Abort landing in the presence of windshear as a minimax optimal control problem,part 2: Multiple shooting and homotopy

In Part 1 of the paper (Ref. 2), we have shown that the necessary conditions for the optimal control problem of the abort landing of a passenger aircraft in the presence of windshear result in a multipoint boundary-value problem. This boundary-value problem is especially well suited for numerical treatment by the multiple shooting method. Since this method is basically a Newton iteration, initial guesses of all variables are needed and assumptions about the switching structure have to be made. These are big obstacles, but both can be overcome by a so-called homotopy strategy where the problem is imbedded into a one-parameter family of subproblems in such a way that (at least) the first problem is simple to solve. The solution data to the first problem may serve as an initial guess for the next problem, thus resulting in a whole chain of problems. This process is to be continued until the objective problem is reached.Techniques are presented here on how to handle the various changes of the switching structure during the homotopy run. The windshear problem, of great interest for safety in aviation, also serves as an excellent benchmark problem: Nearly all features that can arise in optimal control appear when solving this problem. For example, the candidate for an optimal trajectory of the minimax optimal control problem shows subarcs with both bang-bang and singular control functions, boundary arcs and touch points of two state constraints, one being of first order and the other being of third order, etc. Therefore, the results of this paper may also serve as some sort of user's guide for the solution of complicated real-life optimal control problems by multiple shooting.The candidate found for an optimal trajectory is discussed and compared with an approximate solution already known (Refs. 3–4). Besides the known necessary conditions, additional sharp necessary conditions based on sign conditions of certain multipliers are also checked. This is not possible when using direct methods.An extended abstract of this paper was presented at the 8th IFAC Workshop on Control Applications of Nonlinear Programming and Optimization, Paris, France, 1989 (see Ref. 1).This paper is dedicated to Professor Hans J. Stetter on the occasion of his 60th birthday.     

Decentralized and dynamic group formation of reconfigurable agents

Adaptive group formation in dynamic environments performed by heterogeneous swarms of simple agents is an interesting research topic. In this paper we consider an unsupervised scenario where the individuals of the swarm have limited information about their environment as well as limited communication capabilities. The particular case of a multi-agent model with self-organized reconfigurable agents where the agents are confronted with a resource collection task, different movement, and group formation tactics are analyzed experimentally. It is shown that cooperation in groups is profitable for the group members and the optimal group size depends on environmental parameters. Moreover, a simple strategy based on the agents ability to measure their own workload results in an adaptive behavior that influences the size of the groups and increases the performance of the overall system.     

Clustering of search trajectory and its application to parameter tuning

This paper is concerned with automated classification of Combinatorial Optimization Problem instances for instance-specific parameter tuning purpose. We propose the CluPaTra Framework, a generic approach to CLUster instances based on similar PAtterns according to search TRAjectories and apply it on parameter tuning. The key idea is to use the search trajectory as a generic feature for clustering problem instances. The advantage of using search trajectory is that it can be obtained from any local-search based algorithm with small additional computation time. We explore and compare two different search trajectory representations, two sequence alignment techniques (to calculate similarities) as well as two well-known clustering methods. We report experiment results on two classical problems: Travelling Salesman Problem and Quadratic Assignment Problem and industrial case study.     

A hybrid multiagent approach for global trajectory optimization

In this paper we consider a global optimization method for space trajectory design problems. The method, which actually aims at finding not only the global minimizer but a whole set of low-lying local minimizers (corresponding to a set of different design options), is based on a domain decomposition technique where each subdomain is evaluated through a procedure based on the evolution of a population of agents. The method is applied to two space trajectory design problems and compared with existing deterministic and stochastic global optimization methods.     

