Superlinear convergence of the control reduced interior point method for PDE constrained optimization

A thorough convergence analysis of the Control Reduced Interior Point Method in function space is performed. This recently proposed method is a primal interior point pathfollowing scheme with the special feature that the control variable is eliminated from the optimality system. Apart from global linear convergence we show that this method converges locally superlinearly, if the optimal solution satisfies a certain non-degeneracy condition. In numerical experiments we observe that a prototype implementation of our method behaves as predicted by our theoretical results. Supported by the DFG Research Center Matheon “Mathematics for key technologies”.     

Second-order sufficient conditions for control problems with mixed control-state constraints

References 1–4 develop second-order sufficient conditions for local minima of optimal control problems with state and control constraints. These second-order conditions tighten the gap between necessary and sufficient conditions by evaluating a positive-definiteness criterion on the tangent space of the active constraints. The purpose of this paper is twofold. First, we extend the methods in Refs. 3, 4 and include general boundary conditions. Then, we relate the approach to the two-norm approach developed in Ref. 5. A direct sufficiency criterion is based on a quadratic function that satisfies a Hamilton-Jacobi inequality. A specific form of such a function is obtained by applying the second-order sufficient conditions to a parametric optimization problem. The resulting second-order positive-definiteness conditions can be verified by solving Riccati equations.The authors wish to thank K. Malanowski for helpful discussions.     

Regularity of solutions for an optimal control problem with mixed control-state constraints

A class of optimal control problems for a semilinear elliptic partial differential equation with mixed control-state constraints is considered. Existence results of an optimal control and necessary optimality conditions are stated. Moreover, a projection formula is derived that is equivalent to the necessary optimality conditions. As main result, the Lipschitz continuity of the optimal control is obtained.     

Lipschitz stability for elliptic optimal control problems with mixed control-state constraints

A family of linear-quadratic optimal control problems with pointwise mixed state-control constraints governed by linear elliptic partial differential equations is considered. All data depend on a vector parameter of perturbations. Lipschitz stability with respect to perturbations of the optimal control, the state and adjoint variables, and the Lagrange multipliers is established.     

Solving elliptic control problems with interior point and SQP methods: control and state constraints

We study optimal control problems for semilinear elliptic equations subject to control and state inequality constraints. Both boundary control and distributed control problems are considered with boundary conditions of Dirichlet or Neumann type. By introducing suitable discretization schemes, the control problem is transcribed into a nonlinear programming problem. Necessary conditions of optimality are discussed both for the continuous and the discretized control problem. It is shown that the recently developed interior point method LOQO of [35] is capable of solving these problems even for high discretizations. Four numerical examples with Dirichlet and Neumann boundary conditions are provided that illustrate the performance of the algorithm for different types of controls including bang–bang controls.     

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.     

Solving nested-constraint resource allocation problems with an interior point method

Efficient methods for convex resource allocation problems usually exploit algebraic properties of the objective function. For problems with nested constraints, we show that constraint sparsity structure alone allows rapid solution with a general interior point method. The key is a special-purpose linear system solver requiring only linear time in the problem dimensions. Computational tests show that this approach outperforms the previous best algebraically specialized methods.     

Solution differentiability for parametric nonlinear control problems with control-state constraints

This paper considers parametric nonlinear control problems subject to mixed control-state constraints. The data perturbations are modeled by a parameterp of a Banach space. Using recent second-order sufficient conditions (SSC), it is shown that the optimal solution and the adjoint multipliers are differentiable functions of the parameter. The proof blends numerical shooting techniques for solving the associated boundary-value problem with theoretical methods for obtaining SSC. In a first step, a differentiable family of extremals for the underlying parameteric boundary-value problem is constructed by assuming the regularity of the shooting matrix. Optimality of this family of extremals can be established in a second step when SSC are imposed. This is achieved by building a bridge between the variational system corresponding to the boundary-value problem, solutions of the associated Riccati ODE, and SSC.Solution differentiability provides a theoretical basis for performing a numerical sensitivity analysis of first order. Two numerical examples are worked out in detail that aim at reducing the considerable deficit of numerical examples in this area of research.This paper is dedicated to Professor J. Stoer on the occasion of his 60th birthday.The authors are indebted to K. Malanowski for helpful discussions.     

Sensitivity analysis for parametric control problems with control-state constraints

Parametric nonlinear control problems subject to vector-valued mixed control-state constraints are investigated. The model perturbations are implemented by a parameter p of a Banach-space P. We prove solution differentiability in the sense that the optimal solution and the associated adjoint multiplier function are differentiable functions of the parameter. The main assumptions for solution differentiability are composed by regularity conditions and recently developed second-order sufficient conditions (SSC). The analysis generalizes the approach in [16, 20] and establishes a link between (1) shooting techniques for solving the associated boundary value problem (BVP) and (2) SSC. We shall make use of sensitivity results from finite-dimensional parametric programming and exploit the relationships between the variational system associated to BVP and its corresponding Riccati equation.Solution differentiability is the theoretical backbone for any numerical sensitivity analysis. A numerical example with a vector-valued control is presented that illustrates sensitivity analysis in detail.     

