Reliable Real-Time Optimization of Nonconvex Systems Described by Parametrized Partial Differential Equations

The solution of a single optimization problem often requires computationally-demanding evaluations; this is especially true in optimal design of engineering components and systems described by partial differential equations. We present a technique for the rapid and reliable optimization of systems characterized by linear-functional outputs of partial differential equations with affine parameter dependence. The critical ingredients of the method are: (i) reduced-basis techniques for dimension reduction in computational requirements; (ii) an "off-line/on-line" computational decomposition for the rapid calculation of outputs of interest and respective sensitivities in the limit of many queries; (iii) a posteriori error bounds for rigorous uncertainty and feasibility control; (iv) Interior Point Methods (IPMs) for efficient solution of the optimization problem; and (v) a trust-region Sequential Quadratic Programming (SQP) interpretation of IPMs for treatment of possibly non-convex costs and constraints.Singapore-MIT Alliance (SMA

A Partially Randomized Approach to Trajectory Planning and Optimization for Mobile Robots with Flat Dynamics

Motion planning problems are characterized by huge search spaces and complex obstacle structures with no concise mathematical expression. The fixed-wing airplane application considered in this thesis adds differential constraints and point-wise bounds, i. e. an infinite number of equality and inequality constraints. An optimal trajectory planning approach is presented, based on the randomized Rapidly-exploring Random Trees framework (RRT*). The local planner relies on differential flatness of the equations of motion to obtain tree branch candidates that automatically satisfy the differential constraints. Flat output trajectories, in this case equivalent to the airplane's flight path, are designed using Bézier curves. Segment feasibility in terms of point-wise inequality constraints is tested by an indicator integral, which is evaluated alongside the segment cost functional. Although the RRT* guarantees optimality in the limit of infinite planning time, it is argued by intuition and experimentation that convergence is not approached at a practically useful rate. Therefore, the randomized planner is augmented by a deterministic variational optimization technique. To this end, the optimal planning task is formulated as a semi-infinite optimization problem, using the intermediate result of the RRT(*) as an initial guess. The proposed optimization algorithm follows the feasible flavor of the primal-dual interior point paradigm. Discretization of functional (infinite) constraints is deferred to the linear subproblems, where it is realized implicitly by numeric quadrature. An inherent numerical ill-conditioning of the method is circumvented by a reduction-like approach, which tracks active constraint locations by introducing new problem variables. Obstacle avoidance is achieved by extending the line search procedure and dynamically adding obstacle-awareness constraints to the problem formulation. Experimental evaluation confirms that the hybrid approach is practically feasible and does indeed outperform RRT*'s built-in optimization mechanism, but the computational burden is still significant.Bewegungsplanungsaufgaben sind typischerweise gekennzeichnet durch umfangreiche Suchräume, deren vollständige Exploration nicht praktikabel ist, sowie durch unstrukturierte Hindernisse, für die nur selten eine geschlossene mathematische Beschreibung existiert. Bei der in dieser Arbeit betrachteten Anwendung auf Flächenflugzeuge kommen differentielle Randbedingungen und beschränkte Systemgrößen erschwerend hinzu. Der vorgestellte Ansatz zur optimalen Trajektorienplanung basiert auf dem Rapidly-exploring Random Trees-Algorithmus (RRT*), welcher die Suchraumkomplexität durch Randomisierung beherrschbar macht. Der spezifische Beitrag ist eine Realisierung des lokalen Planers zur Generierung der Äste des Suchbaums. Dieser erfordert ein flaches Bewegungsmodell, sodass differentielle Randbedingungen automatisch erfüllt sind. Die Trajektorien des flachen Ausgangs, welche im betrachteten Beispiel der Flugbahn entsprechen, werden mittels Bézier-Kurven entworfen. Die Einhaltung der Ungleichungsnebenbedingungen wird durch ein Indikator-Integral überprüft, welches sich mit wenig Zusatzaufwand parallel zum Kostenfunktional berechnen lässt. Zwar konvergiert der RRT*-Algorithmus (im probabilistischen Sinne) zu einer optimalen Lösung, jedoch ist die Konvergenzrate aus praktischer Sicht unbrauchbar langsam. Es ist daher naheliegend, den Planer durch ein gradientenbasiertes lokales Optimierungsverfahren mit besseren Konvergenzeigenschaften zu unterstützen. Hierzu wird die aktuelle Zwischenlösung des Planers als Initialschätzung für ein kompatibles semi-infinites Optimierungsproblem verwendet. Der vorgeschlagene Optimierungsalgorithmus erweitert das verbreitete innere-Punkte-Konzept (primal dual interior point method) auf semi-infinite Probleme. Eine explizite Diskretisierung der funktionalen Ungleichungsnebenbedingungen ist nicht erforderlich, denn diese erfolgt implizit durch eine numerische Integralauswertung im Rahmen der linearen Teilprobleme. Da die Methode an Stellen aktiver Nebenbedingungen nicht wohldefiniert ist, kommt zusätzlich eine Variante des Reduktions-Ansatzes zum Einsatz, bei welcher der Vektor der Optimierungsvariablen um die (endliche) Menge der aktiven Indizes erweitert wird. Weiterhin wurde eine Kollisionsvermeidung integriert, die in den Teilschritt der Liniensuche eingreift und die Problemformulierung dynamisch um Randbedingungen zur lokalen Berücksichtigung von Hindernissen erweitert. Experimentelle Untersuchungen bestätigen, dass die Ergebnisse des hybriden Ansatzes aus RRT(*) und numerischem Optimierungsverfahren der klassischen RRT*-basierten Trajektorienoptimierung überlegen sind. Der erforderliche Rechenaufwand ist zwar beträchtlich, aber unter realistischen Bedingungen praktisch beherrschbar

An interior-point method for mpecs based on strictly feasible relaxations.

