A new augmented penalty function technique for optimal control problems

This note presents an extension of the Miele—Cragg-Iyer-Levy augmented function method for finite-dimensional optimization problems to optimal control problems. A numerical study is provided.     

A combination of penalty function and multiplier methods for solving optimal control problems

The properties of combined multiplier and penalty function methods are investigated using a second-order expansion and results known for the Riccati equation. It is shown that the lower bound of the values of the penalty constant necessary to obtain a minimum is given by a certain Riccati equation. The convergence rate of a common updating rule for the multipliers is shown to be linear.This work has been supported by the Swedish Institute of Applied Mathematics.     

A second-order smooth penalty function algorithm for constrained optimization problems

This paper introduces a second-order differentiability smoothing technique to the classical l ₁ exact penalty function for constrained optimization problems(COP). Error estimations among the optimal objective values of the nonsmooth penalty problem, the smoothed penalty problem and the original optimization problem are obtained. Based on the smoothed problem, an algorithm for solving COP is proposed and some preliminary numerical results indicate that the algorithm is quite promising.     

A multigrid method for constrained optimal control problems

We consider the fast and efficient numerical solution of linear-quadratic optimal control problems with additional constraints on the control. Discretization of the first-order conditions leads to an indefinite linear system of saddle point type with additional complementarity conditions due to the control constraints. The complementarity conditions are treated by a primal-dual active set strategy that serves as outer iteration. At each iteration step, a KKT system has to be solved. Here, we develop a multigrid method for its fast solution. To this end, we use a smoother which is based on an inexact constraint preconditioner.We present numerical results which show that the proposed multigrid method possesses convergence rates of the same order as for the underlying (elliptic) PDE problem. Furthermore, when combined with a nested iteration, the solver is of optimal complexity and achieves the solution of the optimization problem at only a small multiple of the cost for the PDE solution.     

A virtual control concept for state constrained optimal control problems

A linear elliptic control problem with pointwise state constraints is considered. These constraints are given in the domain. In contrast to this, the control acts only at the boundary. We propose a general concept using virtual control in this paper. The virtual control is introduced in objective, state equation, and constraints. Moreover, additional control constraints for the virtual control are investigated. An error estimate for the regularization error is derived as main result of the paper. The theory is illustrated by numerical tests.     

Improved convergence order for augmented penalty algorithms

We refine the speed of convergence analysis for the quadratic augmented penalty algorithm. We improve the convergence order from 4/3 to 3/2 for the first order multiplier iteration. For the second order iteration, we generalize the analysis, and consider a primal–dual variant which asymptotically reduces to a Newton step for the optimality conditions.     

A globally convergent constrained quasi-Newton method with an augmented lagrangian type penalty function

The recently proposed quasi-Newton method for constrained optimization has very attractive local convergence properties. To force global convergnce of the method, a descent method which uses Zangwill's penalty function and an exact line search has been proposed by Han. In this paper a new method which adopts a differentiable penalty function and an approximate line is presented. The proposed penalty function has the form of the augmented Lagrangian function. An algorithm for updating parameters which appear in the penalty function is described. Global convergence of the given method is proved.     

Solving some optimal control problems using the barrier penalty function method

In this paper we present a new approach to solve a two-level optimization problem arising from an approximation by means of the finite element method of optimal control problems governed by unilateral boundary-value problems. The problem considered is to find a minimum of a functional with respect to the control variablesu. The minimized functional depends on control variables and state variablesx. The latter are the optimal solution of an auxiliary quadratic programming problem, whose parameters depend onu.Our main idea is to replace this QP problem by its dual and then apply the barrier penalty method to this dual QP problem or to the primal one if it is in an appropriate form. As a result we obtain a problem approximating the original one. Its good property is the differentiable dependence of state variables with respect to the control variables. Furthermore, we propose a method for finding an approximate solution of a penalized lower-level problem if the optimal solution of the original QP problem is known. We apply the result obtained to some optimal shape design problems governed by the Dirichlet-Signorini boundary-value problem.This research was supported by the Academy of Finland and the Systems Research Institute of the Polish Academy of Sciences.     

Regularized gap function as penalty term for constrained minimization problems

By using the regularized gap function for variational inequalities, Li and Peng introduced a new penalty function Pα(x) for the problem of minimizing a twice continuously differentiable function in closed convex subset of the n-dimensional space Rn. Under certain assumptions, they proved that the original constrained minimization problem is equivalent to unconstrained minimization of Pα(x). The main purpose of this paper is to give an in-depth study of those properties of the objective function that can be extended from the feasible set to the whole Rn by Pα(x). For example, it is proved that the objective function has bounded level sets (or is strongly coercive) on the feasible set if and only if Pα(x) has bounded level sets (or is strongly coercive) on Rn. However, the convexity of the objective function does not imply the convexity of Pα(x) when the objective function is not quadratic, no matter how small α is. Instead, the convexity of the objective function on the feasible set only implies the invexity of Pα(x) on Rn. Moreover, a characterization for the invexity of Pα(x) is also given.     

An exact penalty method for solving optimal control problems

This paper discusses an algorithm for solving optimal control problems. An optimal control problem is presented where the final time is unknown. The algorithm consists of an integrator and a minimizer; the latter is an exact penalty function used to solve constrained nonlinear programming problems. Essentially, the optimal control problem is converted to a mathematical programming problem such that a point satisfying the differential equations via the integrator is provided to the minimizer, a lower performance index is obtained, the integrator is reinitiated, etc., until a suitable stopping criterion is satisfied.     

