A GENERALIZED REDUCED GRADIENT ALGORITHM FOR SOLVING LARGE-SCALE DISCRETE-TIME NONLINEAR OPTIMAL CONTROL PROBLEMS

Abstract

Nonlinear dynamical control systems are considered in a unified approach as discrete-time nonlinear optimal control problems with time delays and inequality constraints on the state and the control variables (usually double bounds). These problems are in general large-scale and efficient numerical solutions can be obtained by constrained Nonlinear Programming methods if reliable techniques able to deal with these numerical difficulties are employed. A generalized reduced gradient algorithm is proposed exploiting the staircase structure of the jacobian matrix of the dynamic equations by using some priority principles on the partition of the variables into basic and independent sets for each time period when a reenversion is needed, and for choosing a substitute basic variable when change-of-basis occur for regularity reasons. Factorized representations of the basic matrix facilitate the resolutions of the linear systems of equations in different parts of the algorithm. Gaussian eliminations (LU decompositions) of the diagonal blocks of the main factor of the representation improve the numerical stability of these processes. In the reduced dimension space of the bounded independent variables we have an unconstrained differentiable nonlinear objective function. The search directions can be computed by adapted unconstrained optimization methods with memory limitation as Conjugate Gradients. If some amount of extra storage is available, limited-storage methods combining properties of the quasi- Newton BFGS and conjugate gradients methods present superior convergence rates. These alternative search directions can improve the convergence of the algorithm. Time delays in the nonlinear dynamic equations can be considered by some specific sparse matrix techniques with no influence on the algorithm main strategy. A computer code has been designed and numerical experiments with different optimization models for applications to electric power generation planning, macroeconomy and fishery management have been solved with encouraging results.