Solving the continuous nonlinear resource allocation problem with an interior point method

Resource allocation problems are usually solved with specialized methods exploiting their general sparsity and problem-specific algebraic structure. We show that the sparsity structure alone yields a closed-form Newton search direction for the generic primal-dual interior point method. Computational tests show that the interior point method consistently outperforms the best specialized methods when no additional algebraic structure is available.     

On the interplay between interior point approximation and parametric sensitivities in optimal control

Infinite-dimensional parameter-dependent optimization problems of the form ‘minJ(u;p) subject to g(u)?0’ are studied, where u is sought in an L_∞ function space, J is a quadratic objective functional, and g represents pointwise linear constraints. This setting covers in particular control constrained optimal control problems. Sensitivities with respect to the parameter p of both, optimal solutions of the original problem, and of its approximation by the classical primal-dual interior point approach are considered. The convergence of the latter to the former is shown as the homotopy parameter μ goes to zero, and error bounds in various Lq norms are derived. Several numerical examples illustrate the results.     

Superlinear and quadratic convergence of some primal-dual interior point methods for constrained optimization

This paper proves local convergence rates of primal-dual interior point methods for general nonlinearly constrained optimization problems. Conditions to be satisfied at a solution are those given by the usual Jacobian uniqueness conditions. Proofs about convergence rates are given for three kinds of step size rules. They are: (i) the step size rule adopted by Zhang et al. in their convergence analysis of a primal-dual interior point method for linear programs, in which they used single step size for primal and dual variables; (ii) the step size rule used in the software package OB1, which uses different step sizes for primal and dual variables; and (iii) the step size rule used by Yamashita for his globally convergent primal-dual interior point method for general constrained optimization problems, which also uses different step sizes for primal and dual variables. Conditions to the barrier parameter and parameters in step size rules are given for each case. For these step size rules, local and quadratic convergence of the Newton method and local and superlinear convergence of the quasi-Newton method are proved. A preliminary version of this paper was presented at the conference “Optimization-Models and Algorithms” held at the Institute of Statistical Mathematics, Tokyo, March 1993.     

No-gap optimality conditions for an optimal control problem with pointwise control-state constraints

An optimal control problem with pointwise mixed constraints of the instationary three-dimensional Navier–Stokes–Voigt equations is considered. We derive second-order optimality conditions and show that there is no gap between second-order necessary optimality conditions and second-order sufficient optimality conditions. In addition, the second-order sufficient optimality conditions for the problem where the objective functional does not contain a Tikhonov regularization term are also discussed.     

A new efficient short-step projective interior point method for linear programming

In this paper, we are interested in the performance of Karmarkar’s projective algorithm for linear programming. We propose a new displacement step to accelerate and improve the convergence of this algorithm. This purpose is confirmed by numerical experimentations showing the efficiency and the robustness of the obtained algorithm over Schrijver’s one for small problem dimensions.     

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.     

A direct approach to second order conditions for mixed equality constraints

In this paper we consider an optimal control problem posed over piecewise continuous controls and involving state-control (mixed) equality constraints. We provide an explicit derivation of second order necessary conditions simpler than others available in the literature, yielding a clear understanding of how to define a set of “differentially admissible variations” where a certain quadratic form is nonnegative.     

A solution method for regular optimal control problems with state constraints

A method of region analysis is developed for solving a class of optimal control problems with one state and one control variable, including state and control constraints. The performance index is strictly convex with respect to the control variable, while this variable appears only linearly in the state equation. The convexity or linearity assumption of the performance index or the state equation with respect to the state variable is not required.The author would like to express his sincere gratitude to Prof. R. Klötzler, Prof. E. Zeidler, Prof. H. Schumann, Prof. J. Focke, and other colleagues of the Department of Mathematics, Karl Marx University, Leipzig, GDR, for their support during his stay in Leipzig.     

Non-negatively constrained image deblurring with an inexact interior point method

Nonlinear image deblurring procedures based on probabilistic considerations have been widely investigated in the literature. This approach leads to model the deblurring problem as a large scale optimization problem, with a nonlinear, convex objective function and non-negativity constraints on the sign of the variables. The interior point methods have shown in the last years to be very reliable in nonlinear programs. In this paper we propose an inexact Newton interior point (IP) algorithm designed for the solution of the deblurring problem. The numerical experience compares the IP method with another state-of-the-art method, the Lucy Richardson algorithm, and shows a significant improvement of the processing time.     

Direct solution of the normal equations in interior point methods for convex transportation problems

Recent research has shown that very large convex transportation problems can be solved efficiently with interior point methods by finding a good pre-conditioner for an iterative linear-equation solver. In this article, it is demonstrated that the specific structure of the constraints allows use of a direct method for solving those linear equations with the same order of worst-case time complexity.