Abstract. This paper introduces the Mesh Adaptive Direct Search (MADS) class of algorithms for nonlinear optimization. MADS extends the Generalized Pattern Search (GPS) class by allowing local exploration, called polling, in an asymptotically dense set of directions in the space of optimization variables. This means that under certain hypotheses, including a weak constraint qualification due to Rockafellar, MADS can treat constraints by the extreme barrier approach of setting the objective to infinity for infeasible points and treating the problem as unconstrained. The main GPS convergence result is to identify limit points ˆx, where the Clarke generalized derivatives are nonnegative in a finite set of directions, called refining directions. Although in the unconstrained case, nonnegative combinations of these directions span the whole space, the fact that there can only be finitely many GPS refining directions limits rigorous justification of the barrier approach to finitely many linear constraints for GPS. The main result of this paper is that the MADS algorithms can generate an asymptotically dense set of refining directions. For LTMADS, an implementable instance of MADS, the refining directions are dense in the hypertangent cone at ˆx with probability 1. This result holds if the iterates associated with the refining directions converge to a single ˆx. We compare LTMADS to versions of GPS on some test problems. We also illustrate the limitation of our results with examples. Key words. Mesh adaptive direct search algorithms (MADS), convergence analysis, constrained optimization, nonsmooth analysis, Clarke derivatives, hypertangent, contingent cone.

This paper presents an algorithm for sampling and triangulating a smooth surface Σ ⊂ R 3 where the triangulation is homeomorphic to Σ. The only assumption we make is that the input surface representation is amenable to certain types of computations, namely computations of the intersection points of a line with the surface, computations of the critical points of some height functions defined on the surface and its restriction to a plane, and computations of some silhouette points. The algorithm ensures bounded aspect ratio, size optimality, and smoothness of the output triangulation. Unlike previous algorithms, this algorithm does not need to compute the local feature size for generating the sample points which was a major bottleneck. Experiments show the usefulness of the algorithm in remeshing and meshing CAD surfaces that are piecewise smooth. 1

In recent years, there has been a great deal of interest in metaheuristics in the optimization community. Tabu Search (TS) represents a popular class of metaheuristics. However, compared with other metaheuristics like genetic algorithm and simulated annealing, contributions of TS that deals with continuous problems are still very limited. In this paper, we introduce a continuous TS called Directed Tabu Search (DTS) method. In the DTS method, direct-search-based strategies are used to direct a tabu search. These strategies are based on the well-known Nelder-Mead method and a new pattern search procedure called adaptive pattern search. Moreover, we introduce a new tabu list conception with anti-cycling rules called Tabu Regions and Semi-Tabu Regions. In addition, Diversification and Intensification search schemes are employed. Numerical results show that the proposed method is promising and produces high quality solutions.

Abstract. We discuss an implementation of a derivative-free generating set search method for linearly constrained minimization with no assumption of nondegeneracy placed on the constraints. The convergence guarantees for generating set search methods require that the set of search directions possesses certain geometrical properties that allow it to approximate the feasible region near the current iterate. In the hard case, the calculation of the search directions corresponds to finding the extreme rays of a cone with a degenerate vertex at the origin, a difficult problem. We discuss here how state-of-the-art computational geometry methods make it tractable to solve this problem in connection with generating set search. We also discuss a number of other practical issues of implementation, such as the careful treatment of equality constraints and the desirability of augmenting the set of search directions beyond the theoretically minimal set. We illustrate the behavior of the implementation on several problems from the CUTEr test suite. We have found it to be successful on problems with several hundred variables and linear constraints.

... without addressing the so-called measurement-origin uncertainty. Part I and Part II deal with target motion models. Part III covers measurement models and associated techniques. Part IV is concerned with tracking techniques that are based on decisions regarding target maneuvers. This part surveys the multiple-model methods---the use of multiple models (and filters) simultaneously---which is the prevailing approach to maneuvering target tracking in the recent years. The survey is presented in a structured way, centered around three generations of algorithms: autonomous, cooperating, and variable structure. It emphasizes on the underpinning of each algorithm and covers various issues in algorithm design, application, and performance.

