A Shooting Algorithm for Optimal Control Problems with Singular Arcs

In this article, we propose a shooting algorithm for a class of optimal control problems for which all control variables appear linearly. The shooting system has, in the general case, more equations than unknowns and the Gauss–Newton method is used to compute a zero of the shooting function. This shooting algorithm is locally quadratically convergent, if the derivative of the shooting function is one-to-one at the solution. The main result of this paper is to show that the latter holds whenever a sufficient condition for weak optimality is satisfied. We note that this condition is very close to a second order necessary condition. For the case when the shooting system can be reduced to one having the same number of unknowns and equations (square system), we prove that the mentioned sufficient condition guarantees the stability of the optimal solution under small perturbations and the invertibility of the Jacobian matrix of the shooting function associated with the perturbed problem. We present numerical tests that validate our method.     

求时滞Volterra积分系统最优控制问题近似解的一个迭代算法

对一类典型的复杂系统—含多个非线性时滞的Volterra积分系统,给出其控制作用受限(硬约束)最优控制问题近似解的一个迭代算法,证明了该算法是well-defined的,并在一定的条件下证得了算法的收敛性.     

Homotopy Algorithm for Optimal Control Problems with a Second-order State Constraint

This paper deals with optimal control problems with a regular second-order state constraint and a scalar control, satisfying the strengthened Legendre-Clebsch condition. We study the stability of structure of stationary points. It is shown that under a uniform strict complementarity assumption, boundary arcs are stable under sufficiently smooth perturbations of the data. On the contrary, nonreducible touch points are not stable under perturbations. We show that under some reasonable conditions, either a boundary arc or a second touch point may appear. Those results allow us to design an homotopy algorithm which automatically detects the structure of the trajectory and initializes the shooting parameters associated with boundary arcs and touch points.     

Feasible Direction Algorithm for Optimal Control Problems with State and Control Constraints: Implementation

In this paper, we describe the implementation aspects of an optimization algorithm for optimal control problems with control, state, and terminal constraints presented in our earlier paper. The important aspect of the implementation is that, in the direction-finding subproblems, it is necessary only to impose the state constraint at relatively few points in the time involved. This contributes significantly to the algorithmic efficiency. The algorithm is applied to solve several optimal control problems, including the problem of the abort landing of an aircraft in the presence of windshear.     

An Efficient Monte Carlo Method for Optimal Control Problems with Uncertainty

A general framework is proposed for what we call the sensitivity derivative Monte Carlo (SDMC) solution of optimal control problems with a stochastic parameter. This method employs the residual in the first-order Taylor series expansion of the cost functional in terms of the stochastic parameter rather than the cost functional itself. A rigorous estimate is derived for the variance of the residual, and it is verified by numerical experiments involving the generalized steady-state Burgers equation with a stochastic coefficient of viscosity. Specifically, the numerical results show that for a given number of samples, the present method yields an order of magnitude higher accuracy than a conventional Monte Carlo method. In other words, the proposed variance reduction method based on sensitivity derivatives is shown to accelerate convergence of the Monte Carlo method. As the sensitivity derivatives are computed only at the mean values of the relevant parameters, the related extra cost of the proposed method is a fraction of the total time of the Monte Carlo method.     

An Adaptive Dual Parametrization Algorithm for Quadratic Semi-infinite Programming Problems

The so called dual parametrization method for quadratic semi-infinite programming (SIP) problems is developed recently for quadratic SIP problems with a single infinite constraint. A dual parametrization algorithm is also proposed for numerical solution of such problems. In this paper, we consider quadratic SIP problems with positive definite objective and multiple linear infinite constraints. All the infinite constraints are supposed to be continuously dependent on their index variable on a compact set which is defined by a number equality and inequalities. We prove that in the multiple infinite constraint case, the minimu parametrization number, just as in the single infinite constraint case, is less or equal to the dimension of the SIP problem. Furthermore, we propose an adaptive dual parametrization algorithm with convergence result. Compared with the previous dual parametrization algorithm, the adaptive algorithm solves subproblems with much smaller number of constraints. The efficiency of the new algorithm is shown by solving a number of numerical examples.     

An Adaptive Greedy Algorithm for Solving Large RBF Collocation Problems

The solution of operator equations with radial basis functions by collocation in scattered points leads to large linear systems which often are nonsparse and ill-conditioned. But one can try to use only a subset of the data for the actual collocation, leaving the rest of the data points for error checking. This amounts to finding sparse approximate solutions of general linear systems arising from collocation. This contribution proposes an adaptive greedy method with proven (but slow) linear convergence to the full solution of the collocation equations. The collocation matrix need not be stored, and the progress of the method can be controlled by a variety of parameters. Some numerical examples are given.     

An Approximation Scheme for Uncertain Minimax Optimal Control Problems

In this work, we address an uncertain minimax optimal control problem with linear dynamics where the objective functional is the expected value of the supremum of the running cost over a time interval. By taking an independently drawn random sample, the expected value function is approximated by the corresponding sample average function. We study the epi-convergence of the approximated objective functionals as well as the convergence of their global minimizers. Then we define an Euler discretization in time of the sample average problem and prove that the value of the discrete time problem converges to the value of the sample average approximation. In addition, we show that there exists a sequence of discrete problems such that the accumulation points of their minimizers are optimal solutions of the original problem. Finally, we propose a convergent descent method to solve the discrete time problem, and show some preliminary numerical results for two simple examples.     

A Generalization of Mixed Problems with an Application to Multiobjective Optimal Control

This paper generalizes to multiobjective optimization the notion of mixed problems as Philippe Michel calls it for single-objective optimization. This notion is then applied to a multiobjective control problem under constraints in the discrete time framework to obtain strong Pontryagin maximum principles in the finite-horizon case. The infinite-horizon case is also treated with conditions ensuring that the multipliers associated to the objective functions are not all zero.     

