Parametric dual regularization for an optimal control problem with pointwise state constraints

The perturbation method is used in the dual regularization theory for a linear convex optimal control problem with a strongly convex objective functional and pointwise state constraints understood as ones in L ₂. Primary attention is given to the qualitative properties of the dual regularization method, depending on the differential properties of the value function (S-function) in the optimization problem. It is shown that the convergence of the method is closely related to the Lagrange principle and the Pontryagin maximum principle. The dual regularization scheme is shown to provide a new method for proving the maximum principle in the problem with pointwise state constraints understood in L ₂ or C. The regularized Lagrange principle in nondifferential form and the regularized Pontryagin maximum principle are discussed. Illustrative examples are presented.     

An optimal design problem with perimeter penalization

We study the optimal design problem of finding the minimal energy configuration for a mixture of two conducting materials when a perimeter penalization of the unknown domain is added. We show that in this situation an optimal domain exists and that, under suitable assumptions on the data, it is an open set.This work is part of the project EURHomogenization, contract SC1-CT91-0732 of the program SCIENCE of the Commission of the European Communities.     

An iterative Bregman regularization method for optimal control problems with inequality constraints

We study an iterative regularization method of optimal control problems with control constraints. The regularization method is based on generalized Bregman distances. We provide convergence results under a combination of a source condition and a regularity condition on the active sets. We do not assume attainability of the desired state. Furthermore, a priori regularization error estimates are obtained.     

On optimal control problems with state constraints

This paper is concerned with necessary conditions for a general optimal control problem developed by Russak and Tan. It is shown that, in most cases, a further relation between the multipliers holds. This result is of interest in particular for the investigation of perturbations of the state constraint.     

An optimal design problem for submerged bodies

The problem of finding the shape of a smooth body submerged in a fluid of finite depth which minimizes added mass or damping is considered. The optimal configuration is sought in a suitably constrained class so as to be physically meaningful and for which the mathematical problem of a submerged body with linearized free surface condition is uniquely solvable. The problem is formulated as a constrained optimization problem whose cost functional (e.g. added mass) is a domain functional. Continuity of the solution of the boundary value problem with respect to variations of the boundary is established in an appropriate function space setting and this is used to establish existence of an optimal solution. A variational inequality is derived for the optimal shape and it is shown how finite dimensional approximate solutions may be found.     

Approximation of optimal control problems with state constraints

We study the approximation of control problems governed by elliptic partial differential equations with pointwise state constraints. For a finite dimensional approximation of the control set and for suitable perturbations of the state constraints, we prove that the corresponding sequence of discrete control problems converges to a relaxed problem. A similar analysis is carried out for problems in which the state equation is discretized by a finite element method.     

An efficient algorithm for the Lagrangean dual of nonlinear knapsack problems with additional nested constraints

This paper presents an efficient algorithm for solving the Lagrangean dual of nonlinear knapsack problems with additional nested constraints. The dual solution provides a feasible primal solution (if it exists) and associated lower and upper bounds on the optimal objective function value of the primal problem. Computational experience is cited indicating computation time, number of dual iterations, and “tightness” of the bounds.     

Maximum principle for optimal control problems with intermediate constraints

We consider optimal control problems with constraints at intermediate points of the trajectory. A natural technique (propagation of phase and control variables) is applied to reduce these problems to a standard optimal control problem of Pontryagin type with equality and inequality constraints at the trajectory endpoints. In this way we derive necessary optimality conditions that generalize the Pontryagin classical maximum principle. The same technique is applied to so-called variable structure problems and to some hybrid problems. The new optimality conditions are compared with the results of other authors and five examples illustrating their application are presented.     

On some elliptic optimal control problems with state constraints

In this paper optimality conditions will be derived for elliptic optimal control problems with a restriction on the state or on the gradient of the state. Essential tools are the method of transposition and generalized trace theorems and green's formulas from the theory of elliptic differential equations.     

Exterior penalty in optimal control problem with state-control constraints

We consider an abstract optimal control problem with additional equality and inequality state and control constraints, we use the exterior penalty function to transform the constrained optimal control problem into a sequence of unconstrained optimal control problems, under conditions in control lie in L ₁, the sequence of the solution to the unconstrained problem contains a subsequence converging of the solution of constrained problem, this convergence is strong when the problemis non convex, and is weak if the problemis convex in control. This generalizes the results of P.Nepomiastcthy [4] where he considered the control in the Hilbert space L ²(I,? ^m ).     

Some results on optimal control with unilateral state constraints

In this paper, we study the problem of quadratic optimal control with state variables unilateral constraints, for linear time-invariant systems. The necessary conditions are formulated as a linear invariant system with complementary slackness conditions. Some structural properties of this system are examined. Then it is shown that the problem can benefit from the higher order Moreau’s sweeping process, that is, a specific distributional differential inclusion, and from ten Dam’s geometric theory [A.A. ten Dam, K.F. Dwarshuis, J.C. Willems, The contact problem for linear continuous-time dynamical systems: A geometric approach, IEEE Trans. Automat. Control 42 (4) (1997) 458–472; A.A. ten Dam, Unilaterally Constrained Dynamical Systems, Ph.D. Thesis, Rijsuniversiteit Groningen, NL, available at http://irs.ub.rug.nl/ppn/159407869, 1997] for partitioning of the admissible domain boundary (in particular for the case of multivariable systems). In fact, the first step may be also seen as follows: does the higher order Moreau’s sweeping process (developed in Acary et al. [V. Acary, B. Brogliato, D. Goeleven, Higher order Moreau’s sweeping process: Mathematical formulation and numerical simulation, Math. Programm. A 113 (2008) 133–217]) correspond to the necessary conditions of some optimal control problem with an extended integral action? The knowledge of the qualitative behaviour of optimal trajectories at junction times is improved with the approach, which also paves the way towards efficient time-stepping numerical algorithms to solve the optimal control boundary value problem.     

