By Vandenbussche D., Nemhauser G. L.
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Extra resources for A branch-and-cut algorithm for nonconvex quadratic programs with box constraints
The Search Operations In some problems, the search space G may be identical to the problem space X. If we go back to our previous examples, for instance, we will ﬁnd that there exist a lot of optimization strategies that work directly on vectors of real numbers. When minimizing a real function, we could use such an approach (Evolution Strategies, for instance, see Chapter 5 on page 229) and set G = X = R. Also, the conﬁgurations of cars may be represented as bit strings: Assume that such a conﬁguration consists of k features, which can either be included or excluded from an oﬀer to the customer.
Penalty Functions Maybe one of the most popular approach for dealing with constraints, especially in the area of single-objective optimization, goes back to Courant  who introduced the idea of penalty functions in 1943. Here, the constraints are combined with the objective function f , resulting in a new function f ′ which is then actually optimized. The basic idea is that this combination is done in a way which ensures that an infeasible solution candidate has always a worse f ′ -value than a feasible one with the same objective values.
A non-dominated element is, as the name says, not dominated by any other solution candidate. These elements are Pareto optimal and have a domination-count of zero. 9, there are four such areas X⋆1 , X⋆2 , X⋆3 , and X⋆4 . 9: Optimization using the Pareto Frontier approach (second example). 2 What is an optimum? 4, we can see that hills in the domination space occur at positions where both, f3 and f4 have high values. Conversely, regions of the problem space where both functions have small values are dominated by very few elements.