An efficient method for multiobjective optimal control and optimal control subject to integral constraints

We introduce a new and efficient numerical method for multicriterion optimal control and single criterion optimal control under integral constraints. The approach is based on extending the state space to include information on a "budget" remaining to satisfy each constraint; the augmented Hamilton-Jacobi-Bellman PDE is then solved numerically. The efficiency of our approach hinges on the causality in that PDE, i.e., the monotonicity of characteristic curves in one of the newly added dimensions. A semi-Lagrangian "marching" method is used to approximate the discontinuous viscosity solution efficiently. We compare this to a recently introduced "weighted sum" based algorithm for the same problem. We illustrate our method using examples from flight path planning and robotic navigation in the presence of friendly and adversarial observers.Comment: The final version accepted by J. Comp. Math. : 41 pages, 14 figures. Since the previous version: typos fixed, formatting improved, one mistake in bibliography correcte

Efficient and robust numerical treatment of a gradient-enhanced damage model at large deformations

The modeling of damage processes in materials constitutes an ill-posed mathematical problem which manifests in mesh-dependent finite element results. The loss of ellipticity of the discrete system of equations is counteracted by regularization schemes of which the gradient enhancement of the strain energy density is often used. In this contribution, we present an extension of the efficient numerical treatment, which has been proposed by Junker et al. in 2019, to materials that are subjected to large deformations. Along with the model derivation, we present a technique for element erosion in the case of severely damaged materials. Efficiency and robustness of our approach is demonstrated by two numerical examples including snapback and springback phenomena. © 2021 The Authors. International Journal for Numerical Methods in Engineering published by John Wiley & Sons Ltd

Adaptive Sampling for Geometric Approximation

Geometric approximation of multi-dimensional data sets is an essential algorithmic component for applications in machine learning, computer graphics, and scientific computing. This dissertation promotes an algorithmic sampling methodology for a number of fundamental approximation problems in computational geometry. For each problem, the proposed sampling technique is carefully adapted to the geometry of the input data and the functions to be approximated. In particular, we study proximity queries in spaces of constant dimension and mesh generation in 3D. We start with polytope membership queries, where query points are tested for inclusion in a convex polytope. Trading-off accuracy for efficiency, we tolerate one-sided errors for points within an epsilon-expansion of the polytope. We propose a sampling strategy for the placement of covering ellipsoids sensitive to the local shape of the polytope. The key insight is to realize the samples as Delone sets in the intrinsic Hilbert metric. Using this intrinsic formulation, we considerably simplify state-of-the-art techniques yielding an intuitive and optimal data structure. Next, we study nearest-neighbor queries which retrieve the most similar data point to a given query point. To accommodate more general measures of similarity, we consider non-Euclidean distances including convex distance functions and Bregman divergences. Again, we tolerate multiplicative errors retrieving any point no farther than (1+epsilon) times the distance to the nearest neighbor. We propose a sampling strategy sensitive to the local distribution of points and the gradient of the distance functions. Combined with a careful regularization of the distance minimizers, we obtain a generalized data structure that essentially matches state-of-the-art results specific to the Euclidean distance. Finally, we investigate the generation of Voronoi meshes, where a given domain is decomposed into Voronoi cells as desired for a number of important solvers in computational fluid dynamics. The challenge is to arrange the cells near the boundary to yield an accurate surface approximation without sacrificing quality. We propose a sampling algorithm for the placement of seeds to induce a boundary-conforming Voronoi mesh of the correct topology, with a careful treatment of sharp and non-manifold features. The proposed algorithm achieves significant quality improvements over state-of-the-art polyhedral meshing based on clipped Voronoi cells

Computational Methods for Nonlinear Optimization Problems: Theory and Applications

