Predictive Entropy Search for Bayesian optimization with unknown constraints

Unknown constraints arise in many types of expensive black-box optimization problems. Several methods have been proposed recently for performing Bayesian optimization with constraints, based on the expected improvement (EI) heuristic. However, EI can lead to pathologies when used with constraints. For example, in the case of decoupled constraints---i.e., when one can independently evaluate the objective or the constraints---EI can encounter a pathology that prevents exploration. Additionally, computing EI requires a current best solution, which may not exist if none of the data collected so far satisfy the constraints. By contrast, information-based approaches do not suffer from these failure modes. In this paper, we present a new information-based method called Predictive Entropy Search with Constraints (PESC). We analyze the performance of PESC and show that it compares favorably to EI-based approaches on synthetic and benchmark problems, as well as several real-world examples. We demonstrate that PESC is an effective algorithm that provides a promising direction towards a unified solution for constrained Bayesian optimization.José Miguel Hernández-Lobato acknowledges support from the Rafael del Pino Foundation. Zoubin Ghahramani acknowledges support from Google Focused Research Award and EPSRC grant EP/I036575/1. Matthew W. Hoffman acknowledges support from EPSRC grant EP/J012300/1.This is the final published version. It first appeared at http://jmlr.org/proceedings/papers/v37/hernandez-lobatob15.html

A General Framework for Constrained Bayesian Optimization using Information-based Search

We present an information-theoretic framework for solving global black-box optimization problems that also have black-box constraints. Of particular interest to us is to efficiently solve problems with $\textit{decoupled}$ constraints, in which subsets of the objective and constraint functions may be evaluated independently. For example, when the objective is evaluated on a CPU and the constraints are evaluated independently on a GPU. These problems require an acquisition function that can be separated into the contributions of the individual function evaluations. We develop one such acquisition function and call it Predictive Entropy Search with Constraints (PESC). PESC is an approximation to the expected information gain criterion and it compares favorably to alternative approaches based on improvement in several synthetic and real-world problems. In addition to this, we consider problems with a mix of functions that are fast and slow to evaluate. These problems require balancing the amount of time spent in the meta-computation of PESC and in the actual evaluation of the target objective. We take a bounded rationality approach and develop a partial update for PESC which trades off accuracy against speed. We then propose a method for adaptively switching between the partial and full updates for PESC. This allows us to interpolate between versions of PESC that are efficient in terms of function evaluations and those that are efficient in terms of wall-clock time. Overall, we demonstrate that PESC is an effective algorithm that provides a promising direction towards a unified solution for constrained Bayesian optimization.Rafael del Pino Foundation, Google Focused Research Award, Engineering and Physical Sciences Research Council (Grant IDs: EP/I036575/1, EP/J012300/1

Time-Fractional Optimal Control of Initial Value Problems on Time Scales

We investigate Optimal Control Problems (OCP) for fractional systems involving fractional-time derivatives on time scales. The fractional-time derivatives and integrals are considered, on time scales, in the Riemann--Liouville sense. By using the Banach fixed point theorem, sufficient conditions for existence and uniqueness of solution to initial value problems described by fractional order differential equations on time scales are known. Here we consider a fractional OCP with a performance index given as a delta-integral function of both state and control variables, with time evolving on an arbitrarily given time scale. Interpreting the Euler--Lagrange first order optimality condition with an adjoint problem, defined by means of right Riemann--Liouville fractional delta derivatives, we obtain an optimality system for the considered fractional OCP. For that, we first prove new fractional integration by parts formulas on time scales.Comment: This is a preprint of a paper accepted for publication as a book chapter with Springer International Publishing AG. Submitted 23/Jan/2019; revised 27-March-2019; accepted 12-April-2019. arXiv admin note: substantial text overlap with arXiv:1508.0075

Set optimization - a rather short introduction

Recent developments in set optimization are surveyed and extended including various set relations as well as fundamental constructions of a convex analysis for set- and vector-valued functions, and duality for set optimization problems. Extensive sections with bibliographical comments summarize the state of the art. Applications to vector optimization and financial risk measures are discussed along with algorithmic approaches to set optimization problems

Structural and Optical Properties of Diluted Magnetic Ga1−xMnxAs–AlAs Quantum Wells Grown on High-Index GaAs Planes

We report on the structural and optical properties of Ga₁₋ᵪ Mn ᵪ As-AlAs quantum wells (QWs) with χ=0.1% grown by molecular beam epitaxy (MBE) on semi-insulating GaAs substrates with orientations (100), (110), (311)B and (411)B. Atomic force microscopy (AFM), X-ray diffraction (XRD) and photoluminescence (PL) techniques were used to investigate these QWs. AFM results have evidenced the formation of Mn-induced islands, which are randomly distributed on the surface. These islands tend to segregate for samples grown on (110) and (411)B planes, while no clear segregation was observed for samples grown on (100) and (311)B orientations. Results show that the PL line width increases with Mn segregation. XRD measurements were used to determine 2θ,d and cell parameters

Serum 25-hydroxyvitamin D, parathyroid hormone, calcium intake, and bone mineral density in Spanish adults

Summary Vitamin D insufficiency is very common among Spanish community-dwelling adult subjects. A threshold of serum 25(OH)D around 30 ng/ml would be necessary for the prevention of secondary hyperparathyroidism and hip bone loss in our population, regardless of the dairy calcium ingestion. Introduction This study aims to assess 25-hydroxyvitamin D?25(OH)D?status in Spanish adult subjects and to analyze its relationships with serum PTH levels, calcium intake, and bone mineral density (BMD). Methods A total of 1811 individuals (1154 postmenopausal women and 657 men) aged 44?93 years participated in the study. Serum 25(OH)D, intact parathyroid hormone (PTH), aminoterminal propeptide of type I collagen (P1NP), and Cterminal telopeptide of type I collagen (?-CTX) levels were measured by electrochemiluminescence. BMD was determined by dual x-ray absorptiometry (DXA) at lumbar spine, femoral neck, and total hip. Results Serum 25(OH)D levels were below 10, 20, and 30 ng/ml in 5, 40, and 83%of participants, respectively. There was a significant seasonal difference in mean serum 25(OH)D, with higher levels in summer?autumn. In multivariate analysis, 25(OH)D levels were negatively correlated with age, serum PTH and creatinine, body mass index, smoking, alcohol intake, and a number of chronic diseases, but positively with dairy calcium intake. The magnitude of the difference in serum PTH according to 25(OH)D quartiles was not influenced by calcium intake. A threshold of serum 25(OH)D around 30 ng/ml was observed for serum PTH and hip BMD. Conclusions Vitamin D insufficiency is very common among Spanish community-dwelling adult subjects. A threshold of serum 25(OH)D around 30 ng/ml would be necessary for the prevention of secondary hyperparathyroidism and hip bone loss in our population, regardless of the dairy calcium ingestion. Programs to improve vitamin D status may be required in our country

On stationary points of nonexpansive set-valued mappings

In this paper we deal with stationary points (also known as endpoints) of nonexpansive set-valued mappings and show that the existence of such points under certain conditions follows as a consequence of the existence of approximate stationary sequences. In particular we provide abstract extensions of well-known fixed point theorems.Dirección General de Enseñanza SuperiorJunta de Andalucí