Lipschitz optimisation for Lipschitz Interpolation

Understanding the problem space in single-objective numerical optimization using exploratory landscape analysis

The Guided Genetic Algorithm (GCA) is a hybrid of Genetic Algorithm and Guided Local Search, a meta-heuristic search algorithm. As the search progresses, GGA modifies both the fitness function and fitness template of candidate solutions based on feedback from constraints. The fitness template is then used to bias crossover and mutation. The Radio Link Frequency Assignment Problem (RLFAP) is a class of problem that has practical relevance to both military and civil applications. In this paper, we show how GGA can be applied to the RLFAP. We focus on an abstraction of a real life military application that involves the assigning of frequencies to radio links. GGA was tested on a set of eleven benchmark problems provided by the French military. This set of problems has been studied intensively by a number of prominent groups in Europe. It covers a variety of needs in military applications, including the satisfaction of constraints, finding optimal solutions that satisfy all the constraints and optimization of some objective functions whenever no solution exist (“partial constraint satisfaction”). Not only do these benchmark problems vary in problem nature, they are reasonably large for military applications (up to 916 variables, and up to 5548 constraints). This makes them a serious challenge to the generality, reliability as well as efficiency of algorithms. We show in this paper that GGA is capable of producing excellent results reliably in the whole set of benchmark problems.

Complexity of pattern classes and the Lipschitz property

A function f : D ? R has Lipschitz constant c if dR(f(x),f(y)) = c· dD(x,y) for all x,y in D, where dR and dD denote the distance functions on the range and domain of f, respectively. We say a function is Lipschitz if it has Lipschitz constant 1. (Note that rescaling by a factor of 1/c converts a function with a Lipschitz constant c into a Lipschitz function.) In other words, Lipschitz functions are not very sensitive to small changes in the input. We initiate the study of testing and local reconstruction of the Lipschitz property of functions. A property tester has to distinguish functions with the property (in this case, Lipschitz) from functions that are e -far from having the property, that is, differ from every function with the property on at least an e fraction of the domain. A local filter reconstructs an arbitrary function f to ensure that the reconstructed function g has the desired property (in this case, is Lipschitz), changing f only when necessary. A local filter is given a function f and a query x and, after looking up the value of f on a small number of points, it has to output g(x) for some function g, which has the desired property and does not depend on x. If f has the property, g must be equal to f. We consider functions over domains {0,1}d, {1,..., n} and {1,..., n}d, equipped with l1 distance. We design efficient testers of the Lipschitz property for functions of the form f:{0,1}d? d Z, where d ? (0,1] and d Z is the set of integer multiples of d, and of the form f: {1,..., n} ? R, where R is (discretely) metrically convex. In the first case, the tester runs in time O(d· min{d,r}/d e ), where r is the diameter of the image of f, in the second, in time O((log n)/e ). We give corresponding lower bounds of O (d) and O (log n) on the query complexity (in the second case, only for nonadaptive 1-sided error testers). Our lower bound for functions over {0,1}dis tight for the case of the {0,1,2} range and constant e. The first tester implies an algorithm for functions of the form f:{0,1}d? R that distinguishes Lipschitz functions from functions that are e -far from (1+d )-Lipschitz. We also present a local filter of the Lipschitz property for functions of the form f: {1,..., n}d ? R with lookup complexity O((log n+1)d). For functions of the form {0,1}d, we show that every nonadaptive local filter has lookup complexity exponential in d. The testers that we developed have applications to programs analysis. The reconstructors have applications to data privacy. For the first application, the Lipschitz property of the function computed by a program corresponds to a notion of robustness to noise in the data. The application to privacy is based on the fact that a function f of entries in a database of sensitive information can be released with noise of magnitude proportional to a Lipschitz constant of f, while preserving the privacy of individuals whose data is stored in the database (Dwork, McSherry, Nissim and Smith, TCC 2006). We give a differentially private mechanism, based on local filters, for releasing a function f when a Lipschitz constant of f is provided by a distrusted client. We show that when no reliable Lipschitz constant of f is given, previously known differentially private mechanisms either have a substantially higher running time or have a higher expected error for a large class of symmetric functions f.

A selection hyper-heuristic is used to minimise the objective functions of a well-known set of benchmark problems. The resulting sequences of low level heuristic selections and objective function values are used to generate a database of heuristic selections. The sequences in the database are broken down into subsequences and the mathematical concept of a logarithmic return is used to discriminate between “effective” subsequences, which tend to decrease the objective value, and “disruptive” subsequences, which tend to increase the objective value. These subsequences are then employed in a sequenced based hyper-heuristic and evaluated on an unseen set of benchmark problems. Empirical results demonstrate that the “effective” subsequences perform significantly better than the “disruptive” subsequences across a number of problem domains with 99% confidence. The identification of subsequences of heuristic selections that can be shown to be effective across a number of problems or problem domains could have important implications for the design of future sequence based hyper-heuristics.