Continuation method for nonlinear complementarity problems via normal maps

In a recent paper by Chen and Mangasarian (C. Chen, O.L. Mangasarian, A class of smoothing functions for nonlinear and mixed complementarity problems, Computational Optimization and Applications 2 (1996), 97–138) a class of parametric smoothing functions has been proposed to approximate the plus function present in many optimization and complementarity related problems. This paper uses these smoothing functions to approximate the normal map formulation of nonlinear complementarity problems (NCP). Properties of the smoothing function are investigated based on the density functions that defines the smooth approximations. A continuation method is then proposed to solve the NCPs arising from the approximations. Sufficient conditions are provided to guarantee the boundedness of the solution trajectory. Furthermore, the structure of the subproblems arising in the proposed continuation method is analyzed for different choices of smoothing functions. Computational results of the continuation method are reported.     

A framework for finding robust optimal solutions over time

Dynamic optimization problems (DOPs) are those whose specifications change over time, resulting in changing optima. Most research on DOPs has so far concentrated on tracking the moving optima (TMO) as closely as possible. In practice, however, it will be very costly, if not impossible to keep changing the design when the environment changes. To address DOPs more practically, we recently introduced a conceptually new problem formulation, which is referred to as robust optimization over time (ROOT). Based on ROOT, an optimization algorithm aims to find an acceptable (optimal or sub-optimal) solution that changes slowly over time, rather than the moving global optimum. In this paper, we propose a generic framework for solving DOPs using the ROOT concept, which searches for optimal solutions that are robust over time by means of local fitness approximation and prediction. Empirical investigations comparing a few representative TMO approaches with an instantiation of the proposed framework are conducted on a number of test problems to demonstrate the advantage of the proposed framework in the ROOT context.     

Empirical analysis of evolutionary algorithms with immigrants schemes for dynamic optimization

In recent years, there has been a growing interest in studying evolutionary algorithms (EAs) for dynamic optimization problems (DOPs). Among approaches developed for EAs to deal with DOPs, immigrants schemes have been proven to be beneficial. Immigrants schemes for EAs on DOPs aim at maintaining the diversity of the population throughout the run via introducing new individuals into the current population. In this paper, we carefully examine the mechanism of generating immigrants, which is the most important issue among immigrants schemes for EAs in dynamic environments. We divide existing immigrants schemes into two types, namely the direct immigrants scheme and the indirect immigrants scheme, according to the way in which immigrants are generated. Then experiments are conducted to understand the difference in the behaviors of different types of immigrants schemes and to compare their performance in dynamic environments. Furthermore, a new immigrants scheme is proposed to combine the merits of two types of immigrants schemes. The experimental results show that the interactions between the two types of schemes reveal positive effect in improving the performance of EAs in dynamic environments.     

Efficient and Adaptive Lagrange-Multiplier Methods for Nonlinear Continuous Global Optimization

Lagrangian methods are popular in solving continuous constrained optimization problems. In this paper, we address three important issues in applying Lagrangian methods to solve optimization problems with inequality constraints.First, we study methods to transform inequality constraints into equality constraints. An existing method, called the slack-variable method, adds a slack variable to each inequality constraint in order to transform it into an equality constraint. Its disadvantage is that when the search trajectory is inside a feasible region, some satisfied constraints may still pose some effect on the Lagrangian function, leading to possible oscillations and divergence when a local minimum lies on the boundary of the feasible region. To overcome this problem, we propose the MaxQ method that carries no effect on satisfied constraints. Hence, minimizing the Lagrangian function in a feasible region always leads to a local minimum of the objective function. We also study some strategies to speed up its convergence.Second, we study methods to improve the convergence speed of Lagrangian methods without affecting the solution quality. This is done by an adaptive-control strategy that dynamically adjusts the relative weights between the objective and the Lagrangian part, leading to better balance between the two and faster convergence.Third, we study a trace-based method to pull the search trajectory from one saddle point to another in a continuous fashion without restarts. This overcomes one of the problems in existing Lagrangian methods that converges only to one saddle point and requires random restarts to look for new saddle points, often missing good saddle points in the vicinity of saddle points already found.Finally, we describe a prototype Novel (Nonlinear Optimization via External Lead) that implements our proposed strategies and present improved solutions in solving a collection of benchmarks.     

