A Direct Integral Pseudospectral Method for Solving a Class of Infinite-Horizon Optimal Control Problems Using Gegenbauer Polynomials and Certain Parametric Maps

We present a novel direct integral pseudospectral (PS) method (a direct IPS method) for solving a class of continuous-time infinite-horizon optimal control problems (IHOCs). The method transforms the IHOCs into finite-horizon optimal control problems (FHOCs) in their integral forms by means of certain parametric mappings, which are then approximated by finite-dimensional nonlinear programming problems (NLPs) through rational collocations based on Gegenbauer polynomials and Gegenbauer-Gauss-Radau (GGR) points. The paper also analyzes the interplay between the parametric maps, barycentric rational collocations based on Gegenbauer polynomials and GGR points, and the convergence properties of the collocated solutions for IHOCs. Some novel formulas for the construction of the rational interpolation weights and the GGR-based integration and differentiation matrices in barycentric-trigonometric forms are derived. A rigorous study on the error and convergence of the proposed method is presented. A stability analysis based on the Lebesgue constant for GGR-based rational interpolation is investigated. Two easy-to-implement pseudocodes of computational algorithms for computing the barycentric-trigonometric rational weights are described. Two illustrative test examples are presented to support the theoretical results. We show that the proposed collocation method leveraged with a fast and accurate NLP solver converges exponentially to near-optimal approximations for a coarse collocation mesh grid size. The paper also shows that typical direct spectral/PS- and IPS-methods based on classical Jacobi polynomials and certain parametric maps usually diverge as the number of collocation points grow large, if the computations are carried out using floating-point arithmetic and the discretizations use a single mesh grid whether they are of Gauss/Gauss-Radau (GR) type or equally-spaced.Comment: 33 pages, 19 figure

Effective Simulation and Optimization of a Laser Peening Process

Laser peening (LP) is a surface enhancement technique that has been applied to improve fatigue and corrosion properties of metals. The ability to use a high energy laser pulse to generate shock waves, inducing a compressive residual stress field in metallic materials, has applications in multiple fields such as turbomachinery, airframe structures, and medical appliances. In the past, researchers have investigated the effects of LP parameters experimentally and performed a limited number of simulations on simple geometries. However, monitoring the dynamic, intricate relationships of peened materials experimentally is time consuming, expensive, and challenging. With increasing applications of LP on complex geometries, these limited experimental and simulation capabilities are not sufficient for an effective LP process design. Due to high speed, dynamic process parameters, it is difficult to achieve a consistent residual stress field in each treatment and constrain detrimental effects. With increased computer speed as well as increased sophistication in non-linear finite element analysis software, it is now possible to develop simulations that can consider several LP parameters. In this research, a finite element simulation capability of the LP process is developed. These simulations are validated with the available experimental results. Based on the validated model, simplifications to complex models are developed. These models include quarter symmetric 3D model, a cylindrical coupon, a parametric plate, and a bending coupon model. The developed models can perform simulations incorporating the LP process parameters, such as pressure pulse properties, spot properties, number of shots, locations, sequences, overlapping configurations, and complex geometries. These models are employed in parametric investigations and residual stress profile optimization at single and multiple locations. In parametric investigations, quarter symmetric 3D model is used to investigate temporal variations of pressure pulse, pressure magnitude, and shot shape and size. The LP optimization problem is divided into two parts: single and multiple locations peening optimization. The single-location peening optimization problems have mixed design variables and multiple optimal solutions. In the optimization literature, many researchers have solved problems involving mixed variables or multiple optima, but it is difficult to find multiple solutions for mixed-variable problems. A mixed-variable Niche Particle Swarm Optimization (MNPSO) is proposed that incorporates a mixed-variable handling technique and a niching technique to solve the problem. Designing an optimal residual stress profile for multiple-location peening is a challenging task due to the computational cost and the nonlinear behavior of LP. A Progressive Multifidelity Optimization Strategy (PMOS) is proposed to solve the problem. The three-stage PMOS, combines low- and high- fidelity simulations and respective surrogate models and a mixed-variable handling strategy. This strategy employs comparatively low computational-intensity models in the first two stages to locate the design space that may contain the optimal solution. The third stage employs high fidelity simulation and surrogate models to determine the optimal solution. The overall objective of this research is to employ finite element simulations and effective optimization techniques to achieve optimal residual stress fields

Vademecum-based GFEM (V-GFEM): optimal enrichment for transient problems

This is the accepted version of the following article: [Canales, D., Leygue, A., Chinesta, F., González, D., Cueto, E., Feulvarch, E., Bergheau, J. -M., and Huerta, A. (2016) Vademecum-based GFEM (V-GFEM): optimal enrichment for transient problems. Int. J. Numer. Meth. Engng, 108: 971–989. doi: 10.1002/nme.5240.], which has been published in final form at http://onlinelibrary.wiley.com/doi/10.1002/nme.5240/fullThis paper proposes a generalized finite element method based on the use of parametric solutions as enrichment functions. These parametric solutions are precomputed off-line and stored in memory in the form of a computational vademecum so that they can be used on-line with negligible cost. This renders a more efficient computational method than traditional finite element methods at performing simulations of processes. One key issue of the proposed method is the efficient computation of the parametric enrichments. These are computed and efficiently stored in memory by employing proper generalized decompositions. Although the presented method can be broadly applied, it is particularly well suited in manufacturing processes involving localized physics that depend on many parameters, such as welding. After introducing the vademecum-generalized finite element method formulation, we present some numerical examples related to the simulation of thermal models encountered in welding processes.Peer ReviewedPostprint (author's final draft

