem can be stated as a control problem. The state equation is a partial differential equation and the control appears only in the coefficients of the state equation. Existence and uniqueness (in a suitable sense) of 'an optimal problem' is proved and a convergent iterative method is given to compute the solution.
- Keywords: THEORETICAL MATHEMATICS, (*MINIMAX TECHNIQUE, PARTIAL DIFFERENTIAL EQUATIONS), OPTIMIZATION, ALGORITHMS, ITERATIONS, MATHEMATICAL PROGRAMMING, PROBLEM SOLVING

(Falk, 1973) ⇒ James E. Falk. (1973). “A Linear Max — Min Problem.” In: Mathematical Programming, 5(1). doi:10.1007/BF01580119
- ABSTRACT: We consider a two person max — min problem in which the maximizing player moves first and the minimizing player has perfect information of the outcome of this move. The move of the maximizing player influences not only the objective function but also the constraints of the minimizing player. The joint constraints as well as the objective function are assumed to be linear.
  For this problem it is shown that the familiar inequality min max ges max min is reversed due to the influence of the joint constraints. The problem is characterized as a nonconvex program and a method of solution based on the branch and bound philosophy is given. A small example is presented to illustrate the algorithm.