Convergence Analysis of the Implicit Euler-discretization and Sufficient Conditions for Optimal Control Problems Subject to Index-one Differential-algebraic Equations

For optimal control problems subject to index-one differential-algebraic equations in semi-explicit form we discuss second order sufficient conditions in form of a coercivity condition taking into account the two-norm discrepancy. Furthermore we introduce a related Riccati-type and Legendre-Clebsch condition which are sufficient for the validity of the coercivity condition. Using the implicit Euler-discretization we approximate the optimal control problem and analyze the convergence of solutions of the local minimum principle for the discretized optimal control problem by applying the general convergence framework of Stetter, which requires the discretization method to be continuous, consistent, and stable.

     

Abstract convergence theorem for quasi-convex optimization problems with applications

Abstract

Quasi-convex optimization is fundamental to the modelling of many practical problems in various fields such as economics, finance and industrial organization. Subgradient methods are practical iterative algorithms for solving large-scale quasi-convex optimization problems. In the present paper, focusing on quasi-convex optimization, we develop an abstract convergence theorem for a class of sequences, which satisfy a general basic inequality, under some suitable assumptions on parameters. The convergence properties in both function values and distances of iterates from the optimal solution set are discussed. The abstract convergence theorem covers relevant results of many types of subgradient methods studied in the literature, for either convex or quasi-convex optimization. Furthermore, we propose a new subgradient method, in which a perturbation of the successive direction is employed at each iteration. As an application of the abstract convergence theorem, we obtain the convergence results of the proposed subgradient method under the assumption of the Hölder condition of order p and by using the constant, diminishing or dynamic stepsize rules, respectively. A preliminary numerical study shows that the proposed method outperforms the standard, stochastic and primal-dual subgradient methods in solving the Cobb–Douglas production efficiency problem.     

On the convexity of integrals of multivalued mappings: Applications in control theory

Using the notion of the local convexity index, we characterize in a quantitative way the local convexity of a set in then-dimensional Euclidean space, defined by an integral of a multivalued mapping. We estimate the rate of convergence of the conditional gradient method for solving an abstract optimization problem by means of the convexity index of the constraining set at the solution point. These results are applied to the qualitative analysis of the solutions of time-optimal and Mayer problems for linear control systems, as well as for estimating the convergence rate of algorithms solving these problems.     

Locating a nearest matrix with an eigenvalue of prespecified algebraic multiplicity

The Wilkinson distance of a matrix A is the two-norm of the smallest perturbation E so that A + E has a multiple eigenvalue. Malyshev derived a singular value optimization characterization for the Wilkinson distance. In this work we generalize the definition of the Wilkinson distance as the two-norm of the smallest perturbation so that the perturbed matrix has an eigenvalue of prespecified algebraic multiplicity. We provide a singular value characterization for this generalized Wilkinson distance. Then we outline a numerical technique to solve the derived singular value optimization problems. In particular the numerical technique is applicable to Malyshev’s formula to compute the Wilkinson distance as well as to retrieve a nearest matrix with a multiple eigenvalue.     

Primal-dual interior-point methods for PDE-constrained optimization

This paper provides a detailed analysis of a primal-dual interior-point method for PDE-constrained optimization. Considered are optimal control problems with control constraints in L ^p . It is shown that the developed primal-dual interior-point method converges globally and locally superlinearly. Not only the easier L ^∞-setting is analyzed, but also a more involved L ^q -analysis, q < ∞, is presented. In L ^∞, the set of feasible controls contains interior points and the Fréchet differentiability of the perturbed optimality system can be shown. In the L ^q -setting, which is highly relevant for PDE-constrained optimization, these nice properties are no longer available. Nevertheless, a convergence analysis is developed using refined techniques. In parti- cular, two-norm techniques and a smoothing step are required. The L ^q -analysis with smoothing step yields global linear and local superlinear convergence, whereas the L ^∞-analysis without smoothing step yields only global linear convergence.     