Transformation Method for Solving Hamilton-Jacobi-Bellman Equation for Constrained Dynamic Stochastic Optimal Allocation Problem

In this paper we propose and analyze a method based on the Riccati transformation for solving the evolutionary Hamilton-Jacobi-Bellman equation arising from the stochastic dynamic optimal allocation problem. We show how the fully nonlinear Hamilton-Jacobi-Bellman equation can be transformed into a quasi-linear parabolic equation whose diffusion function is obtained as the value function of certain parametric convex optimization problem. Although the diffusion function need not be sufficiently smooth, we are able to prove existence, uniqueness and derive useful bounds of classical H\"older smooth solutions. We furthermore construct a fully implicit iterative numerical scheme based on finite volume approximation of the governing equation. A numerical solution is compared to a semi-explicit traveling wave solution by means of the convergence ratio of the method. We compute optimal strategies for a portfolio investment problem motivated by the German DAX 30 Index as an example of application of the method

Construction of the Optimal Control Strategy for an Electric-Powered Train

Předložená disertační práce se zabývá popisem charakteru optimální strategie řízení pro elektrický vlak a výpočtem přepínacích okamžiků mezi jednotlivými optimálními jízdními režimy pro standardní typy odporové funkce. S využitím Pontrjaginova principu a souvisejících nástrojů teorie optimálního řízení odvodíme optimální strategii řízení a rovnice pro výpočet přepínacích okamžiků včetně odpovídajících rychlostních profilů. Kromě základního tvaru úlohy o energeticky optimální jízdě vlaku budeme uvažovat i její modifikace zahrnující globální rychlostní omezení, sklon trati i časově-energeticky optimální řízení vlaku. Navíc uvedeme i analýzu řešení s využitím teorie nelineární parametrické optimalizace. Důraz je kladen na exaktní tvar řešení s minimálním využitím numerických metod.This thesis deals with the description of the nature of optimal driving strategy for an electric-powered train as well as the calculation of switching times of optimal driving regimes for standard types of resistance function. We apply the Pontryagin principle and related tools of optimal control theory to develop the optimal driving strategy and to derive equations for computation of switching times and the corresponding speed profiles. Besides the basic form of the energy efficient train control problem we consider also its modifications including the global speed constraint, track gradient as well as time-energy efficient train control. Moreover, we analyse also the solution with use of the theory of nonlinear parametric programming. The emphasize is put on exact forms of solutions with a minimal use of numerical methods.

Data-Driven Estimation in Equilibrium Using Inverse Optimization

Equilibrium modeling is common in a variety of fields such as game theory and transportation science. The inputs for these models, however, are often difficult to estimate, while their outputs, i.e., the equilibria they are meant to describe, are often directly observable. By combining ideas from inverse optimization with the theory of variational inequalities, we develop an efficient, data-driven technique for estimating the parameters of these models from observed equilibria. We use this technique to estimate the utility functions of players in a game from their observed actions and to estimate the congestion function on a road network from traffic count data. A distinguishing feature of our approach is that it supports both parametric and \emph{nonparametric} estimation by leveraging ideas from statistical learning (kernel methods and regularization operators). In computational experiments involving Nash and Wardrop equilibria in a nonparametric setting, we find that a) we effectively estimate the unknown demand or congestion function, respectively, and b) our proposed regularization technique substantially improves the out-of-sample performance of our estimators.Comment: 36 pages, 5 figures Additional theorems for generalization guarantees and statistical analysis adde

Suboptimal Solution Path Algorithm for Support Vector Machine

We consider a suboptimal solution path algorithm for the Support Vector Machine. The solution path algorithm is an effective tool for solving a sequence of a parametrized optimization problems in machine learning. The path of the solutions provided by this algorithm are very accurate and they satisfy the optimality conditions more strictly than other SVM optimization algorithms. In many machine learning application, however, this strict optimality is often unnecessary, and it adversely affects the computational efficiency. Our algorithm can generate the path of suboptimal solutions within an arbitrary user-specified tolerance level. It allows us to control the trade-off between the accuracy of the solution and the computational cost. Moreover, We also show that our suboptimal solutions can be interpreted as the solution of a \emph{perturbed optimization problem} from the original one. We provide some theoretical analyses of our algorithm based on this novel interpretation. The experimental results also demonstrate the effectiveness of our algorithm.Comment: A shorter version of this paper is submitted to ICML 201

Branch-and-lift algorithm for deterministic global optimization in nonlinear optimal control

This paper presents a branch-and-lift algorithm for solving optimal control problems with smooth nonlinear dynamics and potentially nonconvex objective and constraint functionals to guaranteed global optimality. This algorithm features a direct sequential method and builds upon a generic, spatial branch-and-bound algorithm. A new operation, called lifting, is introduced, which refines the control parameterization via a Gram-Schmidt orthogonalization process, while simultaneously eliminating control subregions that are either infeasible or that provably cannot contain any global optima. Conditions are given under which the image of the control parameterization error in the state space contracts exponentially as the parameterization order is increased, thereby making the lifting operation efficient. A computational technique based on ellipsoidal calculus is also developed that satisfies these conditions. The practical applicability of branch-and-lift is illustrated in a numerical example. © 2013 Springer Science+Business Media New York