This thesis develops an iterated linear programming (LP) strategy which automatically deals with inequality constraints and variable bounds. This method reduces to Newton's method in the absence of inequalities and thus converges quadratically in the neighborhood of the solution. Moreover, this LP approach can be shown to generate a descent direction for the L{dollar}sb1{dollar} norm of the constraint violations. Therefore, by applying a line search in the algorithm, one can guarantee progress toward the solution for each nonzero solution to the LP. Other advantages to this approach are that it can deal with many classes of singular points, and the path toward the solution is always contained within the variable bounds.; We further extend this strategy to handle a general class of nonsmooth problems characterized by explicit nondifferentiable operators such as max and absolute value. Two related LP-based formulations are developed, one using continuous variables and the other involving binary variables. These approaches are demonstrated using a number of examples, including pipeline network flow problems containing check valves which may either be on or off, equidistribution problems which minimize the absolute value of the approximation error, and transition flow from laminar to turbulent.; Finally, we consider an approach for the solution of a subclass of implicitly nondifferentiable systems which are characterized by regions of continuous operation having nonsmooth boundaries. The proposed strategy uses a penalty function approach to accommodate the nonsmooth nature of the system. We extend the theoretical results of the iterated approach to characterize descent and convergence properties. The performance of this formulation is demonstrated for process models involving phase equilibrium, such as transitions from one and two phases in flash and distillation problems, where mass and energy balances must be satisfied but the phase equilibrium expression can be relaxed. Isothermal flash problems with ideal and nonideal phase equilibrium relations are considered as well as a case which exhibits retrograde condensation behavior near the critical point. We also examine limiting distillation cases including columns operating below minimum operational reflux ratio (resulting in dry trays) and below minimum operational reboiler heat duty (resulting in flooded trays).
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