Benchmark problems for numerical implementations of phase field models

We study local filters for two properties of functions f:{0,1} d → ℝ: the Lipschitz property and monotonicity. A local filter with additive error a is a randomized algorithm that is given black-box access to a function f and a query point x in the domain of f. Its output is a value F(x), such that (i) the reconstructed function F(x) satisfies the property (in our case, is Lipschitz or monotone) and (ii) if the input function f satisfies the property, then for every point x in the domain (with high constant probability) the reconstructed value F(x) differs from f(x) by at most a. Local filters were introduced by Saks and Seshadhri (SICOMP 2010) and the relaxed definition we study is due to Bhattacharyya et al.(RANDOM 2010), except that we further relax it by allowing additive error. Local filters for Lipschitz and monotone functions have applications to areas such as data privacy.We show that every local filter for Lipschitz or monotone functions runs in time exponential in the dimension d, even when the filter is allowed significant additive error. Prior lower bounds (for local filters with no additive error, i.e., with a = 0) applied only to more restrictive class of filters, e.g., nonadaptive filters. To prove our lower bounds, we construct families of hard functions and show that lookups of a local filter on these functions are captured by a combinatorial object that we call a c-connector. Then we present a lower bound on the maximum outdegree of a c-connector, and show that it implies the desired bounds on the running time of local filters. Our lower bounds, in particular, imply the same bound on the running time for a class of privacy mechanisms.

When designing a benchmark problem set, it is important to create a set of benchmark problems that are a good generalization of the set of all possible problems. One possible way of easing this difficult task is by using artificially generated problems. In this paper, one such single-objective continuous problem generation approach is analyzed and compared with the COCO benchmark problem set, a well know problem set for benchmarking numerical optimization algorithms. Using Exploratory Landscape Analysis and Singular Value Decomposition, we show that such representations allow us to further explore the relations between the problems by applying visualization and correlation analysis techniques, with the goal of decreasing the bias in benchmark problem assessment.

Software effort data contains a large amount of missing values of project attributes. The problem of absent features, which occurred recently in machine learning, is often neglected by researchers of software engineering when handling the missingness in software effort data. In essence, absent features (structural missingness) and unobserved values (unstructured missingness) are different cases of missingness although their appearance in the data set are the same. This paper attempts to clarify the root cause of missingness of software effort data. When regarding missingness as absent features, we develop Max-margin regression to predict real effort of software projects. When regarding missingness as unobserved values, we use existing imputation techniques to impute missing values. Then, ε – SVR is used to predict real effort of software projects with the input data sets. Experiments on ISBSG (International Software Benchmarking Standard Group) and CSBSG (Chinese Software Benchmarking Standard Group) data sets demonstrate that, with the tasks of effort prediction, the treatment regarding missingness in software effort data set as unobserved values can produce more desirable performance than that of regarding missingness as absent features. This paper is the first to introduce the concept of absent features to deal with missingness of software effort data.

A quick artificial bee colony (qABC) algorithm and its performance on optimization problems

We propose an Artificial Bee Colony to solve Inventory Routing Problem with Backlogging/Backordering (IRPB). The network that we consider is a one-to-many distribution network which consists of a single depot and multiple customers. A fleet of heterogeneous vehicle delivers a single product to fulfill the customers' demands over the planning horizon. Here, we assume that the depot has enough supply to cover all the demand throughout the planning horizon. Backordering is considered only when it is more economical or when there is insufficient vehicle capacity. The objective of IRPB is to find solutions such that inventory cost, backorder cost and transportation cost is optimal resulting in the minimization of the overall total cost. Artificial Bee Colony (ABC) proposed is modified from [10] where we embed inventory information and also incorporate the inventory updating mechanism to handle both inventory and backorder. We also apply 2-opt∗ as local search to improve the routing and the algorithm is tested on a set of benchmark problems.

This paper presents an equivalence between feasible kinodynamic planning and optimal kinodynamic planning, in that any optimal planning problem can be transformed into a series of feasible planning problems in a state-cost space, whose solutions approach the optimum. This transformation yields a meta-algorithm that produces an asymptotically optimal planner, given any feasible kinodynamic planner as a subroutine. The meta-algorithm is proven to be asymptotically optimal and a formula is derived relating expected running time and solution suboptimality. It is directly applicable to a wide range of optimal planning problems because it does not resort to the use of steering functions or numerical boundary-value problem solvers. On a set of benchmark problems, it is demonstrated to perform, using the expansive space tree (EST) and rapidly-exploring random tree (RRT) algorithms as subroutines, at a level that is superior or comparable to related planners.

Particle Swarm Optimization (PSO), classified as a swarm intelligence technique, mimics the well-informed swarming behavior of social species. A simple and effective searching strategy declares PSO as a potential member for solving various optimization problems. The present study embeds the concept of chaos at different stages of PSO, intending to enhance the convergence speed while trying to avoid stagnation and maintaining the solution quality. The proposed PSO variant is termed as “Totally disturbed PSO (TDPSO)”. The algorithm starts with a disturbed (chaotic) population, generated by considered chaotic system. Thereafter, when a certain number of iterations have elapsed and the searching process approaches equilibrium state, a relative velocity index is calculated for each particle to evaluate its present state and to decide whether or not the particle needs perturbation. The efficacy of proposed algorithm is tested against a set of benchmark problems and results are compared with existing Chaotic PSO and a standard PSO variant. Numerical results manifest that TDPSO works better over considered existing variants by effectively enhancing the searching capability and precision as well.

An evidence-based robust optimization method for pulsed laser orbital debris removal (LODR) is presented. Epistemic type uncertainties due to limited knowledge are considered. The objective of the design optimization is set to minimize the debris lifetime while at the same time maximizing the corresponding belief value. The Dempster–Shafer theory of evidence (DST), which merges interval-based and probabilistic uncertainty modeling, is used to model and compute the uncertainty impacts. A Kriging based surrogate is used to reduce the cost due to the expensive numerical life prediction model. Effectiveness of the proposed method is illustrated by a set of benchmark problems. Based on the method, a numerical simulation of the removal of Iridium 33 with pulsed lasers is presented, and the most robust solutions with minimum lifetime under uncertainty are identified using the proposed method.

Network cost minimization using threshold-based discounting