Sufficient quadratic conditions of extremum for discontinuous controls in optimal control problems with mixed constraints

We derive second-order sufficient optimality conditions for discontinuous controls in optimal control problems of ordinary differential equations with initial-final state constraints and mixed state-control constraints of equality and inequality type. Under the assumption that the gradients with respect to the control of active mixed constraints are linearly independent, the sufficient conditions imply a bounded strong minimum in the problem.     

An Exact Penalty Function Method for Continuous Inequality Constrained Optimal Control Problem

In this paper, we consider a class of optimal control problems subject to equality terminal state constraints and continuous state and control inequality constraints. By using the control parametrization technique and a time scaling transformation, the constrained optimal control problem is approximated by a sequence of optimal parameter selection problems with equality terminal state constraints and continuous state inequality constraints. Each of these constrained optimal parameter selection problems can be regarded as an optimization problem subject to equality constraints and continuous inequality constraints. On this basis, an exact penalty function method is used to devise a computational method to solve these optimization problems with equality constraints and continuous inequality constraints. The main idea is to augment the exact penalty function constructed from the equality constraints and continuous inequality constraints to the objective function, forming a new one. This gives rise to a sequence of unconstrained optimization problems. It is shown that, for sufficiently large penalty parameter value, any local minimizer of the unconstrained optimization problem is a local minimizer of the optimization problem with equality constraints and continuous inequality constraints. The convergent properties of the optimal parameter selection problems with equality constraints and continuous inequality constraints to the original optimal control problem are also discussed. For illustration, three examples are solved showing the effectiveness and applicability of the approach proposed.     

Dynamic optimization and event location in atmospheric chemistry

A model coupling differential equations and an optimization problem with equality and inequality constraints is described. The first order optimality conditions are coupled with the differential equations to form a differential-algebraic system. Inequality constraints for the optimization variables induce discontinuities in the first derivatives of the solution in time. Tracking techniques based on sensitivity analysis and second order multistep methods are proposed to locate the discontinuities. Application to the dynamics of organic atmospheric aerosol particles is given to illustrate the detection of the activation of constraints. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Necessary quadratic conditions of extremum for discontinuous controls in optimal control problems with mixed constraints

We derive necessary second-order optimality conditions for discontinuous controls in optimal control problems of ordinary differential equations with initial-final state constraints and mixed state-control constraints of equality and inequality type. Under the assumption that the gradients withrespect to the control of active mixed constraints are linearly independent, the necessary conditions follows from a Pontryagin minimum in the problem. Together with sufficient second-order conditions [70], the necessary conditions of the present paper constitute a pair of no-gap conditions.     

A transformation technique for optimal control problems with partially linear state inequality constraints

This paper considers optimal control problems involving the minimization of a functional subject to differential constraints, terminal constraints, and a state inequality constraint. The state inequality constraint is of a special type, namely, it is linear in some or all of the components of the state vector.A transformation technique is introduced, by means of which the inequality-constrained problem is converted into an equality-constrained problem involving differential constraints, terminal constraints, and a control equality constraint. The transformation technique takes advantage of the partial linearity of the state inequality constraint so as to yield a transformed problem characterized by a new state vector of minimal size. This concept is important computationally, in that the computer time per iteration increases with the square of the dimension of the state vector.In order to illustrate the advantages of the new transformation technique, several numerical examples are solved by means of the sequential gradient-restoration algorithm for optimal control problems involving nondifferential constraints. The examples show the substantial savings in computer time for convergence, which are associated with the new transformation technique.This research was supported by the Office of Scientific Research, Office of Aerospace Research, United States Air Force, Grant No. AF-AFOSR-76-3075, and by the National Science Foundation, Grant No. MCS-76-21657.     