A nonlinear augmented Lagrangian for constrained minimax problems

A novel smooth nonlinear augmented Lagrangian for solving minimax problems with inequality constraints, is proposed in this paper, which has the positive properties that the classical Lagrangian and the penalty function fail to possess. The corresponding algorithm mainly consists of minimizing the nonlinear augmented Lagrangian function and updating the Lagrange multipliers and controlling parameter. It is demonstrated that the algorithm converges Q-superlinearly when the controlling parameter is less than a threshold under the mild conditions. Furthermore, the condition number of the Hessian of the nonlinear augmented Lagrangian function is studied, which is very important for the efficiency of the algorithm. The theoretical results are validated further by the preliminary numerical experiments for several testing problems reported at last, which show that the nonlinear augmented Lagrangian is promising.     

Nonlinearly constrained time-delayed optimal control problems

A computational algorithm for solving a class of optimal control problems involving terminal and continuous state constraints of inequality type was developed in Ref. 1. In this paper, we extend the results of Ref. 1 to a more general class of constrained time-delayed optimal control problems, which involves terminal state equality constraints as well as terminal state inequality constraints and continuous state constraints. Two examples have been solved to illustrate the efficiency of the method.     

非光滑最优控制问题的一种数值解法

针对非光滑最优控制问题提出一种分段数值解法.首先对问题进行全局拟谱离散,然后选取分点,将时间区域进行剖分,在每段区域上对问题进行离散,离散过程采用Chebyshev-Legendre拟谱方法,可以有效借助快速Legendre变换提高算法的运算效率,比现有算法在很大程度上节省了计算时间.给出了相关的理论分析,数值结果表明方法的高精度和有效性.     

A numerical method for nonconvex multi-objective optimal control problems

A numerical method is proposed for constructing an approximation of the Pareto front of nonconvex multi-objective optimal control problems. First, a suitable scalarization technique is employed for the multi-objective optimal control problem. Then by using a grid of scalarization parameter values, i.e., a grid of weights, a sequence of single-objective optimal control problems are solved to obtain points which are spread over the Pareto front. The technique is illustrated on problems involving tumor anti-angiogenesis and a fed-batch bioreactor, which exhibit bang–bang, singular and boundary types of optimal control. We illustrate that the Bolza form, the traditional scalarization in optimal control, fails to represent all the compromise, i.e., Pareto optimal, solutions.     

Asymptotic expansions for interior penalty solutions of control constrained linear-quadratic problems

We consider a quadratic optimal control problem governed by a nonautonomous affine ordinary differential equation subject to nonnegativity control constraints. For a general class of interior penalty functions, we provide a first order expansion for the penalized states and adjoint states around the state and adjoint state of the original problem. Our main argument relies on the following fact: if the optimal control satisfies strict complementarity conditions for its Hamiltonian except for a set of times with null Lebesgue measure, the functional estimates for the penalized optimal control problem can be derived from the estimates of a related finite dimensional problem. Our results provide several types of efficiency measures of the penalization technique: error estimates of the control for L ^s norms (s in [1, +∞]), error estimates of the state and the adjoint state in Sobolev spaces W ^1,s (s in [1, +∞)) and also error estimates for the value function. For the L ¹ norm and the logarithmic penalty, the sharpest results are given, by establishing an error estimate for the penalized control of order ${O(\varepsilon|\log\epsilon|)}$ where ${\varepsilon >0 }$ is the (small) penalty parameter.     

Multi-mesh adaptive finite element algorithms for constrained optimal control problems governed by semi-linear parabolic equations

In this paper, we derive a posteriori error estimators for the constrained optimal control problems governed by semi-linear parabolic equations under some assumptions. Then we use them to construct reliable and efficient multi-mesh adaptive finite element algorithms for the optimal control problems. Some numerical experiments are presented to illustrate the theoretical results.     

An improved penalty function method for solving constrained parameter optimization problems

An effective algorithm is described for solving the general constrained parameter optimization problem. The method is quasi-second-order and requires only function and gradient information. An exterior point penalty function method is used to transform the constrained problem into a sequence of unconstrained problems. The penalty weightr is chosen as a function of the pointx such that the sequence of optimization problems is computationally easy. A rank-one optimization algorithm is developed that takes advantage of the special properties of the augmented performance index. The optimization algorithm accounts for the usual difficulties associated with discontinuous second derivatives of the augmented index. Finite convergence is exhibited for a quadratic performance index with linear constraints; accelerated convergence is demonstrated for nonquadratic indices and nonlinear constraints. A computer program has been written to implement the algorithm and its performance is illustrated in fourteen test problems.     

The Lagrange-Newton method for state constrained optimal control problems

Local convergence of the Lagrange-Newton method for optimization problems with two-norm discrepancy in abstract Banach spaces is investigated. Based on stability analysis of optimization problems with two-norm discrepancy, sufficient conditions for local superlinear convergence are derived. The abstract results are applied to optimal control problems for nonlinear ordinary differential equations subject to control and state constraints.This research was completed while the second author was a visitor at the University of Bayreuth, Germany, supported by grant No. CIPA3510CT920789 from the Commission of the European Communities.     

Rationalized Haar approach for nonlinear constrained optimal control problems

This paper presents a numerical method for solving nonlinear optimal control problems including state and control inequality constraints. The method is based upon rationalized Haar functions. The differential and integral expressions which arise in the system dynamics, the performance index and the boundary conditions are converted into some algebraic equations which can be solved for the unknown coefficients. Illustrative examples are included to demonstrate the validity and applicability of the technique.