A Study Of Optimization And Optimal Control Computation:Exact Penalty Function Approach

In this thesis, We propose new computational algorithms and methods for solving four classes of constrained optimization and optimal control problems.In Chapter1, we present a brief review on optimization and optimal control.In Chapter2, we consider a class of continuous inequality constrained optimiza-tion problems. The continuous inequality constraints are first approximated by smooth function in integral form. Then, we construct a new exact penalty function, where the summation of all these approximate smooth functions in integral form, called the con-straint violation, is appended to the objective function. In this way, we obtain a sequence of approximate unconstrained optimization problems. It is shown that if the value of the penalty parameter is sufficiently large, then any local minimizer of the corresponding unconstrained optimization problem is a local minimizer of the original problem. For illustration, three examples are solved using the proposed method. From the solutions obtained, we observe that the values of their objective functions are amongst the smallest when compared with those obtained by other existing methods available in the literature. More importantly, our method finds solutions which satisfy the continuous inequality constraints.In Chapter3, we investigate the optimal design of allpass variable fractional delay (VFD) filters with coefficients expressed as sums of signed powers-of-two terms, where the weighted integral squared error is minimized. A new optimization procedure is pro-posed to generate a reduced discrete search region. Then, a new exact penalty function method is developed to solve the optimal design of allpass variable fractional delay filter with signed powers-of-two coefficients. Design examples show that the proposed method is highly effective. Compared with the conventional quantization method, the solutions obtained by our method are of much higher accuracy. Furthermore, the computational complexity is low.In Chapter4, we consider an optimal control problem in which the control takes values from a discrete set and the state and control are subject to continuous inequality constraints. By introducing auxiliary controls and applying a time-scaling transforma-tion, we transform this optimal control problem into an equivalent optimal control prob-lem subject to original constraints and additional linear and quadratic constraints, where the decision variables are taking values from a feasible region, which is the union of some continuous sets. However, due to the new quadratic constraints, standard optimization techniques do not perform well when they are applied to solve the transformed problem directly. We introduce a novel exact penalty function to penalize constraint violation-s, and then append this penalty function to the objective function, forming a penalized objective function. This leads to a sequence of approximate optimal control problem-s, each of which can be solved by using optimal control techniques, and consequently, many optimal control software packages, such as MISER3.4, can be used. Convergence results show that when the penalty parameter is sufficiently large, any local solution of the approximate problem is also a local solution of the original problem. We conclude this chapter with some numerical results for two train control problems.In Chapter5, we consider a class of nonlinear time-delay optimal control problems with canonical equality and inequality constraints. We propose a new computational ap-proach, which combines the well-known control parameterization technique with a new hybrid time-scaling strategy, for solving such time-delay optimal control problems. The proposed approach involves approximating the control by a piecewise constant function whose heights and switching times are decision variables to be optimized. Then, the resulting discretized problem is transformed, via the hybrid time-scaling transformation into an equivalent problem that is easier to solve. The new time-scaling transformation is hybrid in the sense that it yields two doubled time-delay systems-one defined in terms of the original time variable, the other defined in terms of new time variable in which the switching times are fixed. This is different from the conventional time-scaling transfor-mation in the literature, which is not applicable to time-delay systems. To demonstrate the effectiveness of the proposed approach, we solve two numerical examples. The re-sults show that our new method produces suboptimal controls with lower cost and less switches compared with existing methods.In Chapter6, some concluding remarks and suggestions for future research direc-tions are made.