Abstract. This paper presents a taxonomy of existing approaches for using response surfaces for global optimization. Each method is illustrated with a simple numerical example that brings out its advantages and disadvantages. The central theme is that methods that seem quite reasonable often have non-obvious failure modes. Understanding these failure modes is essential for the development of practical algorithms that fulfill the intuitive promise of the response surface approach. Key words: global optimization, response surface, kriging, splines 1.

We introduce a method that aims to find the global minimum of a continuous nonconvex function on a compact subset of R^d. It is assumed that function evaluations are expensive and that no additional information is available. Radial basis function interpolation is used to define a utility function. The maximizer of this function is the next point where the objective function is evaluated. We show that, for most types of radial basis functions that are considered in this paper, convergence can be achieved without further assumptions on the objective function. Besides, it turns out that our method is closely related to a statistical global optimization method, the P-algorithm. A general framework for both methods is presented. Finally, a few numerical examples show that on the set of Dixon-Szego test functions our method yields favourable results in comparison to other global optimization methods.

INTRODUCTION In this paper we consider Simulated Annealing algorithms (SA in what follows) applied to continuous global optimization problems, i.e. problems with the following form f = min x2X f(x); (1.1) where X ` ! n is a continuous domain, often assumed to be compact, which, combined with the continuity or lower semicontinuity of f , guarantees the existence of the minimum value f . SA algorithms are based on an analogy with a physical phenomenon: while at high temperatures the molecules in a liquid move freely, if the temperature is slowly decreased the thermal mobility of the molecules is lost and they form a pure crystal which also corresponds to a state of minimum energy. If the temperature is decreased too quickly (the so called quenching) a liquid metal rather ends up in a polycrystalline or amorphous state with

Finding the global optimum on a large, multimodal, complex, and discontinuous (or nondifferentiable) landscape is usually very hard, even using the evolutionary approach. However, some of these complex landscapes can be approximated and smoothened without changing the nature of the problem, i.e., without modifying the global optimum and its location. The approximated and smoothened landscape is often much easier to search than the original one. In this paper, we propose a new algorithm using landscape approximation and hybrid evolutionary and local search. We also list several algorithm design principles. Following the basic algorithm, an example algorithm is given from our previous work of the combination of landscape approximation and local search (LALS). Furthermore, we develop a novel evolutionary algorithm with n-dimensional approximation (EANA), which shares the same rules as the basic algorithm, but remedies some of the drawbacks found in the LALS. Comparisons with evo...

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this paper |produces a user-speci ed number of suggested evaluation points in each step; |proceeds by successive partitioning of the box (branch) and building local quadratic models ( t); |combines local and global search and allows the user to determine which of both should be emphasized; |handles local search from the best point with the aid of trust regions; |allows for hidden constraints and assigns to such points a function value based on the function values of nearby feasible points

Abstract — The development of optimized motions of humanoid robots that guarantee a fast and also stable walking is an important task especially in the context of autonomous soccer playing robots in RoboCup. We present a walking motion optimization approach for the humanoid robot prototype HR18 which is equipped with a low dimensional parameterized walking trajectory generator, joint motor controller and an internal stabilization. The robot is included as hardware-in-the-loop to define a low dimensional black-box optimization problem. In contrast to previously performed walking optimization approaches we apply a sequential surrogate optimization approach using stochastic approximation of the underlying objective function and sequential quadratic programming to search for a fast and stable walking motion. This is done under the conditions that only a small number of physical walking experiments should have to be carried out during the online optimization process. For the identified walking motion for the considered 55 cm tall humanoid robot we measured a forward walking speed of more than 30 cm/sec. With a modified version of the robot even more than 40 cm/sec could be achieved in permanent operation.

This paper proposes a simulation-based active policy learning algorithm for finite-horizon, partially-observed sequential decision processes. The algorithm is tested in the domain of robot navigation and exploration under uncertainty. In such a setting, the expected cost, that must be minimized, is a function of the belief state (filtering distribution). This filtering distribution is in turn nonlinear and subject to discontinuities, which arise because constraints in the robot motion and control models. As a result, the expected cost is non-differentiable and very expensive to simulate. The new algorithm overcomes the first difficulty and reduces the number of required simulations as follows. First, it assumes that we have carried out previous simulations which returned values of the expected cost for different corresponding policy parameters. Second, it fits a Gaussian process (GP) regression model to these values, so as to approximate the expected cost as a function of the policy parameters. Third, it uses the GP predicted mean and variance to construct a statistical measure that determines which policy parameters should be used in the next simulation. The process is then repeated using the new parameters and the newly gathered expected cost observation. Since the objective is to find the policy parameters that minimize the expected cost, this iterative active learning approach effectively trades-off between exploration (in regions where the GP variance is large) and exploitation (where the GP mean is low). In our experiments, a robot uses the proposed algorithm to plan an optimal path for accomplishing a series of tasks, while maximizing the information about its pose and map estimates. These estimates are obtained with a standard filter for simultaneous localization and mapping. Upon gathering new observations, the robot updates the state estimates and is able to replan a new path in the spirit of open-loop feedback control.

We consider a Bayesian ranking and selection problem with independent normal rewards and a correlated multivariate normal belief on the mean values of these rewards. Because this formulation of the ranking and selection problem models dependence between alternatives’ mean values, algorithms may utilize this dependence to perform efficiently even when the number of alternatives is very large. We propose a fully sequential sampling policy called the knowledge-gradient policy, which is provably optimal in some special cases and has bounded suboptimality in all others. We then demonstrate how this policy may be applied to efficiently maximize a continuous function on a continuous domain while constrained to a fixed number of noisy measurements.

Recent advances in machine learning make it possible to design efficient prediction algorithms for data sets with huge numbers of parameters. This article describes a new technique for ‘hedging ’ the predictions output by many such algorithms, including support vector machines, kernel ridge regression, kernel nearest neighbours, and by many other state-of-the-art methods. The hedged predictions for the labels of new objects include quantitative measures of their own accuracy and reliability. These measures are provably valid under the assumption of randomness, traditional in machine learning: the objects and their labels are assumed to be generated independently from the same probability distribution. In particular, it becomes possible to control (up to statistical fluctuations) the number of erroneous predictions by selecting a suitable confidence level. Validity being achieved automatically, the remaining goal of hedged prediction is efficiency: taking full account of the new objects ’ features and other available information to produce as accurate predictions as possible. This can be done successfully using the powerful machinery of modern machine learning. 1