An interior-point method for solving mathematical programs with equilibrium constraints (MPECs) is proposed. At each iteration of the algorithm, a single primaldual step is computed from each subproblem of a sequence. Each subproblem is defined as a relaxation of the MPEC with a nonempty strictly feasible region. In contrast to previous approaches, the proposed relaxation scheme preserves the nonempty strict feasibility of each subproblem even in the limit. Local and superlinear convergence of the algorithm is proved even with a less restrictive strict complementarity condition than the standard one. Moreover, mechanisms for inducing global convergence in practice are proposed. Numerical results on the MacMPEC test problem set demonstrate the fast-local convergence properties of the algorithm

GIPC: Fast and stable Gauss-Newton optimization of IPC barrier energy

Barrier functions are crucial for maintaining an intersection and inversion free simulation trajectory but existing methods which directly use distance can restrict implementation design and performance. We present an approach to rewriting the barrier function for arriving at an efficient and robust approximation of its Hessian. The key idea is to formulate a simplicial geometric measure of contact using mesh boundary elements, from which analytic eigensystems are derived and enhanced with filtering and stiffening terms that ensure robustness with respect to the convergence of a Project-Newton solver. A further advantage of our rewriting of the barrier function is that it naturally caters to the notorious case of nearly-parallel edge-edge contacts for which we also present a novel analytic eigensystem. Our approach is thus well suited for standard second order unconstrained optimization strategies for resolving contacts, minimizing nonlinear nonconvex functions where the Hessian may be indefinite. The efficiency of our eigensystems alone yields a 3x speedup over the standard IPC barrier formulation. We further apply our analytic proxy eigensystems to produce an entirely GPU-based implementation of IPC with significant further acceleration

Nonlinear programming without a penalty function or a filter

A new method is introduced for solving equality constrained nonlinear optimization problems. This method does not use a penalty function, nor a barrier or a filter, and yet can be proved to be globally convergent to first-order stationary points. It uses different trust-regions to cope with the nonlinearities of the objective function and the constraints, and allows inexact SQP steps that do not lie exactly in the nullspace of the local Jacobian. Preliminary numerical experiments on CUTEr problems indicate that the method performs well

Interior-point methods for PDE-constrained optimization

In applied sciences PDEs model an extensive variety of phenomena. Typically the final goal of simulations is a system which is optimal in a certain sense. For instance optimal control problems identify a control to steer a system towards a desired state. Inverse problems seek PDE parameters which are most consistent with measurements. In these optimization problems PDEs appear as equality constraints. PDE-constrained optimization problems are large-scale and often nonconvex. Their numerical solution leads to large ill-conditioned linear systems. In many practical problems inequality constraints implement technical limitations or prior knowledge. In this thesis interior-point (IP) methods are considered to solve nonconvex large-scale PDE-constrained optimization problems with inequality constraints. To cope with enormous fill-in of direct linear solvers, inexact search directions are allowed in an inexact interior-point (IIP) method. This thesis builds upon the IIP method proposed in [Curtis, Schenk, Wächter, SIAM Journal on Scientific Computing, 2010]. SMART tests cope with the lack of inertia information to control Hessian modification and also specify termination tests for the iterative linear solver. The original IIP method needs to solve two sparse large-scale linear systems in each optimization step. This is improved to only a single linear system solution in most optimization steps. Within this improved IIP framework, two iterative linear solvers are evaluated: A general purpose algebraic multilevel incomplete L D L^T preconditioned SQMR method is applied to PDE-constrained optimization problems for optimal server room cooling in three space dimensions and to compute an ambient temperature for optimal cooling. The results show robustness and efficiency of the IIP method when compared with the exact IP method. These advantages are even more evident for a reduced-space preconditioned (RSP) GMRES solver which takes advantage of the linear system's structure. This RSP-IIP method is studied on the basis of distributed and boundary control problems originating from superconductivity and from two-dimensional and three-dimensional parameter estimation problems in groundwater modeling. The numerical results exhibit the improved efficiency especially for multiple PDE constraints. An inverse medium problem for the Helmholtz equation with pointwise box constraints is solved by IP methods. The ill-posedness of the problem is explored numerically and different regularization strategies are compared. The impact of box constraints and the importance of Hessian modification on the optimization algorithm is demonstrated. A real world seismic imaging problem is solved successfully by the RSP-IIP method