An alternating direction method of multipliers for elliptic equation constrained optimization problem

We propose an alternating direction method of multipliers (ADMM) for solving the state constrained optimization problems governed by elliptic equations. The unconstrained as well as box-constrained cases of the Dirichlet boundary control, Robin boundary control, and right-hand side control problems are considered here. These continuous optimization problems are transformed into discrete optimization problems by the finite element method discretization, then are solved by ADMM. The ADMM is an efficient first order algorithm with global convergence, which combines the decomposability of dual ascent with the superior convergence properties of the method of multipliers. We shall present exhaustive convergence analysis of ADMM for these different type optimization problems. The numerical experiments are performed to verify the efficiency of the method.     

改进种群多样性的双变异差分进化算法

差分进化算法(DE)是一种基于种群的启发式随机搜索技术,对于解决连续性优化问题具有较强的鲁棒性.然而传统差分进化算法存在种群多样性和收敛速度之间的矛盾,一种改进种群多样性的双变异差分进化算法(DADE),通过引入BFS-best机制(基于排序的可行解选取递减策略)改进变异算子"DE/current-to-best",将其与DE/rand/1构成双变异策略来改善DE算法中种群多样性减少的问题.同时,每个个体的控制参数基于排序自适应更新.最后,利用多个CEC2013标准测试函数对改进算法进行测试,实验结果表明,改进后的算法能有效改善种群多样性,较好地提高了算法的全局收敛能力和收敛速度.     

Sourcewise representability conditions and power estimates of convergence rate in Tikhonov’s scheme for solving ill-posed extremal problems

We study the rate of the convergence of approximations generated by the Tikhonov scheme for solving ill-posed optimization problems with smooth functionals given in a general form in a Hilbert space. We establish sourcewise representability conditions which are necessary and sufficient for the convergence of approximations at a power rate. Sufficient conditions are related to the estimate of the discrepancy with respect to the objective functional, while the necessary ones are formulated for the estimate with respect to the argument. We specify certain cases when sufficient and necessary conditions coincide in essence.     

Sensitivity of solutions to systems of optimality conditions under the violation of constraint qualifications

A survey is given of old and new results on the sensitivity of solutions to systems of optimality conditions with respect to parametric perturbations. Results of this kind play a key role in subtle convergence analysis of various constrained optimization algorithms. General systems of optimality conditions for problems with abstract constraints, Karush-Kuhn-Tucker systems for mathematical programs, and Lagrange systems for problems with equality constraints are examined. Special attention is given to the cases where the traditional constraint qualifications are violated.     

On the convergence of inexact augmented Lagrangian methods for problems with convex constraints

We consider the augmented Lagrangian method (ALM) for constrained optimization problems in the presence of convex inequality and convex abstract constraints. We focus on the case where the Lagrangian sub-problems are solved up to approximate stationary points, with increasing accuracy. We analyze two different criteria of approximate stationarity for the sub-problems and we prove the global convergence to stationary points of ALM in both cases.     

Convergence analysis of two-phase methods for approximating the Edgeworth-Pareto hull in nonlinear multicriteria optimization problems

The convergence of two-phase methods for approximating the Edgeworth-Pareto hull (EPH) in nonlinear multicriteria optimization problems is analyzed. The methods are based on the iterative supplement of the finite set of feasible criteria vectors (approximation basis) whose EPH approximates the desired set. A feature of two-phase methods is that the criteria images of randomly generated points of the decision space approach the Pareto frontier via local optimization of adaptively chosen convolutions of criteria. The convergence of two-phase methods is proved for both an abstract form of the algorithm and for a two-phase method based on the Germeier convolution.     

State-constrained optimal control of nonlinear elliptic variational inequalities

An optimization control problem for systems described by abstract variational inequalities with state constraints is considered. The solvability of this problem is proved. The problem is approximated by the penalty method. The convergence of this method is proved. Necessary conditions of optimality for the approximation problem are obtained. Its solution is an approximate optimal control of the initial problem.     

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.     

