Modeling and Solution of Continuous Set Covering Problems by Means of semi-infinite Optimization

The task of designing product portfolios in technical contexts motivates a new perspective on optimal product portfolio design. From a mathematical point of view, a new optimization problem, the continuous set covering problem, is developed. This formulation in fact is a semi-infinite optimization problem (SIP). A solution approach combining adaptive discretization of the infinite index set with regularization of the non-smooth constraint function is suggested. Besides, the lower level problem of the SIP is analyzed. Unfortunately, it is not a convex optimization problem, which makes necessary global optimization difficult in general. Yet a characterization for continuous set covering data is developed that allows the identification of global maximum points of the lower level problem as solutions of a finite number of lower dimensional and thus potentially easier optimization problems. Two situations are presented, where the lower level problem can be solved even analytically using procedures from the field of computational geometry. Finally, numerical examples based on questions from pump industry show that the presented approach is capable to cope with real-world.