Our work presents a novel approach to shape optimization, with the twofold objective to improve the efficiency of global optimization algorithms while promoting the generation of high-quality designs during the optimization process free of geometrical anomalies. This is accomplished by reducing the number of the original design variables defining a new reduced subspace where the geometrical variance is maximized and modeling the underlying generative process of the data via probabilistic linear latent variable models such as factor analysis and probabilistic principal component analysis. We show that the data follows approximately a Gaussian distribution when the shape modification method is linear and the design variables are sampled uniformly at random, due to the direct application of the central limit theorem. The degree of anomalousness is measured in terms of Mahalanobis distance, and the paper demonstrates that abnormal designs tend to exhibit a high value of this metric. This enables the definition of a new optimization model where anomalous geometries are penalized and consequently avoided during the optimization loop. The procedure is demonstrated for hull shape optimization of the DTMB 5415 model, extensively used as an international benchmark for shape optimization problems. The global optimization routine is carried out using Bayesian optimization and the DIRECT algorithm. From the numerical results, the new framework improves the convergence of global optimization algorithms, while only designs with high-quality geometrical features are generated through the optimization routine thereby avoiding the wastage of precious computationally expensive simulations.

This paper presents a practical global optimization algorithm for the K-center clustering problem, which aims to select K samples as the cluster centers to minimize the maximum within-cluster distance. This algorithm is based on a reduced-space branch and bound scheme and guarantees convergence to the global optimum in a finite number of steps by only branching on the regions of centers. To improve efficiency, we have designed a two-stage decomposable lower bound, the solution of which can be derived in a closed form. In addition, we also propose several acceleration techniques to narrow down the region of centers, including bounds tightening, sample reduction, and parallelization. Extensive studies on synthetic and real-world datasets have demonstrated that our algorithm can solve the K-center problems to global optimal within 4 hours for ten million samples in the serial mode and one billion samples in the parallel mode. Moreover, compared with the state-of-the-art heuristic methods, the global optimum obtained by our algorithm can averagely reduce the objective function by 25.8% on all the synthetic and real-world datasets.

Basin hopping is a global optimization algorithm. It was developed to solve problems in chemical physics, although it is an effective algorithm suited for nonlinear objective functions with multiple optima. In this tutorial, you will discover the basin hopping global optimization algorithm. Basin Hopping Optimization in Python Photo by Pedro Szekely, some rights reserved. Basin Hopping is a global optimization algorithm developed for use in the field of chemical physics.

As teams apply optimization earlier and more frequently in the modeling process, they develop high-performing models at a faster pace. This virtuous cycle increases the number of models that make it into production, which amplifies the impact of these models on the business. At the Deep Learning Summit in San Francisco, SigOpt will be showcasing their model optimization software and how they automate model tuning to accelerate the model development process and amplify the impact of models in production at scale. We spoke to Nick Payton, B2B Marketing Lead at SigOpt to learn more. SigOpt's mission is to empower experts.

The learning algorithm of a neural network tries to optimize the neural network's weights until some stopping condition has been met. This condition is typically either when the error of the network reaches an acceptable level of accuracy on the training set, when the error of the network on the validation set begins to deteriorate, or when the specified computational budget has been exhausted. The most common learning algorithm for neural networks is back-propagation, an algorithm that uses stochastic gradient descent, which was discussed earlier on in this series. The are some problems with this approach. Adjusting all the weights at once can result in a significant movement of the neural network in weight space, the gradient descent algorithm is quite slow, and the gradient descent algorithm is susceptible to local minima.

The goal of the paper is to design sequential strategies which lead to efficient optimization of an unknown function under the only assumption that it has a finite Lipschitz constant. We first identify sufficient conditions for the consistency of generic sequential algorithms and formulate the expected minimax rate for their performance. We introduce and analyze a first algorithm called LIPO which assumes the Lipschitz constant to be known. Consistency, minimax rates for LIPO are proved, as well as fast rates under an additional H\"older like condition. An adaptive version of LIPO is also introduced for the more realistic setup where the Lipschitz constant is unknown and has to be estimated along with the optimization. Similar theoretical guarantees are shown to hold for the adaptive LIPO algorithm and a numerical assessment is provided at the end of the paper to illustrate the potential of this strategy with respect to state-of-the-art methods over typical benchmark problems for global optimization.

Efficient global optimization is the problem of minimizing an unknown function f, using as few evaluations f(x) as possible. It can be considered as a continuum-armed bandit problem, with noiseless data and simple regret. Expected improvement is perhaps the most popular method for solving this problem; the algorithm performs well in experiments, but little is known about its theoretical properties. Implementing expected improvement requires a choice of Gaussian process prior, which determines an associated space of functions, its reproducing-kernel Hilbert space (RKHS). When the prior is fixed, expected improvement is known to converge on the minimum of any function in the RKHS. We begin by providing convergence rates for this procedure. The rates are optimal for functions of low smoothness, and we modify the algorithm to attain optimal rates for smoother functions. For practitioners, however, these results are somewhat misleading. Priors are typically not held fixed, but depend on parameters estimated from the data. For standard estimators, we show this procedure may never discover the minimum of f. We then propose alternative estimators, chosen to minimize the constants in the rate of convergence, and show these estimators retain the convergence rates of a fixed prior.

The fundamental laws of quantum world upsets the logical foundation of classic physics. They are completely counter-intuitive with many bizarre behaviors. However, this paper shows that they may make sense from the perspective of a general decision-optimization principle for cooperation. This principle also offers a generalization of Nash equilibrium, a key concept in game theory, for better payoffs and stability of game playing.