Fast Iterative Solution of Saddle Point Problems in Optimal Control Based on Wavelets

In this paper, wavelet techniques are employed for the fast numerical solution of a control problem governed by an elliptic boundary value problem with boundary control. A quadratic cost functional involving natural norms of the state and the control is to be minimized. Firstly the constraint, the elliptic boundary value problem, is formulated in an appropriate weak form that allows to handle varying boundary conditions explicitly: the boundary conditions are treated by Lagrange multipliers, leading to a saddle point problem. This is combined with a fictitious domain approach in order to cover also more complicated boundaries.Deviating from standard approaches, we then use (biorthogonal) wavelets to derive an equivalent infinite discretized control problem which involves only ₂-norms and -operators. Classical methods from optimization yield the corresponding optimality conditions in terms of two weakly coupled (still infinite) saddle point problems for which a unique solution exists. For deriving finite-dimensional systems which are uniformly invertible, stability of the discretizations has to be ensured. This together with the ₂-setting circumvents the problem of preconditioning: all operators have uniformly bounded condition numbers independent of the discretization.In order to numerically solve the resulting (finite-dimensional) linear system of the weakly coupled saddle point problems, a fully iterative method is proposed which can be viewed as an inexact gradient scheme. It consists of a gradient algorithm as an outer iteration which alternatingly picks the two saddle point problems, and an inner iteration to solve each of the saddle point problems, exemplified in terms of the Uzawa algorithm. It is proved here that this strategy converges, provided that the inner systems are solved sufficiently well. Moreover, since the system matrix is well-conditioned, it is shown that in combination with a nested iteration strategy this iteration is asymptotically optimal in the sense that it provides the solution on discretization level J with an overall amount of arithmetic operations that is proportional to the number of unknows N _J on that level.Finally, numerical results are provided.     

Mesh-independent convergence of penalty methods applied to optimal control with partial differential equations

In this paper, optimal control problems with elliptic state equations and constraints on controls are considered. Also state constraints are briefly discussed. Barrier-penalty methods are applied to treat the occurring restrictions. In the case of finite-dimensional optimization problems, the considered methods have a linear rate of convergence in dependence of the penalty parameter. However, in the case of infinite-dimensional problems, as studied in this article, the direct application of finite-dimensional theory, as given in Grossmann and Zadlo [A general class of penalty/barrier path-following Newton methods for nonlinear programming, Optimization 54 (2005), pp. 161–190], would lead to mesh-dependent order one estimates that deteriorate if the discretization is refined. In this article a first rigorous proof is given for inequality constrained problems that in the case of quadratic penalties a mesh-independence principle holds, i.e. the first-order convergence estimate holds for the continuous problem as well as for discretized problems independently of the discretization step size. The penalty techniques rest upon the control approximate reduction as discussed, e.g. in Grossmann et al. [C. Grossmann, H. Kunz, and R. Meischner, Elliptic control by penalty techniques with control reduction, in System Modeling and Optimization, IFIP Advances in Information and Communication Technology, Vol. 312, Springer, Berlin, 2009, pp. 251–267; M. Hinze, A variational discretization concept in control constrained optimization: the linear-quadratic case, Comput. Optim. Appl. 30 (2005), pp. 45–61]. For the discretization conforming linear element discretization is applied. Some numerical examples illustrate and confirm the theoretical results.     

The Convergence of a Levenberg–Marquardt Method for Nonlinear Inequalities

In this paper, we consider the least l ₂-norm solution for a possibly inconsistent system of nonlinear inequalities. The objective function of the problem is only first-order continuously differentiable. By introducing a new smoothing function, the problem is approximated by a family of parameterized optimization problems with twice continuously differentiable objective functions. Then a Levenberg–Marquardt algorithm is proposed to solve the parameterized smooth optimization problems. It is proved that the algorithm either terminates finitely at a solution of the original inequality problem or generates an infinite sequence. In the latter case, the infinite sequence converges to a least l ₂-norm solution of the inequality problem. The local quadratic convergence of the algorithm was produced under some conditions.     

Mean-field stochastic control with elephant memory in finite and infinite time horizon

ABSTRACT

Our purpose of this paper is to study stochastic control problems for systems driven by mean-field stochastic differential equations with elephant memory, in the sense that the system (like the elephants) never forgets its history. We study both the finite horizon case and the infinite time horizon case.

