Constraint AggregationConstraint AggregationConstraint AggregationConstraint AggregationConstraint AggregationConstraint AggregationConstraint AggregationConstraint AggregationConstraint AggregationConstraint AggregationConstraint AggregationConstraint AggregationConstraint AggregationConstraint AggregationConstraint AggregationConstraint Aggregation• 1Constraint Aggregation• 2Constraint Aggregation•A possible motivation is, refering to the Capital Budgeting problem, that there are two kinds of costs associated with each item, and the budgets for the different types come from different sources, so they may have to be obeyed separately. •Or, in the knapsack terminology, each item has a size and a weight parameter, and the knapsack can carry only a certain total size and a certain total weight, so we have to satisfy both constraints.3Constraint Aggregation•Let us first convert the inequality constraints into equations, by introducing slack variables. •Let xN+1 be the slack variable in the first inquality, •and xN+2 in the second. •Further, let aN+1 = 1; bN+1 = 0; bN+2 = 1.• In this way we can simply include the slack variables in the summation, without extra notation.4Constraint Aggregation•Moreover, we can observe that the size of the “gap” the slack variable has to fill can be at most A in the first constraint, and at most B in the second.5Constraint Aggregation• 6Constraint Aggregation•Now the question is this: can we replace the two equality constraints above by a single one, so that this single constraint is equivalent to the joint effect of the two? •By equivalence we mean that they generate the same set of feasible solutions.7Constraint Aggregation•The answer to the above question is yes, even though it may seem counter-intuitive at first. Let us define the parameters of the new constraint by:ci = ai + Mbifor every i, andC = A + MBwhere M is a constant that we are going to choose later.8Constraint Aggregation• 9Constraint Aggregation• 10Constraint Aggregation• 11Constraint Aggregation• 12Constraint Aggregation• 13Constraint Aggregation14•Constraint
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