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Mathematical Programming vs. Constraint Programming

Mathematical programming and constraint programming are two technologies critical to solving complex planning and scheduling problems. At IBM, we've found that knowing both technologies is important in addressing some of the most difficult optimization problems. To make the most of each, it is important to understand their similarities and differences.

General model structure

A constraint programming optimization model has the same structure as a mathematical programming model: a set of decision variables, an objective function to maximize or minimize, and a set of constraints.

Modeling differences:

Optimization engine differences:

Mathematical programming / Constraint programming Comparison Table

Mathematical programming Constraint programming
Relaxation Yes  
Lower bound and optimality gap measure Yes  
Optimality proof Yes   Yes  
Modeler support Yes   Yes
Model-and-run Yes   Yes
Modeling limitations Restricted to linear and quadratic problems Restricted to discrete problems
Specialized constraints
Yes  
Logical constraints Yes   Yes  
Theoretical basis Algebra Logical inferences

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