An Exact Penalty Method for Free Terminal Time Optimal Control Problem with Continuous Inequality Constraints

In this paper, we consider a class of optimal control problems with free terminal time and continuous inequality constraints. First, the problem is approximated by representing the control function as a piecewise-constant function. Then the continuous inequality constraints are transformed into terminal equality constraints for an auxiliary differential system. After these two steps, we transform the constrained optimization problem into a penalized problem with only box constraints on the decision variables using a novel exact penalty function. This penalized problem is then solved by a gradient-based optimization technique. Theoretical analysis proves that this penalty function has continuous derivatives, and for a sufficiently large and finite penalty parameter, its local minimizer is feasible in the sense that the continuous inequality constraints are satisfied. Furthermore, this local minimizer is also the local minimizer of the constrained problem. Numerical simulations on the range maximization for a hypersonic vehicle reentering the atmosphere subject to a heating constraint demonstrate the effectiveness of our method.     

Stable barrier-projection and barrier-Newton methods in linear programming

The present paper is devoted to the application of the space transformation techniques for solving linear programming problems. By using a surjective mapping the original constrained optimization problem is transformed to a problem in a new space with only equality constraints. For the numerical solution of the latter problem the stable version of the gradient-projection and Newton's methods are used. After an inverse transformation to the original space a family of numerical methods for solving optimization problems with equality and inequality constraints is obtained. The proposed algorithms are based on the numerical integration of the systems of ordinary differential equations. These algorithms do not require feasibility of the starting and current points, but they preserve feasibility. As a result of a space transformation the vector fields of differential equations are changed and additional terms are introduced which serve as a barrier preventing the trajectories from leaving the feasible set. A proof of a convergence is given.Dedicated to Professor George B. Dantzig on the occasion of his eightieth birthday.Research was supported by the grant N93-012-450 from Russian Scientific Fund.     

Second-Order Necessary Optimality Conditions for Some State-Constrained Control Problems of Semilinear Elliptic Equations

In this paper we are concerned with some optimal control problems governed by semilinear elliptic equations. The case of a boundary control is studied. We consider pointwise constraints on the control and a finite number of equality and inequality constraints on the state. The goal is to derive first- and second-order optimality conditions satisfied by locally optimal solutions of the problem. Accepted 6 May 1997     

A partial proximal point algorithm for nuclear norm regularized matrix least squares problems

We introduce a partial proximal point algorithm for solving nuclear norm regularized matrix least squares problems with equality and inequality constraints. The inner subproblems, reformulated as a system of semismooth equations, are solved by an inexact smoothing Newton method, which is proved to be quadratically convergent under a constraint non-degeneracy condition, together with the strong semi-smoothness property of the singular value thresholding operator. Numerical experiments on a variety of problems including those arising from low-rank approximations of transition matrices show that our algorithm is efficient and robust.     

An algorithm for linear least squares problems with equality and nonnegativity constraints

We present a new algorithm for solving a linear least squares problem with linear constraints. These are equality constraint equations and nonnegativity constraints on selected variables. This problem, while appearing to be quite special, is the core problem arising in the solution of the general linearly constrained linear least squares problem. The reduction process of the general problem to the core problem can be done in many ways. We discuss three such techniques.The method employed for solving the core problem is based on combining the equality constraints with differentially weighted least squares equations to form an augmented least squares system. This weighted least squares system, which is equivalent to a penalty function method, is solved with nonnegativity constraints on selected variables.Three types of examples are presented that illustrate applications of the algorithm. The first is rank deficient, constrained least squares curve fitting. The second is concerned with solving linear systems of algebraic equations with Hilbert matrices and bounds on the variables. The third illustrates a constrained curve fitting problem with inconsistent inequality constraints.     

Optimal multigrid preconditioned semi‐monotonic augmented Lagrangians applied to the Stokes problem

We propose an optimal computational complexity algorithm for the solution of quadratic programming problems with equality constraints arising from partial differential equations. The algorithm combines a variant of the semi‐monotonic augmented Lagrangian (SMALE) method with adaptive precision control and a multigrid preconditioning for the Hessian of the cost function and for the inner product on the space of Lagrange variables. The update rule for penalty parameter acts as preconditioning of constraints. The optimality of the algorithm is theoretically proven and confirmed by numerical experiments for the two‐dimensional Stokes problem. Copyright © 2007 John Wiley & Sons, Ltd.     

Error bounds for attainability sets of control systems with phase constraints

The problem of the error bounds for attainability sets of control systems described by ordinary differential equations under discretization of the phase constraints is studied. The peculiarity of the problems investigated in this paper is the phase constraints in the form of equality.     

