SEQUENTIAL QUADRATIC PROGRAMMING METHODS FOR OPTIMAL CONTROL PROBLEMS WITH STATE CONSTRAINTS

A Kind of direct methods is presented for the solution of optimal control problems with state constraints.These methods are sequential quadratic programming methods.At every iteration a quadratic programming which is obtained by quadratic approximation to Lagrangian function and Linear approximations to constraints is solved to get a search direction for a merit function.The merit function is formulated by augmenting the Lagrangian funetion with a penalty term.A line search is carried out along the search direction to determine a step length such that the merit function is decreased.The methods presented in this paper include continuous sequential quadratic programming methods and discreate sequential quadrade programming methods.     

Second order sufficient conditions and sensitivity analysis for the optimal control of a container crane under state constraints

Stability and sensitivity analysis of parametric control problems has recently been elaborated for optimal control problems subject to pure state constraints. This paper illustrates the numerical aspects of sensitivity analysis for a complex practical example: the optimal control of a container crane with a state constraint on the vertical velocity. The multiple shooting method is used to determine a nominal solution satisfying first order necessary conditions. Second order sufficient conditions are checked by showing that an associated Riccati equation has a bounded solution. Sensitivity differentials of optimal solutions an computed with respect to variations in the swing angle     

Second order sensitivity analysis and asymptotic theory of parametrized nonlinear programs

In this paper we study second-order differential properties of an optimal-value function(x). It is shown that under certain conditions(x) possesses second-order directional derivatives, which can be calculated by solving corresponding quadratic programs. Also upper and lower bounds on these derivatives are introduced under weaker assumptions. In particular we show that the second-order directional derivative is infinite if the corresponding quadratic program is unbounded. Finally sensitivity results are applied to investigate asymptotics of estimators in parametrized nonlinear programs.     

A computational study of active set strategies in nonlinear programming with linear constraints

An essential part of many iterative methods for linearly constrained nonlinear programming problems is a procedure for determining those inequality constraints which will be active (that is, satisfied as equalities) at each iteration. We discuss experiments in which we used several strategies for identifying active constraints in conjunction with two well-known algorithms for linearly constrained optimization. The results indicate that in most cases a strategy which keeps the number of constraints in the active set as small as possible is computationally most efficient.Sponsored by the United States Army under Contract No. DAAG29-75-C-0024.     

A practicable way for computing the directional derivative of the optimal value function in convex programming

Formulas for computing the directional derivative of the optimal value function or of lower or upper bounds of it are well-known from literature. Because they have as a rule a minmax structure, methods from nondifferentiable optimization are required.

Considering a fully parametrized convex problem, in the paper the mentioned minmax formulas are transformed into usual programming problems. Although they are nonconvex in general, the computational effort is much lower than that for minmax problems. In several special cases, for instance, for linear least squares problems, linear programming problems arise.     

The application of optimal control methodology to nonlinear programming problems

Dynamic programming techniques have proven to be more successful than alternative nonlinear programming algorithms for solving many discrete-time optimal control problems. The reason for this is that, because of the stagewise decomposition which characterizes dynamic programming, the computational burden grows approximately linearly with the numbern of decision times, whereas the burden for other methods tends to grow faster (e.g.,n ³ for Newton's method). The idea motivating the present study is that the advantages of dynamic programming can be brought to bear on classical nonlinear programming problems if only they can somehow be rephrased as optimal control problems.As shown herein, it is indeed the case that many prominent problems in the nonlinear programming literature can be viewed as optimal control problems, and for these problems, modern dynamic programming methodology is competitive with respect to processing time. The mechanism behind this success is that such methodology achieves quadratic convergence without requiring solution of large systems of linear equations.     

An implementation of some implicit function theorems with applications to sensitivity analysis

This paper presents an implementation of some recent results of Bigelow and Shapiro [1]. These implicit function theorems are shown to provide a convenient means of performing certain types of sensitivity analysis, in particular updating the lagrange multipliers, associated with particular classes of problems. As a result we extend the usual sensitivity analysis results to include improving estimates of the effect of changing the right-hand sides of constraints. Examples of chemical equilibrium and entropy maximization models are used to illustrate the results.