Inverse Stackelberg Public Goods Game with Multiple Hierarchies Under Global and Local Information Structures

This paper studies the inverse Stackelberg game with multiple hierarchies under global and local information structures, where the players have discrete strategy spaces. For the classic public goods game, we solve the pure-strategy inverse Stackelberg equilibria under three typical hierarchical structures. The results reveal some counterintuitive characteristics within the systems with hierarchies, such as that the cooperation does not increase with the return rate at the equilibria. Furthermore, by defining a local information structure, we give an estimate of the fewest hierarchies required for full cooperation, which can be a constant multiple of the logarithm or square root of the population size or of the population size itself, according to different information structures and return rates. This paper proposes a novel mechanism to play the game and promote cooperation. Both the formulation and analysis method are different from existing works, and the results can find their ample implications in practice, which might help decision making in hierarchical systems.     

Generalized Trajectory Methods for Finding Multiple Extrema and Roots of Functions

Two generalized trajectory methods are combined to provide a novel and powerful numerical procedure for systematically finding multiple local extrema of a multivariable objective function. This procedure can form part of a strategy for global optimization in which the greatest local maximum and least local minimum in the interior of a specified region are compared to the largest and smallest values of the objective function on the boundary of the region. The first trajectory method, a homotopy scheme, provides a globally convergent algorithm to find a stationary point of the objective function. The second trajectory method, a relaxation scheme, starts at one stationary point and systematically connects other stationary points in the specified region by a network of trjectories. It is noted that both generalized trajectory methods actually solve the stationarity conditions, and so they can also be used to find multiple roots of a set of nonlinear equations.     

Parallel radial basis function methods for the global optimization of expensive functions

We introduce a master–worker framework for parallel global optimization of computationally expensive functions using response surface models. In particular, we parallelize two radial basis function (RBF) methods for global optimization, namely, the RBF method by Gutmann [Gutmann, H.M., 2001a. A radial basis function method for global optimization. Journal of Global Optimization 19(3), 201–227] (Gutmann-RBF) and the RBF method by Regis and Shoemaker [Regis, R.G., Shoemaker, C.A., 2005. Constrained global optimization of expensive black box functions using radial basis functions, Journal of Global Optimization 31, 153–171] (CORS-RBF). We modify these algorithms so that they can generate multiple points for simultaneous evaluation in parallel. We compare the performance of the two parallel RBF methods with a parallel multistart derivative-based algorithm, a parallel multistart derivative-free trust-region algorithm, and a parallel evolutionary algorithm on eleven test problems and on a 6-dimensional groundwater bioremediation application. The results indicate that the two parallel RBF algorithms are generally better than the other three alternatives on most of the test problems. Moreover, the two parallel RBF algorithms have comparable performances on the test problems considered. Finally, we report good speedups for both parallel RBF algorithms when using a small number of processors.     

Abort landing in the presence of windshear as a minimax optimal control problem,part 1: Necessary conditions

The landing of a passenger aircraft in the presence of windshear is a threat to aviation safety. The present paper is concerned with the abort landing of an aircraft in such a serious situation. Mathematically, the flight maneuver can be described by a minimax optimal control problem. By transforming this minimax problem into an optimal control problem of standard form, a state constraint has to be taken into account which is of order three. Moreover, two additional constraints, a first-order state constraint and a control variable constraint, are imposed upon the model. Since the only control variable appears linearly, the Hamiltonian is not regular. Thus, well-known existence theorems about the occurrence of boundary arcs and boundary points cannot be applied. Numerically, this optimal control problem is solved by means of the multiple shooting method in connection with an appropriate homotopy strategy. The solution obtained here satisfies all the sharp necessary conditions including those depending on the sign of certain multipliers. The trajectory consists of bang-bang and singular subarcs, as well as boundary subarcs induced by the two state constraints. The occurrence of boundary arcs is known to be impossible for regular Hamiltonians and odd-ordered state constraints if the order exceeds two. Additionally, a boundary point also occurs where the third-order state constraint is active. Such a situation is known to be the only possibility for odd-ordered state constraints to be active if the order exceeds two and if the Hamiltonian is regular. Because of the complexity of the optimal control, this single problem combines many of the features that make this kind of optimal control problems extremely hard to solve. Moreover, the problem contains nonsmooth data arising from the approximations of the aerodynamic forces and the distribution of the wind velocity components. Therefore, the paper can serve as some sort of user's guide to solve inequality constrained real-life optimal control problems by multiple shooting.An extended abstract of this paper was presented at the 8th IFAC Workshop on Control Applications of Nonlinear Programming and Optimization, Paris, France, 1989 (see Ref. 1).This paper is dedicated to Professor Hans J. Stetter on the occasion of his 60th birthday.     

