An exact piecewise differentiable parameter-free penalty function

Abstract

In this dissertation, the problem of optimizing a nonlinear objective function subject to linear and/or nonlinear constraints is considered. When the constraints are nonlinear, it is particularly advantageous to transform the constrained problem into one or more unconstrained problem(s). Historically, the approach to solve constrained problems by such transformations involved determining a positive parameter, minimizing the transformation; decreasing the parameter, minimizing the transformation associated with this new parameter value; and so on. Thus, a sequence of subproblems must be solved, where each subproblem corresponds to a parameter value. Under appropriate conditions, the sequence of optimal solutions to the corresponding subproblems approaches the optimal solution to the original constrained problem as the parameter values approach zero. This method of solution is called a sequential unconstrained transformation (penalty) technique. More recently, the appealing idea of solving a single unconstrained problem has appeared in the literature. Here, one appropriate parameter value is calculated, so that, under certain conditions, the solution to the constrained problem is obtained by solving the one unconstrained problem corresponding to this parameter value. This method of solution is known as an exact penalty function technique, and the appropriate parameter is called an optimal parameter. ...