A reformulation–linearization–convexification algorithm for optimal correction of an inconsistent system of linear constraints

Abstract In this paper, an algorithm is introduced to find an optimal solution for an optimization problem that arises in total least squares with inequality constraints, and in the correction of infeasible linear systems of inequalities. The stated problem is a nonconvex program with a special structure that allows the use of a reformulation-linearization-convexification technique for its solution. A branch-and-bound method for finding a global optimum for this problem is introduced based on this technique. Some computational experiments are included to highlight the efficacy of the proposed methodology

A global method for mixed categorical optimization with catalogs

In this article, we propose an algorithmic framework for globally solving mixed problems with continuous variables and categorical variables whose properties are available from a catalog. It supports catalogs of arbitrary size and properties of arbitrary dimension, and does not require any modeling effort from the user. Our tree search approach, similar to spatial branch and bound methods, performs an exhaustive exploration of the range of the properties of the categorical variables ; branching, constraint programming and catalog lookup phases alternate to discard inconsistent values. A novel catalog-based contractor guarantees consistency between the categorical properties and the existing catalog items. This results in an intuitive generic approach that is exact and easy to implement. We demonstrate the validity of the approach on a numerical example in which a categorical variable is described by a two-dimensional property space

Inner Regions and Interval Linearizations for Global Optimization

International audienceResearchers from interval analysis and constraint (logic) programming communities have studied intervals for their ability to manage infinite solution sets of numerical constraint systems. In particular, inner regions represent subsets of the search space in which all points are solutions. Our main contribution is the use of recent and new inner region extraction algorithms in the upper bounding phase of constrained global optimization. Convexification is a major key for efficiently lower bounding the objective function. We have adapted the convex interval taylorization proposed by Lin & Stadtherr for producing a reliable outer and inner polyhedral approximation of the solution set and a linearization of the objective function. Other original ingredients are part of our optimizer, including an efficient interval constraint propagation algorithm exploiting monotonicity of functions. We end up with a new framework for reliable continuous constrained global optimization. Our interval B&B is implemented in the interval-based explorer Ibex and extends this free C++ library. Our strategy significantly outperforms the best reliable global optimizers

Optimal Correction of Infeasible System in Linear Equality via Genetic Algorithm

This work is focused on the optimal correction of infeasible system of linear equality. In this paper, for correcting this system, we will make the changes just in the coefficient matrix by using l norm and show that solving this problem is equivalent to solving a fractional quadratic problem. To solve this problem, we use the genetic algorithm. Some examples are provided to illustrate the efficiency and validity of the proposed method

Optimal Guidance and Control with Nonlinear Dynamics Using Sequential Convex Programming

This paper presents a novel method for expanding the use of sequential convex programming (SCP) to the domain of optimal guidance and control problems with nonlinear dynamics constraints. SCP is a useful tool in obtaining real-time solutions to direct optimal control, but it is unable to adequately model nonlinear dynamics due to the linearization and discretization required. As nonlinear program solvers are not yet functioning in real-time, a tool is needed to bridge the gap between satisfying the nonlinear dynamics and completing execution fast enough to be useful. Two methods are proposed, sequential convex programming with nonlinear dynamics correction (SCPn) and modified SCPn (M-SCPn), which mixes SCP and SCPn to reduce runtime and improve algorithmic robustness. Both methods are proven to generate optimal state and control trajectories that satisfy the nonlinear dynamics. Simulations are presented to validate the efficacy of the methods as compared to SCP

Optimal Guidance and Control with Nonlinear Dynamics Using Sequential Convex Programming

This paper presents a novel method for expanding the use of sequential convex programming (SCP) to the domain of optimal guidance and control problems with nonlinear dynamics constraints. SCP is a useful tool in obtaining real-time solutions to direct optimal control, but it is unable to adequately model nonlinear dynamics due to the linearization and discretization required. As nonlinear program solvers are not yet functioning in real-time, a tool is needed to bridge the gap between satisfying the nonlinear dynamics and completing execution fast enough to be useful. Two methods are proposed, sequential convex programming with nonlinear dynamics correction (SCPn) and modified SCPn (M-SCPn), which mixes SCP and SCPn to reduce runtime and improve algorithmic robustness. Both methods are proven to generate optimal state and control trajectories that satisfy the nonlinear dynamics. Simulations are presented to validate the efficacy of the methods as compared to SCP

Copositivity and constrained fractional quadratic programs

Abstract We provide Completely Positive and Copositive Optimization formulations for the Constrained Fractional Quadratic Problem (CFQP) and Standard Fractional Quadratic Problem (StFQP). Based on these formulations, Semidefinite Programming (SDP) relaxations are derived for finding good lower bounds to these fractional programs, which can be used in a global optimization branch-and-bound approach. Applications of the CFQP and StFQP, related with the correction of infeasible linear systems and eigenvalue complementarity problems are also discussed

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

Optimization-based Analysis and Training of Human Decision Making

In the research domain Complex Problem Solving (CPS) in psychology, computer-supported tests are used to analyze complex human decision making and problem solving. The approach is to use computer-based microworlds and to evaluate the performance of participants in such test-scenarios and correlate it to certain characteristics. However, these test-scenarios have usually been defined on a trial-and-error basis, until certain characteristics became apparent. The more complex models become, the more likely it is that unforeseen and unwanted characteristics emerge in studies. In this thesis,we use mathematical optimization methods as an analysis and training tool for Complex Problem Solving, but also show how optimization should be used in the design stage of new complex problem scenarios in the future. We present the IWR Tailorshop, a novel test scenario with functional relations and model parameters that have been formulated based on optimization results. The IWR Tailorshop is the first CPS test-scenario designed for the application of optimization and is based on the economic framing of another famous microworld, the Tailorshop. We describe an optimization-based analysis approach and extend it to optimization-based feedback with different approaches for both feedback computation and feedback presentation. Additionally, we investigate differentiable reformulations for an unavoidable minimum expression and show the according numerical results. To address the difficulties of computing globally optimal solutions for this test-scenario, which yields non-convex mixed-integer optimization problems, we present a decomposition approach for the IWR Tailorshop. The new test-scenario has been implemented in a web-based interface together with an analysis software for collected data, which both are available as open-source software and allow for an easy adaption to other test-scenarios. In this work, we also apply our methodology in a web-based feedback study using the IWR Tailorshop in which participants are trained to control the microworld by optimization-based feedback. In this study with 148 participants, we show that such a feedback can significantly improve participants’ performance in a complex microworld with a possibly huge difference to a control group. However, the performance improvement depends on the representation of the feedback. We give a detailed analysis of the study and report on new insights about human decision making which only have been possible through the IWR Tailorshop and our optimization-based analysis and training approach