双人合作的在线雪橇租赁问题

以往的文献只研究了单人雪橇租赁问题,本文将雪橇租赁问题扩展到了双人合作情形.研究了两个在线决策者的合作博弈模型,给出了TBS策略和BCS策略,并求出了双方收益分配的纳什均衡解.结论显示,TBS策略具有最小竞争比,但基于该策略的合作却不稳定,需要契约维持;BCS策略不具有最小竞争比,却是占优策略,基于该策略的合作是稳定的。因此存在合作可能的情况下,选择BCS策略的合作总比非合作要好。文章第4节详细的比较了TBS策略和BCS策略。   此外,文章还得到了一个有意思的发现,随着参与人的增加,竞争比是有可能不上升的.这一发现与经典的在线问题(如k-server问题)的结论不一样,在k-server问题中,随着参与者(服务器)的增加,竞争比会呈线性提高》。     

Trajectory Modeling of Robot Manipulators in the Presence of Obstacles

This paper presents two different strategies for the problem of the optimal trajectory planning of robot manipulators in the presence of fixed obstacles. The first strategy is related to the situation where the trajectory must pass through a given number of points. The second strategy corresponds to the case where only the initial and final points are given. The optimal traveling time and the minimum mechanical energy of the actuators are considered together to build a multiobjective function. The trajectories are defined using spline functions and are obtained through offline computation for online operation. Sequential unconstrained minimization techniques (SUMT) have been used for the optimization. The obstacles are considered as three-dimensional objects sharing the same workspace performed by the robot. The obstacle avoidance is expressed in terms of the distances between potentially colliding parts. Simulation results are presented and show the efficiency of the general methodology used in this paper.     

Comparison of Fast Iterative Methods for Symmetric Systems

The multigrid method is compared to ICCG/MICCG methods for solvingsymmetric systems of linear equations arising from approximationsto differential equations with jump discontinuities in the coefficients.An optimal multigrid algorithm for these types of problems isdeveloped. It includes pattern relaxation and acceleration.Optimization of ICCG/MICCG algorithms is investigated. Thisincludes the effect of adding extra (up to ten) bands to theapproximate factorization and of different grid ordering schemes.Numerical results are presented comparing the scalar work ofthe algorithms. For large problems the multigrid algorithm issuperior. The optimal multigrid scheme can be highly vectorized.     

Topology optimization methods for guided flow – comparison of optimality criteria vs. phase-field method

Optimization of guided flow problems is an important task for industrial applications especially those with high Reynolds numbers. There exist several optimization methods to increase the energy efficiency of these problems. Different optimization methods are shown bei Klimetzek [1], Hinterberger [2] and Pingen [3]. In recent years the phase-field method has been shown to be an applicable method for different kinds of topology optimization [4, 5]. We present results of topology optimization methods with optimality criterion and by using a phase-field model in the area of guided fluid flow problems. The two methods aim on the same main target reducing the pressure drop between the inlet and outlet of the flow domain. The first method is based on local optimality criterion, preventing the backflow in the flow domain [1, 6, 7]. The second method is based on a phase field model, which describes a minimization problem and uses a specially constructed driving force to minimize the total energy of the system [4, 5]. We investigate the capabilities and limits of both methods and present examples of different resulting geometries. The initial configurations are prepared in a way that the same optimization problem is solved with both methods. We discuss these results regarding the shape of the improved flow geometry. (© 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Constrained Global Optimization of Expensive Black Box Functions Using Radial Basis Functions