A trust region interior point algorithm for infinite dimensional nonlinear programming

A trust region interior point algorithm for infinite dimensional nonlinear problem, which is motivated by the application of black-box approach to the distributed parameter system optimal control problem with equality and inequality constraints on states and controls, and with bounds on the controls is formulated. By introducing a proper functional which is analogous to the Lagrange function, both equality and inequality constraints can be treated identically and the first order optimality condition is given, then based on the works of Coleman, Ulbrich and Heinkenschloss, the trust region interior point algorithm which is employed to solve the optimization problem under consideration is presented.     

A sparse nonlinear optimization algorithm

One of the most effective numerical techniques for solving nonlinear programming problems is the sequential quadratic programming approach. Many large nonlinear programming problems arise naturally in data fitting and when discretization techniques are applied to systems described by ordinary or partial differential equations. Problems of this type are characterized by matrices which are large and sparse. This paper describes a nonlinear programming algorithm which exploits the matrix sparsity produced by these applications. Numerical experience is reported for a collection of trajectory optimization problems with nonlinear equality and inequality constraints.The authors wish to acknowledge the insightful contributions of Dr. William Huffman.     

Euler-Lagrange equation in the case of nonregular equality constraints

An optimization problem with inequality and equality constraints in Banach spaces is considered in the case when the operators which determine the equality constraints are nonregular. In this case, the classical Euler-Lagrange equation has the degenerate form, i.e., does not depend on the functional to be minimized. Applying some generalization of the Lusternik theorem to the Dubovitskii-Milyutin method, the family of Euler-Lagrange equations is obtained in the nondegenerate form under the assumption of twice Fréchet differentiability of the operators. The Pareto-optimal problem is also considered.The author would like to thank the reviewers for valuable suggestions and remarks. This research was partially supported by the SIUE Research Scholar Award.     

On the optimal control problems with Gâteaux differentiable operator constraints

In this paper the problem of optimal control with mixed equality and inequality operator constraints is considered under the assumption of Gâteaux differentiability. The extremum principle for this kind of problem is obtained by using some specification of the Dubovitskii-Milyutin method for the case of n equality constraints given by Gâteaux differentiable operators.     

High-resolution numerical algorithm for one-dimensional scalar conservation laws with a constrained solution

The CABARET computational algorithm is generalized to one-dimensional scalar quasilinear hyperbolic partial differential equations with allowance for inequality constraints on the solution. This generalization can be used to analyze seepage of liquid radioactive wastes through the unsaturated zone.     

The maximum principle for the nonlinear stochastic optimal control problem of switching systems

The aim of this paper is to present a stochastic maximum principle for an optimal control problem of switching systems. It presents necessary conditions of optimality in the form of a maximum principle for stochastic switching systems, in which the dynamic of the constituent processes takes the form of stochastic differential equations. The restrictions on transitions for the system are described through equality constraints.     

A primal-dual interior point method for optimal zero-forcing beamformer design under per-antenna power constraints

In this paper, we consider an optimal zero-forcing beamformer design problem in multi-user multiple-input multiple-output broadcast channel. The minimum user rate is maximized subject to zero-forcing constraints and power constraint on each base station antenna array element. The natural formulation leads to a nonconvex optimization problem. This problem is shown to be equivalent to a convex optimization problem with linear objective function, linear equality and inequality constraints and quadratic inequality constraints. Here, the indirect elimination method is applied to reduce the convex optimization problem into an equivalent convex optimization problem of lower dimension with only inequality constraints. The primal-dual interior point method is utilized to develop an effective algorithm (in terms of computational efficiency) via solving the modified KKT equations with Newton method. Numerical simulations are carried out. Compared to algorithms based on a trust region interior point method and sequential quadratic programming method, it is observed that the method proposed is much superior in terms of computational efficiency.     

Nonlinear programming applied to state-constrained optimization problems

Necessary conditions for the minimization of a differentiable function subject to differentiable equality and inequality constraints are given. These conditions are applied to a control problem with a state-variable inequality constraint. The connection with earlier work of Neustadt and Jacobson, Lele, and Speyer is given.