• In the finite horizon case, results about existence and uniqueness of solutions of such a system are given. Moreover, we prove sufficient as well as necessary stochastic maximum principles for the optimal control of such systems. We apply our results to solve a mean-field linear quadratic control problem.

• For infinite horizon, we derive sufficient and necessary maximum principles.

  As an illustration, we solve an optimal consumption problem from a cash flow modelled by an elephant memory mean-field system.

     

Linear and Discontinuous Approximations for Optimal Control Problems

Abstract

An optimal control problem for 2D and 3D elliptic equations is investigated with pointwise control constraints. This paper is concerned with the discretization of the control by piecewise linear but discontinuous functions. The state and the adjoint state are discretized by linear finite elements. The paper is focused on similarities and differences to piecewise constant and piecewise linear (continuous) approximation of the controls. Approximation of order h in the L ^∞-norm is proved in the main result.     

Superconvergence analysis of fully discrete finite element methods for semilinear parabolic optimal control problems

We study the superconvergence property of fully discrete finite element approximation for quadratic optimal control problems governed by semilinear parabolic equations with control constraints. The time discretization is based on difference methods, whereas the space discretization is done using finite element methods. The state and the adjoint state are approximated by piecewise linear functions and the control is approximated by piecewise constant functions. First, we define a fully discrete finite element approximation scheme for the semilinear parabolic control problem. Second, we derive the superconvergence properties for the control, the state and the adjoint state. Finally, we do some numerical experiments for illustrating our theoretical results.     

Contraction behaviour of iteration–discretization based on gradient type projections

There is a wide range of iterative methods in infinite dimensional spaces to treat variational equations or variational inequalities. As a rule, computational handling of problems in infinite dimensional spaces requires some discretization. Any useful discretization of the original problem leads to families of problems over finite dimensional spaces. Thus, two infinite techniques, namely discretization and iteration are embedded into each other. In the present paper, the behaviour of truncated iterative methods is studied, where at each discretization level only a finite number of steps is performed. In our study no accuracy dependent a posteriori stopping criterion is used. From an algorithmic point of view, the considered methods are of iteration–discretization type. The major aim here is to provide the convergence analysis for the introduced abstract iteration–discretization methods. A special emphasis is given on algorithms for the treatment of variational inequalities with strongly monotone operators over fixed point sets of quasi-nonexpansive mappings.     

Characterization of kernel functions associated with operator algebraic Riccati equations for linear delay systems

In infinite time quadratic control and stochastic filtering problems for linear delay systems, operator algebraic Riccati equations play a very important role. However, since these are abstract operator equations, it is very useful, in analyzing their structure, to be able to characterize the kernel functions associated with the solutions of the operator Riccati equations. The kernel functions are given by the unique solution of a set of coupled differential equations. By comparing these kernel equations with similar ones available in the literature, it is shown that this characterization result is somewhat stronger than previously known results. Possible extentions to systems with control, observation, as well as state delays are also pointed out.     

A posteriori error estimates of hp spectral element methods for optimal control problems with L2-norm state constraint

In this paper, we investigate a distributed optimal control problem governed by elliptic partial differential equations with L²-norm constraint on the state variable. Firstly, the control problem is approximated by hp spectral element methods, which combines the advantages of the finite element methods with spectral methods; then, the optimality conditions of continuous system and discrete system are presented, respectively. Next, hp a posteriori error estimates are derived for the coupled state and control approximation. In the end, a projection gradient iterative algorithm is given, which solves the optimal control problems efficiently. Numerical experiments are carried out to confirm that the numerical results are in good agreement with the theoretical results.

     

A structure‐preserving doubling algorithm for Lur'e equations

We introduce a numerical method for the numerical solution of the Lur'e equations, a system of matrix equations that arises, for instance, in linear‐quadratic infinite time horizon optimal control. We focus on small‐scale, dense problems. Via a Cayley transformation, the problem is transformed to the discrete‐time case, and the structural infinite eigenvalues of the associated matrix pencil are deflated. The deflated problem is associated with a symplectic pencil with several Jordan blocks of eigenvalue 1 and even size, which arise from the nontrivial Kronecker chains at infinity of the original problem. For the solution of this modified problem, we use the structure‐preserving doubling algorithm. Implementation issues such as the choice of the parameter γ in the Cayley transform are discussed. The most interesting feature of this method, with respect to the competing approaches, is the absence of arbitrary rank decisions, which may be ill‐posed and numerically troublesome. The numerical examples presented confirm the effectiveness of this method. Copyright © 2015 John Wiley & Sons, Ltd.     