New algorithm for optimal parameter estimation with linear constraints

This paper presents a new algorithm for optimal parameter estimation problems with linear constraints. The algorithm developed is based on least absolute-value approximations. The problem is solved first using a least-error-square technique, where we add to the cost function the equality constraints via Lagrange multipliers, to obtain a good estimate for the residuals of the measurements, having gained this information, we choose a number of measurements with the smallest residuals. This number equals the number of parameters to be estimated minus the number of constraints. Using these measurements together with the constraints, we obtain a number of observations equal to the number of parameters to be estimated. By using this technique, we show that there is no need to either iterate or use linear programming to obtain the estimation.This work was supported by the Natural Sciences and Engineering Research Council of Canada, Grant A4146.     

Second-order analysis for optimal control problems with pure state constraints and mixed control-state constraints

This paper deals with the optimal control problem of an ordinary differential equation with several pure state constraints, of arbitrary orders, as well as mixed control-state constraints. We assume (i) the control to be continuous and the strengthened Legendre–Clebsch condition to hold, and (ii) a linear independence condition of the active constraints at their respective order to hold. We give a complete analysis of the smoothness and junction conditions of the control and of the constraints multipliers. This allows us to obtain, when there are finitely many nontangential junction points, a theory of no-gap second-order optimality conditions and a characterization of the well-posedness of the shooting algorithm. These results generalize those obtained in the case of a scalar-valued state constraint and a scalar-valued control.     

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.     

Minimum-weight design of structures with natural-frequency constraints

The problem of minimum-weight design of structures with several natural-frequency constraints is considered in this paper. The problem is solved by using a combined finite element method and sequential linear programming (FEM-SLP) formulation. The unique features of the present approach include the use of the assumed mode reanalysis formulation for the repeated eigensolution and the associated sensitivity analysis. The present approach has been implemented using the general-purpose finite element program . Two examples are included to illustrate the effectiveness of the present approach.     

Sequential gradient-restoration algorithm for optimal control problems with nondifferential constraints

This paper considers the numerical solution of optimal control problems involving a functionalI subject to differential constraints, nondifferential constraints, and terminal constraints. The problem is to find the statex(t), the controlu(t), and the parameter π so that the functional is minimized, while the constraints are satisfied to a predetermined accuracy. The approach taken is a sequence of two-phase processes or cycles, composed of a gradient phase and a restoration phase. The gradient phase involves a single iteration and is designed to decrease the functional, while the constraints are satisfied to first order. The restoration phase involves one or several iterations and is designed to restore the constraints to a predetermined accuracy, while the norm of the variations of the control and the parameter is minimized. The principal property of the algorithm is that it produces a sequence of feasible suboptimal solutions: the functionsx(t),u(t), π obtained at the end of each cycle satisfy the constraints to a predetermined accuracy. Therefore, the functionals of any two elements of the sequence are comparable. The stepsize of the gradient phase is determined by a one-dimensional search on the augmented functionalJ, and the stepsize of the restoration phase by a one-dimensional search on the constraint errorP. If α _g is the gradient stepsize and α _r is the restoration stepsize, the gradient corrections are ofO(α _g ) and the restoration corrections are ofO(α _r α _g ²). Therefore, for α _g sufficiently small, the restoration phase preserves the descent property of the gradient phase: the functionalÎ at the end of any complete gradient-restoration cycle is smaller than the functionalI at the beginning of the cycle. To facilitate the numerical solution on digital computers, the actual time ? is replaced by the normalized timet, defined in such a way that the extremal arc has a normalized time length Δt=1. In this way, variable-time terminal conditions are transformed into fixed-time terminal conditions. The actual time τ at which the terminal boundary is reached is regarded to be a component of the parameter π being optimized. The present general formulation differs from that of Ref. 4 because of the inclusion of the nondifferential constraints to be satisfied everywhere over the interval 0 ≤t ≤ 1. Its importance lies in that (i) many optimization problems arise directly in the form considered here, (ii) problems involving state equality constraints can be reduced to the present scheme through suitable transformations, and (iii) problems involving inequality constraints can be reduced to the present scheme through suitable transformations. The latter statement applies, for instance, to the following situations: (a) problems with bounded control, (b) problems with bounded state, (c) problems with bounded time rate of change of the state, and (d) problems where some bound is imposed on an arbitrarily prescribed function of the parameter, the control, the state, and the time rate of change of the state. Numerical examples are presented for both the fixed-final-time case and the free-final-time case. These examples demonstrate the feasibility as well as the rapidity of convergence of the technique developed in this paper.