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

Upper Bounding in Inner Regions for Global Optimization under Inequality Constraints

International audienceIn deterministic constrained global optimization, upper bounding the objective function generally resorts to local minimization at the nodes of the branch and bound. The local minimization process is sometimes costly when constraints must be respected. We propose in this paper an alternative approach when the constraints are inequalities or relaxed equalities so that the feasible space has a non-null volume. First, we extract an inner region, i.e., an (entirely feasible) convex polyhedron or box in which all points satisfy the constraints. Second, we select a point inside the extracted inner region and update the upper bound with its cost. We use two inner region extraction algorithms implemented in our interval B&B called IbexOpt [7]. This upper bounding shows good performance in medium-sized systems proposed in the COCONUT suite

On the Stability of Nonlinear Receding Horizon Control: A Geometric Perspective

%!TEX root = LCSS_main_max.tex The widespread adoption of nonlinear Receding Horizon Control (RHC) strategies by industry has led to more than 30 years of intense research efforts to provide stability guarantees for these methods. However, current theoretical guarantees require that each (generally nonconvex) planning problem can be solved to (approximate) global optimality, which is an unrealistic requirement for the derivative-based local optimization methods generally used in practical implementations of RHC. This paper takes the first step towards understanding stability guarantees for nonlinear RHC when the inner planning problem is solved to first-order stationary points, but not necessarily global optima. Special attention is given to feedback linearizable systems, and a mixture of positive and negative results are provided. We establish that, under certain strong conditions, first-order solutions to RHC exponentially stabilize linearizable systems. Surprisingly, these conditions can hold even in situations where there may be \textit{spurious local minima.} Crucially, this guarantee requires that state costs applied to the planning problems are in a certain sense `compatible' with the global geometry of the system, and a simple counter-example demonstrates the necessity of this condition. These results highlight the need to rethink the role of global geometry in the context of optimization-based control

Aspects of Unstructured Grids and Finite-Volume Solvers for the Euler and Navier-Stokes Equations

One of the major achievements in engineering science has been the development of computer algorithms for solving nonlinear differential equations such as the Navier-Stokes equations. In the past, limited computer resources have motivated the development of efficient numerical schemes in computational fluid dynamics (CFD) utilizing structured meshes. The use of structured meshes greatly simplifies the implementation of CFD algorithms on conventional computers. Unstructured grids on the other hand offer an alternative to modeling complex geometries. Unstructured meshes have irregular connectivity and usually contain combinations of triangles, quadrilaterals, tetrahedra, and hexahedra. The generation and use of unstructured grids poses new challenges in CFD. The purpose of this note is to present recent developments in the unstructured grid generation and flow solution technology

Optimization of spatiotemporally fractionated radiotherapy treatments with bounds on the achievable benefit

Spatiotemporal fractionation schemes, that is, treatments delivering different dose distributions in different fractions, may lower treatment side effects without compromising tumor control. This is achieved by hypofractionating parts of the tumor while delivering approximately uniformly fractionated doses to the healthy tissue. Optimization of such treatments is based on biologically effective dose (BED), which leads to computationally challenging nonconvex optimization problems. Current optimization methods yield only locally optimal plans, and it has been unclear whether these are close to the global optimum. We present an optimization model to compute rigorous bounds on the normal tissue BED reduction achievable by such plans. The approach is demonstrated on liver tumors, where the primary goal is to reduce mean liver BED without compromising other treatment objectives. First a uniformly fractionated reference plan is computed using convex optimization. Then a nonconvex quadratically constrained quadratic programming model is solved to local optimality to compute a spatiotemporally fractionated plan that minimizes mean liver BED subject to the constraints that the plan is no worse than the reference plan with respect to all other planning goals. Finally, we derive a convex relaxation of the second model in the form of a semidefinite programming problem, which provides a lower bound on the lowest achievable mean liver BED. The method is presented on 5 cases with distinct geometries. The computed spatiotemporal plans achieve 12-35% mean liver BED reduction over the reference plans, which corresponds to 79-97% of the gap between the reference mean liver BEDs and our lower bounds. This indicates that spatiotemporal treatments can achieve substantial reduction in normal tissue BED, and that local optimization provides plans that are close to realizing the maximum potential benefit

Optimization of Gaussian Random Fields

Many engineering systems are subject to spatially distributed uncertainty, i.e. uncertainty that can be modeled as a random field. Altering the mean or covariance of this uncertainty will in general change the statistical distribution of the system outputs. We present an approach for computing the sensitivity of the statistics of system outputs with respect to the parameters describing the mean and covariance of the distributed uncertainty. This sensitivity information is then incorporated into a gradient-based optimizer to optimize the structure of the distributed uncertainty to achieve desired output statistics. This framework is applied to perform variance optimization for a model problem and to optimize the manufacturing tolerances of a gas turbine compressor blade