This dissertation is motivated by the lack of efficient global optimization techniques for polynomial optimization problems. The objective is twofold. First, a new mathematical foundation for obtaining a global or near-global solution will be developed. Second, several case studies will be conducted on a variety of real-world problems. Global optimization, convex relaxation and distributed computation are at the heart of this PhD dissertation. Some of the specific problems to be addressed in this thesis on both the theory and the application of nonlinear optimization are explained below: Graph theoretic algorithms for low-rank optimization problems: There is a rapidly growing interest in the recovery of an unknown low-rank matrix from limited information and measurements. This problem occurs in many areas of engineering and applied science such as machine learning, control, and computer vision. We develop a graph-theoretic technique in Part I that is able to generate a low-rank solution for a sparse Linear Matrix Inequality (LMI), which is directly applicable to a large set of problems such as low-rank matrix completion with many unknown entries. Our approach finds a solution with a guarantee on its rank, using the recent advances in graph theory. Resource allocation for energy systems: The flows in an electrical grid are described by nonlinear AC power flow equations. Due to the nonlinear interrelation among physical parameters of the network, the feasibility region represented by power flow equations may be nonconvex and disconnected. Since 1962, the nonlinearity of the network constraints has been studied, and various heuristic and local-search algorithms have been proposed in order to perform optimization over an electrical grid [Baldick, 2006; Pandya and Joshi, 2008]. Part II is concerned with finding convex formulations of the power flow equations using semidefinite programming (SDP). The potential of SDP relaxation for problems in power systems has been manifested in [Lavaei and Low, 2012], with further studies conducted in [Lavaei, 2011; Sojoudi and Lavaei, 2012]. A variety of graph-theoretic and algebraic methods are developed in Part II in order to facilitate performing fundamental, yet challenging tasks such as optimal power flow (OPF) problem, security-constrained OPF and the classical power flow problem. Synthesis of distributed control systems: Real-world systems mostly consist of many interconnected subsystems, and designing an optimal controller for them pose several challenges to the field of control theory. The area of distributed control is created to address the challenges arising in the control of these systems. The objective is to design a constrained controller whose structure is specified by a set of permissible interactions between the local controllers with the aim of reducing the computation or communication complexity of the overall controller. It has been long known that the design of an optimal distributed (decentralized) controller is a daunting task because it amounts to an NP-hard optimization problem in general [Witsenhausen, 1968; Tsitsiklis and Athans, 1984]. Part III is devoted to study the potential of the SDP relaxation for the optimal distributed control (ODC) problem Our approach rests on formulating each of different variations of the ODC problem as rank-constrained optimization problems from which SDP relaxations can be derived. As the first contribution, we show that the ODC problem admits a sparse SDP relaxation with solutions of rank at most 3. Since a rank-1 SDP matrix can be mapped back into a globally-optimal controller, the low-rank SDP solution may be deployed to retrieve a near-global controller. Parallel computation for sparse semidefinite programs: While small- to medium-sized semidefinite programs are efficiently solvable by second-order-based interior point methods in polynomial time up to any arbitrary precision [Vandenberghe and Boyd, 1996a], these methods are impractical for solving large-scale SDPs due to computation time and memory issues. In Part IV of this dissertation, a parallel algorithm for solving an arbitrary SDP is introduced based on the alternating direction method of multipliers. The proposed algorithm has a guaranteed convergence under very mild assumptions. Each iteration of this algorithm has a simple closed-form solution, and consists of scalar multiplication and eigenvalue decomposition over matrices whose sizes are not greater than the treewdith of the sparsity graph of the SDP problem. The cheap iterations of the proposed algorithm enable solving real-world large-scale conic optimization problems

An Entropic Optimal Transport Numerical Approach to the Reflector Problem

The point source far field reflector design problem is one of the main classic optimal transport problems with a non-euclidean displacement cost [Wang, 2004] [Glimm and Oliker, 2003]. This work describes the use of Entropic Optimal Transport and the associated Sinkhorn algorithm [Cuturi, 2013] to solve it numerically. As the reflector modelling is based on the Kantorovich potentials , several questions arise. First, on the convergence of the discrete entropic approximation and here we follow the recent work of [Berman, 2017] and in particular the imposed discretization requirements therein. Secondly, the correction of the Entropic bias induced by the Entropic OT, as discussed in particular in [Ramdas et al., 2017] [Genevay et al., 2018] [Feydy et al., 2018], is another important tool to achieve reasonable results. The paper reviews the necessary mathematical and numerical tools needed to produce and discuss the obtained numerical results. We find that Sinkhorn algorithm may be adapted, at least in simple academic cases, to the resolution of the far field reflector problem. Sinkhorn canonical extension to continuous potentials is needed to generate continuous reflector approximations. The use of Sinkhorn divergences [Feydy et al., 2018] is useful to mitigate the entropic bias

Relaxations and Approximations for Mixed-Integer Optimal Control

This thesis treats different aspects of the class of Mixed-Integer Optimal Control Problems (MIOCPs). These are optimization problems that combine the difficulties of underlying dynamic processes with combinatorial decisions. Typically, these combinatorial decisions are realized as switching decisions between the system’s different operations modes. During the last decades, direct methods emerged as the state-of-the-art solvers for MIOCPs. The formulation of a valid, tight and dependable integral relaxation, i.e., the formulation of a model for fractional values, plays an important role for these direct solution methods. We give detailed insight into several relaxation approaches for MIOCPs and compare them with regard to their respective structures. In particular, these are the typical solution’s structures and properties as convexity, problem size and numerical behavior. From these structural properties, we deduce some required specifications of a solver. Additionally, the modeling and subsequent limitation of the switching process directly tackle the class-specific typical issue of chattering solutions. One of the relaxation methods for MIOCPs is the outer convexification, where the binary variables only enter affinely. For the approximation of this relaxation’s solution, we took up on the control approximation problem in integral sense derived by Sager as part of a decomposition approach for MIOCPs with affine binary controls. This problem describes the optimal approximation of fractional controls with binary controls such that the corresponding dynamic process is changed as little as possible. For the multi-dimensional problem, we developed a new heuristic, which for the first time gives a bound that only depends on the control grid and not anymore on the number of the system’s controls. For the generalization of the control approximation problem with additional constraints, we derived a tailored branch-and-bound algorithm, which is based on the properties of the Lagrangian relaxation of the one-dimensional problem. This algorithm beats state-of-the-art commercial solvers for Mixed-Integer Linear Programs (MILPs) for this special approximation problem by several orders of magnitude. Overall, we present several, partially new modeling approaches for MIOCPs together with the accompanying structural properties. On this basis, we develop new theories for the approximation of certain relaxed solutions. We discuss the efficient implementation of the resulting structure exploiting algorithms. This leads to a deeper and better understanding of MIOCPs. We show the practicability of the theoretical observations with the help of four prototypical problems. The presented methods and algorithms allow on their basis the direct development of decision support and analysis tools in practice