We present a new strategy for the constrained global optimization of expensive black box functions using response surface models. A response surface model is simply a multivariate approximation of a continuous black box function which is used as a surrogate model for optimization in situations where function evaluations are computationally expensive. Prior global optimization methods that utilize response surface models were limited to box-constrained problems, but the new method can easily incorporate general nonlinear constraints. In the proposed method, which we refer to as the Constrained Optimization using Response Surfaces (CORS) Method, the next point for costly function evaluation is chosen to be the one that minimizes the current response surface model subject to the given constraints and to additional constraints that the point be of some distance from previously evaluated points. The distance requirement is allowed to cycle, starting from a high value (global search) and ending with a low value (local search). The purpose of the constraint is to drive the method towards unexplored regions of the domain and to prevent the premature convergence of the method to some point which may not even be a local minimizer of the black box function. The new method can be shown to converge to the global minimizer of any continuous function on a compact set regardless of the response surface model that is used. Finally, we considered two particular implementations of the CORS method which utilize a radial basis function model (CORS-RBF) and applied it on the box-constrained Dixon–Szegö test functions and on a simple nonlinearly constrained test function. The results indicate that the CORS-RBF algorithms are competitive with existing global optimization algorithms for costly functions on the box-constrained test problems. The results also show that the CORS-RBF algorithms are better than other algorithms for constrained global optimization on the nonlinearly constrained test problem.     

Molp with an interactive assessment of a piecewise linear utility function

The paper presents a methodology for Multi-Objective Linear Programming (MOLP) problems. It relies on three steps: (1) Generation of a subset of feasible efficient solutions (from 10 to 50) as representative as possible of the efficient set. (2) Assessment of an additive utility function using an interactive method (Prefcalc). (3) Optimization of the additive utility function on the original set of feasible alternatives. Following this methodology enables the user to find compromise solutions which can be different from the vertices. It is particularly adapted for large scale linear programs where traditional multiobjective methods would be too costly to use, since the interactive phase is limited to step 2, using Prefcalc on a micro-computer. A micro-computer version of the method (Prefchat) is available.     

Generalized controls: A necessary condition for optimality in a Banach space

A necessary condition is established for optimality in the case of problems where the constraints are simultaneously functions of the trajectory and the control (Problem 3.1, Theorem 5.7); this condition holds for generalized controls with values in a Hausdorff space and for a state space which is a Banach space. To demonstrate the result a new technique is used, based on the differentiability of the trajectory (Theorem 2.6) and the introduction of the notion of pseudosolution (Definition 2.8). These results are then applied to calculus of variations problems in a Banach space (Theorem 6.3).     

Cell-to-cell mapping method for time-optimal trajectory planning of multiple robot arm systems

This paper presents the results obtained by applying the cell-to-cell mapping method to solve the problem of the time-optimal trajectory planning for coordinated multiple robotic arms handling a common object along a specified geometric path. Based on the structure of the time-optimal trajectory control law, the continuous dynamic model of multiple arms is first approximated by a discrete and finite cell-to-cell mapping on a two-dimensional cell space over a phase plane. The optimal trajectory and the corresponding control are then determined by using the cell-to-cell mapping and a simple search algorithm. To further improve the computational efficiency and to allow for parallel computation, a hierarchical search algorithm consisting of a multiple-variable optimization on the top level and a number of cell-to-cell searches on the bottom level is proposed and implemented in the paper. Besides its simplicity, another distinguishing feature of the cell-to-cell mapping methods is the generation of all optimal trajectories for a given final state and all possible initial states through a single searching process. For most of the existing trajectory planning methods, the planning process can be started only when both the initial and final states have been specified. The cell-to-cell method can be generalized to any optimal trajectory planning problem for a multiple robotic arms system.