On the discrepancy principle for iterative and parametric methods to solve linear ill-posed equations

Summary. For the numerical solution of (non-necessarily well-posed) linear equations in Banach spaces we consider a class of iterative methods which contains well-known methods like the Richardson iteration, if the associated resolvent operator fulfils a condition with respect to a sector. It is the purpose of this paper to show that for given noisy right-hand side the discrepancy principle (being a stopping rule for the iteration methods belonging to the mentioned class) defines a regularization method, and convergence rates are proved under additional smoothness conditions on the initial error. This extends similar results obtained for positive semidefinite problems in Hilbert spaces. Then we consider a class of parametric methods which under the same resolvent condition contains the method of the abstract Cauchy problem, and (under a weaker resolvent condition) the iterated method of Lavrentiev. A modified discrepancy principle is formulated for them, and finally numerical illustrations are presented. Received August 29, 1994 / Revised version received September 19, 1995     

A New Conical Regularization for Some Optimization and Optimal Control Problems: Convergence Analysis and Finite Element Discretization

In this article, we study an abstract constrained optimization problem that appears commonly in the optimal control of linear partial differential equations. The main emphasis of the present study is on the case when the ordering cone for the optimization problem has an empty interior. To circumvent this major difficulty, we propose a new conical regularization approach in which the main idea is to replace the ordering cone by a family of dilating cones. We devise a general regularization approach and use it to give a detailed convergence analysis for the conical regularization as well as a related regularization approach. We showed that the conical regularization approach leads to a family of optimization problems that admit regular multipliers. The approach remains valid in the setting of general Hilbert spaces and it does not require any sort of compactness or positivity condition on the operators involved. One of the main advantages of the approach is that it is amenable for numerical computations. We consider four different examples, two of them elliptic control problems with state constraints, and present numerical results that completely support our theoretical results and confirm the numerical feasibility of our approach. The motivation for the conical regularization is to overcome the difficulties associated with the lack of Slater's type constraint qualification, which is a common hurdle in numerous branches of applied mathematics including optimal control, inverse problems, vector optimization, set-valued optimization, sensitivity analysis, variational inequalities, among others.     

The lagrange-newton method for infinite-dimensional optimization problems

This paper investigates local convergence properties of the Lagrange-Newton method for optimization problems in reflexive Banach spaces. Sufficient conditions for quadratic convergence of optimal solutions and Lagrange multipliers are given. The results are applied to optimal control problems.     

Successive approximation and linear stability involving convergent sequences of optimization problems

A certain convergence notion for extended real-valued functions, which has been studied by a number of authors in various applied contexts since the latter 1960s, is examined here in relation to abstract optimization problems in normed linear spaces. The main facts concerning behavior of the optimal values, the optimal solution sets and the -optimal solution sets corresponding to “convergent” sequences of such problems are developed. General linear perturbations are incorporated explicitly into the problems of the sequence, lending a stability-theoretic character to the results. Most of the results apply to nonconvex minimization.     

Covering Pareto Sets by Multilevel Subdivision Techniques

In this work, we present a new set-oriented numerical method for the numerical solution of multiobjective optimization problems. These methods are global in nature and allow to approximate the entire set of (global) Pareto points. After proving convergence of an associated abstract subdivision procedure, we use this result as a basis for the development of three different algorithms. We consider also appropriate combinations of them in order to improve the total performance. Finally, we illustrate the efficiency of these techniques via academic examples plus a real technical application, namely, the optimization of an active suspension system for cars.The authors thank Joachim Lückel for his suggestion to get into the interesting field of multiobjective optimization. Katrin Baptist as well as Frank Scharfeld helped the authors with fruitful discussions. This work was partly supported by the Deutsche Forschungsgemeinschaft within SFB 376 and SFB 614.     

On some optimal control problems and their finite difference approximations and regularization for quasilinear elliptic equations with controls in the coefficients

Mathematical statements of the optimal control problems for quasilinear elliptic equations with the controls in the variable coefficients of the equation of state are considered. Both local and integral constraints on the controls are considered. The objective functionals correspond to the optimization with respect to a certain number of quality indexes. Finite difference approximations of optimization problems are constructed, and estimates of the approximation error with respect to the state and to the objective functional are established. The weak convergence in control is proved. The approximations are regularized after Tikhonov. Interesting examples of some applied optimization problems that naturally lead to the nonlinear optimal control problems examined in this paper are considered.     

Convergence and optimal control problems of nonlinear evolution equations governed by time-dependent operator

We study an abstract nonlinear evolution equation governed by a time-dependent operator of subdifferential type in a real Hilbert space. In this paper we discuss the convergence of solutions to our evolution equations. Also, we investigate the optimal control problems of nonlinear evolution equations. Moreover, we apply our abstract results to a quasilinear parabolic PDE with a mixed boundary condition.