Adaptive Algorithm for Constrained Least-Squares Problems

This paper is concerned with the implementation and testing of an algorithm for solving constrained least-squares problems. The algorithm is an adaptation to the least-squares case of sequential quadratic programming (SQP) trust-region methods for solving general constrained optimization problems. At each iteration, our local quadratic subproblem includes the use of the Gauss–Newton approximation but also encompasses a structured secant approximation along with tests of when to use this approximation. This method has been tested on a selection of standard problems. The results indicate that, for least-squares problems, the approach taken here is a viable alternative to standard general optimization methods such as the Byrd–Omojokun trust-region method and the Powell damped BFGS line search method.     

A Hybrid Differential Dynamic Programming Algorithm for Constrained Optimal Control Problems. Part 2: Application

In the first part of this paper series, a new solver, called HDDP, was presented for solving constrained, nonlinear optimal control problems. In the present paper, the algorithm is extended to include practical safeguards to enhance robustness, and four illustrative examples are used to evaluate the main algorithm and some variants. The experiments involve both academic and applied problems to show that HDDP is capable of solving a wide class of constrained, nonlinear optimization problems. First, the algorithm is verified to converge in a single iteration on a simple multi-phase quadratic problem with trivial dynamics. Successively, more complicated constrained optimal control problems are then solved demonstrating robust solutions to problems with as many as 7 states, 25 phases, 258 stages, 458 constraints, and 924 total control variables. The competitiveness of HDDP, with respect to general-purpose, state-of-the-art NLP solvers, is also demonstrated.     

Sensitivity Properties of Parametric Nonconvex Evolution Inclusions with Application to Optimal Control Problems

Set-Valued and Variational Analysis - The main concern of this paper is to investigate sensitivity properties of parametric evolution systems of first order involving a general class of nonconvex...     

Optimal Battery Aging: An Adaptive Weights Dynamic Programming Algorithm

We present an algorithm to handle the optimization over a long horizon of an electric microgrid including a battery energy storage system. While the battery is an important and costly component of the microgrid, its aging process is often not taken into account by the energy management system, mostly because of modeling and computing challenges. We address the computing aspect by a new approach combining dynamic programming, decomposition and relaxation techniques. We illustrate this adaptive weight’ method with numerical simulations for a toy microgrid model. Compared to a straightforward resolution by dynamic programming, our algorithm decreases the computing time by more than one order of magnitude, can be parallelized, and allows for online implementations. We believe that this approach can be used for other applications presenting fast and slow variables.     

Probabilistic Solutions to Optimal Control Problems

Let ( x 1 ( t ), x 2 ( t )) be a controlled two-dimensional diffusion process. The problem of minimizing, or maximizing, the time spent by ( x 1 ( t ), x 2 ( t )) in a given subset of 2 is solved, in two particular instances, by transforming the optimal control problems into purely probabilistic problems. In Section 2 , ( x 1 ( t ), x 2 ( t )) is a two-dimensional Wiener process and the optimal control is obtained by transforming a nonlinear dynamic programming equation into the Kolmogorov backward equation for a two-dimensional geometric Brownian motion. In Section 3 , the converse problem is solved. The problem of finding the maximal instantaneous reward that we can give for survival in the continuation region is also treated.     

Solutions to Constrained Optimal Control Problems with Linear Systems

This paper is devoted to present solutions to constrained finite-horizon optimal control problems with linear systems, and the cost functional of the problem is in a general form. According to the Pontryagin’s maximum principle, the extremal control of such problem is a function of the costate trajectory, but an implicit function. We here develop the canonical backward differential flows method and then give the extremal control explicitly with the costate trajectory by canonical backward differential flows. Moreover, there exists an optimal control if and only if there exists a unique extremal control. We give the proof of the existence of the optimal solution for this optimal control problem with Green functions.     

An Infinite-Dimensional Semismooth Newton Method for Elasto-Plastic Contact Problems

A Fenchel dualization scheme for the one-step time-discretized elasto-plastic contact problem with kinematic or isotropic hardening is considered. The associated path is induced by a combined Moreau-Yosida / Tichonov regularization of the dual problem. The sequence of solutions to the regularized problems is shown to converge strongly to the solution of the original problem. This property relies on the density of the intersection of certain convex sets. The corresponding conditions are worked out and customary regularization approaches are shown to be valid in this context. It is also argued that without higher regularity assumptions on the data the resulting problems possess Newton differentiable optimality systems in infinite dimensions [2]. Consequently, each regularized subsystem can be solved mesh-independently at a local superlinear rate of convergence [6]. Numerically the problems are solved using conforming finite elements. (© 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

An Algorithm for Optimal Tapered Matching,With Application to Disparities in Survival

In a tapered matched comparison, one group of individuals, called the focal group, is compared to two or more nonoverlapping matched comparison groups constructed from one population in such a way that successive comparison groups increasingly resemble the focal group. An optimally tapered matching solves two problems simultaneously: it optimally divides the single comparison population into nonoverlapping comparison groups and optimally pairs members of the focal group with members of each comparison group. We show how to use the optimal assignment algorithm in a new way to solve the optimally tapered matching problem, with implementation in R. This issue often arises in studies of groups defined by race, gender, or other categorizations such that equitable public policy might require an understanding of the mechanisms that produce disparate outcomes, where certain specific mechanisms would be judged illegitimate, necessitating reform. In particular, we use data from Medicare and the SEER Program of the National Cancer Institute as part of an ongoing study of black-white disparities in survival among women with endometrial cancer.