Nonlinear Programming:CONCLUSIONS

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CONCLUSIONS

Practical optimization problems frequently involve nonlinear behavior that must be taken into account. It is sometimes possible to reformulate these nonlinearities to fit into a lin- ear programming format, as can be done for separable programming problems. However, it is frequently necessary to use a nonlinear programming formulation.

In contrast to the case of the simplex method for linear programming, there is no efficient all-purpose algorithm that can be used to solve all nonlinear programming problems. In fact, some of these problems cannot be solved in a very satisfactory manner by any method. However, considerable progress has been made for some important classes of problems, including quadratic programming, convex programming, and certain special

types of nonconvex programming. A variety of algorithms that frequently perform well are available for these cases. Some of these algorithms incorporate highly efficient pro- cedures for unconstrained optimization for a portion of each iteration, and some use a succession of linear or quadratic approximations to the original problem.

There has been a strong emphasis in recent years on developing high-quality, re- liable software packages for general use in applying the best of these algorithms. For example, several powerful software packages have been developed in the Systems Op- timization Laboratory at Stanford University This chapter also has pointed out the im- pressive capabilities of Solver, ASPE, MPL/Solvers, and LINGO/LINDO. These pack- ages are widely used for solving many of the types of problems discussed in this chapter (as well as linear and integer programming problems). The steady improvements be- ing made in both algorithmic techniques and software now are bringing some rather large problems into the range of computational feasibility.

Research in nonlinear programming remains very active.

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