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Pro c ess O p timization 31

of other methods is measured. See Luenberger and Ye (2008) for
additional information.

3.7.2 Algorithms for Solving Constrained Nonlinear Problems

Many algorithms for constrained problems proceed in two main
steps: (1) The constrained problems are transformed into equivalent
unconstrained ones, and then (2) the modified problems are subjected
to unconstrained search algorithms. Groups of such algorithms
include primal methods, penalty and barrier methods, and primal-
dual methods.
The primal algorithms search for the optimal solution directly
through the feasible region. For a problem with n variables and
m equality constraints, primal methods work within the (n − m)-
dimensional feasible space. Each point in the process is feasible, and
the value of the objective function constantly decreases. Thus, a
feasible solution is guaranteed even if the search is interrupted before
it finishes. Most primal methods do not rely on special problem
structure (such as convexity), so they are applicable to general NLP
problems. The weaknesses of primal algorithms are their requirement
that an initial feasible solution be found or specified and their
slowness (or even failure) to converge in the case of nonlinear
constraints.
Penalty and barrier methods utilize unconstrained approximations
of the constrained optimization problem. Penalty methods add a
penalty term to the objective function, which results in a high cost if
constraints are violated. Barrier methods instead add a term that
favors points interior to the feasible region over those near the
boundary. There are two major issues with applying these methods.
First, it is important to ensure the accuracy with which the
unconstrained NLP problem being solved approximates the true,
constrained one. Second, it is possible that the penalty or barrier term
may dominate the objective function, skewing the problem to such a
degree that the feasible solutions found are not actually optimal.

3.8 Deterministic Methods for Solving Discrete Problems
All integer programming problems (pure IP, MILP, and MINLP)
possess combinatorial features. The first solution methods used
cutting planes (i.e., added constraints to force integrality), and various
algorithms have been devised on this basis. The effective approach
has been to divide the problem into a number of smaller problems, a
method known as branch and bound. This is a strategy of problem
decomposition: it aims to partition the feasible region into more
manageable subdivisions and then, if required, into further subdivided
partitions. The method was proposed by Land and Doig (1960) for
solving LPR problems and has since evolved to a point where it can