We describe an asynchronous parallel derivative-free algorithm for linearly constrained optimization. Generating set search (GSS) is the basis of our method. At each iteration, a GSS algorithm computes a set of search directions and corresponding trial points and then evaluates the objective function value at each trial point. Asynchronous versions of the algorithm have been developed in the unconstrained and bound-constrained cases which allow the iterations to continue (and new trial points to be generated and evaluated) as soon as any other trial point completes. This enables better utilization of parallel resources and a reduction in overall run time, especially for problems where the objective function takes minutes or hours to compute. For linearly constrained GSS, the convergence theory requires that the set of search directions conforms to the nearby boundary. This creates an immediate obstacle for asynchronous methods where the definition of nearby is not well defined. In this paper, we develop an asynchronous linearly constrained GSS method that overcomes this difficulty and maintains the original convergence theory. We describe our implementation in detail, including how to avoid function evaluations by caching function values and using approximate lookups. We test our implementation on every CUTEr test problem with general linear constraints and up to 1000 variables. Without tuning to individual problems, our implementation was able to solve 95% of the test problems with 10 or fewer variables, 73% of the problems with 11-100 variables, and nearly half of the problems with 100-1000 variables. To the best of our knowledge, these are the best results that have ever been achieved with a derivative-free method for linearly constrained optimization. Our asynchronous parallel implementation is freely available as part of the APPSPACK software.

APPSPACK is software for solving unconstrained and bound-constrained optimization problems. It implements an asynchronous parallel pattern search method that has been specifically designed for problems characterized by expensive function evaluations. Using APPSPACK to solve optimization problems has several advantages: No derivative information is needed; the procedure for evaluating the objective function can be executed via a separate program or script; the code can be run serially or in parallel, regardless of whether the function evaluation itself is parallel; and the software is freely available. We describe the underlying algorithm, data structures, and features of APPSPACK version 4.0, as well as how to use and customize the software.

Abstract. The DIRECT algorithm is a deterministic sampling method for bound constrained Lipschitz continuous optimization. We prove a subsequential convergence result for the DIRECT algorithm that quantifies some of the convergence observations in the literature. Our results apply to several variations on the original method, including one that will handle general constraints. We use techniques from nonsmooth analysis, and our framework is based on recent results for the MADS sampling algorithms.

Routing in mobile ad hoc networks remains as a challenging problem given the limited wireless bandwidth, users ’ mobility and potentially large scale. Recently, there has been a thrust of research to address these problems, including on-demand routing [1-2], geographical routing [6-8], virtual coordinates [15], etc. In this paper, we focus on geographical routing, which was shown to achieve good scalability without flooding, but it usually requires location information and can suffer from the severe dead end problem especially in sparse networks. Specifically, we propose a new Hop ID based routing protocol, which does not require any location information, yet achieves comparable performance with the shortest path routing. In addition, we design efficient algorithms for setting up the system and adapt to the node mobility quickly, and can effectively route out of dead ends. The extensive analysis and simulation show that the Hop ID based routing achieves efficient routing for mobile ad hoc networks with various density, irregular topologies and obstacles. 1.

We present a new generating set search (GSS) approach for minimizing functions subject to linear constraints. GSS is a class of direct search optimization methods that includes generalized pattern search. One of our main contributions in this paper is a new condition to define the set of conforming search directions that admits several computational advantages. For continuously differentiable functions we also derive a bound relating a measure of stationarity, which is equivalent to the norm of the gradient of the objective in the unconstrained case, and a parameter used by GSS algorithms to control the lengths of the steps. With the additional assumption that the derivative is Lipschitz, we obtain a big-O bound. As a consequence of this relationship, we obtain subsequence convergence to a KKT point, even though GSS algorithms lack explicit gradient information. Numerical results indicate that the bound provides a reasonable estimate of stationarity.