On differentiability with respect to parameter of solutions to convex optimal control problems subject to state space constraints

A family of convex optimal control problems that depend on a real parameterh is considered. The optimal control problems are subject to state space constraints.It is shown that under some regularity conditions on data the solutions of these problems as well as the associated Lagrange multipliers are directionally-differentiable functions of the parameter.The respective right-derivatives are given as the solution and respective Lagrange multipliers for an auxiliary quadratic optimal control problem subject to linear state space constraints.If a condition of strict complementarity type holds, then directional derivatives become continuous ones.     

Adaptive Wavelet Methods II—Beyond the Elliptic Case

This paper is concerned with the design and analysis of adaptive wavelet methods for systems of operator equations. Its main accomplishment is to extend the range of applicability of the adaptive wavelet-based method developed in [17] for symmetric positive definite problems to indefinite or unsymmetric systems of operator equations. This is accomplished by first introducing techniques (such as the least squares formulation developed in [26]) that transform the original (continuous) problem into an equivalent infinite system of equations which is now well-posed in the Euclidean metric. It is then shown how to utilize adaptive techniques to solve the resulting infinite system of equations. This second step requires a significant modification of the ideas from [17]. The main departure from [17] is to develop an iterative scheme that directly applies to the infinite-dimensional problem rather than finite subproblems derived from the infinite problem. This rests on an adaptive application of the infinite-dimensional operator to finite vectors representing elements from finite-dimensional trial spaces. It is shown that for a wide range of problems, this new adaptive method performs with asymptotically optimal complexity, i.e., it recovers an approximate solution with desired accuracy at a computational expense that stays proportional to the number of terms in a corresponding wavelet-best N -term approximation. An important advantage of this adaptive approach is that it automatically stabilizes the numerical procedure so that, for instance, compatibility constraints on the choice of trial spaces, like the LBB condition, no longer arise.     

Semi-Infinite Programming Approach to Continuously-Constrained Linear-Quadratic Optimal Control Problems

Consider the class of linear-quadratic (LQ) optimal control problems with continuous linear state constraints, that is, constraints imposed on every instant of the time horizon. This class of problems is known to be difficult to solve numerically. In this paper, a computational method based on a semi-infinite programming approach is given. The LQ optimal control problem is formulated as a positive-quadratic infinite programming problem. This can be done by considering the control as the decision variable, while taking the state as a function of the control. After parametrizing the decision variable, an approximate quadratic semi-infinite programming problem is obtained. It is shown that, as we refine the parametrization, the solution sequence of the approximate problems converges to the solution of the infinite programming problem (hence, to the solution of the original optimal control problem). Numerically, the semi-infinite programming problems obtained above can be solved efficiently using an algorithm based on a dual parametrization method.     

Error estimates for the discretization of elliptic control problems with pointwise control and state constraints

A family of elliptic optimal control problems with pointwise constraints on control and state is considered. We are interested in approximation of the optimal solution by a finite element discretization of the involved partial differential equations. The discretization error for a problem with mixed state constraints is estimated in the semidiscrete case and in the fully discrete scheme with the convergence of order h|ln h| and h ^1/2, respectively. However, considering the unregularized continuous problem and the discrete regularized version, and choosing suitable relation between the regularization parameter and the mesh size, i.e., ε∼h ², a convergence order arbitrary close to 1, i.e., h ^1−β is obtained. Therefore, we benefit from tuning the involved parameters.     

Minimizing a Sum of Norms Subject to Linear Equality Constraints

Numerical analysis of a class of nonlinear duality problems is presented. One side of the duality is to minimize a sum of Euclidean norms subject to linear equality constraints (the constrained MSN problem). The other side is to maximize a linear objective function subject to homogeneous linear equality constraints and quadratic inequalities. Large sparse problems of this form result from the discretization of infinite dimensional duality problems in plastic collapse analysis.The solution method is based on the l ₁ penalty function approach to the constrained MSN problem. This can be formulated as an unconstrained MSN problem for which the first author has recently published an efficient Newton barrier method, and for which new methods are still being developed.Numerical results are presented for plastic collapse problems with up to 180000 variables, 90000 terms in the sum of norms and 90000 linear constraints. The obtained accuracy is of order 10^-8 measured in feasibility and duality gap.     

Game-theoretic coupled riccati equations associated to controlled linear differential systems with jump markov perturbations

In this paper we investigate several properties of the stabilizing solution of a class of systems of Riccati type differential equations with indefinite sign associated to controlled systems described by differential equations with Markovian jumping.

We show that the existence of a bounded on R ₊ and stabilizing solution for this class of systems of Riccati type differential equations is equivalent to the solvability of a control-theoretic problem, namely disturbance attenuation problem.

If the coefficients of the considered system are theta;-periodic functions then the stabilizing solution is also theta;-periodic and if the coefficients are asymptotic almost periodic functions, then the stabilizing solution is also asymptotic almost periodic and its almost periodic component is a stabilizing solution for a system of Riccati type differential equations defined on the whole real axis. One proves also that the existence of a stabilizing and bounded on R ₊ solution of a system of Riccati differential equations with indefinite sign is equivalent to the existence of a solution to a corresponding system of matrix inequalities. Finally, a minimality property of the stabilizing solution is derived.     

On the finite volume element method

Summary The finite volume element method (FVE) is a discretization technique for partial differential equations. It uses a volume integral formulation of the problem with a finite partitioning set of volumes to discretize the equations, then restricts the admissible functions to a finite element space to discretize the solution. this paper develops discretization error estimates for general selfadjoint elliptic boundary value problems with FVE based on triangulations with linear finite element spaces and a general type of control volume. We establishO(h) estimates of the error in a discreteH ¹ semi-norm. Under an additional assumption of local uniformity of the triangulation the estimate is improved toO(h ²). Results on the effects of numerical integration are also included.This research was sponsored in part by the Air Force Office of Scientific Research under grant number AFOSR-86-0126 and the National Science Foundation under grant number DMS-8704169. This work was performed while the author was at the University of Colorado at Denver     

A Cutting Plane Approach to Solving Quadratic Infinite Programs on Measure Spaces

We study infinite dimensional quadratic programming (QP) problems of integral type. The decision variable is taken in the space of bounded regular Borel measures on compact Hausdorff spaces. An implicit cutting plane algorithm is developed to obtain an optimal solution of the infinite dimensional QP problem. The major computational tasks in using the implicit cutting plane approach to solve infinite dimensional QP problems lie in finding a global optimizer of a non-linear and non-convex program. We present an explicit scheme to relax this requirement and to get rid of the unnecessary constraints in each iteration in order to reduce the size of the computatioinal programs. A general convergence proof of this approach is also given.     

First main problem of the axisymmetric stress state of a ball with an ellipsoidal cavity

We consider the problem of axisymmetric strain of an elastic ball with an elongated ellipsoidal cavity whose center is at the center of the ball when given forces act on the spherical and the ellipsoidal surfaces. The problem is solved using an approach based on integral representation of p-analytical functions with p-x characteristic by analytical functions. The method of p-analytical functions reduces the solution of the problem to the solution of an infinite quasi-completely regular system of linear algebraic equations in which the free terms are upper bounded and tend to zero as the index increases.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 72, pp. 42–48, 1990.     

A control reduced primal interior point method for a class of control constrained optimal control problems

A primal interior point method for control constrained optimal control problems with PDE constraints is considered. Pointwise elimination of the control leads to a homotopy in the remaining state and dual variables, which is addressed by a short step pathfollowing method. The algorithm is applied to the continuous, infinite dimensional problem, where discretization is performed only in the innermost loop when solving linear equations. The a priori elimination of the least regular control permits to obtain the required accuracy with comparatively coarse meshes. Convergence of the method and discretization errors are studied, and the method is illustrated at two numerical examples. Supported by the DFG Research Center Matheon “Mathematics for key technologies” in Berlin. This paper appeared as ZIB Report 04-38.